Section 2 Photonic Devices

*Recent Advances in Nanophotonics - Fundamentals and Applications*

[74] Patel, S.K., Sorathiya, V., Lavadiya,

polarization-insensitive squared spiralshaped graphene metasurface with negative refractive index. Appl. Phys. B 126, 80 (2020). https://doi.org/10.1007/

[75] Jiang, Q. Zhang, Q. Ma, S. Yan, F. Wu, X. He, Dynamically tunable electromagnetically induced reflection in terahertz complementary graphene metamaterials, Opt. Mater. Express. 5 (2015) 1962. doi:10.1364/ome.5.001962.

[76] G.W. Hanson, Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene, J. Appl. Phys. 103 (2008).

[77] Y. Jiang, W.B. Lu, H.J. Xu, Z.G. Dong, T.J. Cui, A planar electromagnetic "black hole" based on graphene, Phys. Lett. Sect. A Gen. At. Solid State Phys. 376 (2012) 1468-1471. doi:10.1016/j.

[78] Tutorial models for COMSOL Webinar "Simulating Graphene-Based Photonic and Optoelectronic Devices" [Internet]. 2020. Available from: https://www.comsol.co.in/community/ exchange/361/ [Accessed: 2020-08-30]

[79] Alexander V. Kildishev, "Graphene Paves the Way for Next-Generation Plasmonics", [Internet]. 2020. https://www.comsol.com/story/ graphene-paves-the-way-for-nextgeneration-plasmonics-53551 [Accessed:

doi:10.1063/1.2891452.

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[67] Mattiucci N, Trimm R,

D'Aguanno G, et al (2012) Tunable, narrow-band, all-metallic microwave absorber. Appl Phys Lett 101:141115. https://doi.org/10.1063/1.4757282

[68] Bai Y, Zhao L, Ju D, et al (2015) Wide-angle, polarization-independent and dual-band infrared perfect absorber based on L-shaped metamaterial. Opt Express 23:8670. https://doi. org/10.1364/OE.23.008670

[69] Guo Y, Zhang T, Yin WY, Wang XH (2015) Improved hybrid FDTD method for studying tunable graphene frequency-selective surfaces (GFSS) for THz-wave applications. IEEE Trans Terahertz Sci Technol 5:358-367. https:// doi.org/10.1109/TTHZ.2015.2399105

[70] Wang DW, Zhao WS, Xie H, et al (2017) Tunable THz multiband frequency-selective surface based on hybrid metal-graphene structures. IEEE Trans Nanotechnol 16:1132-1137. https:// doi.org/10.1109/TNANO.2017.2749269

[71] Yan R, Arezoomandan S, Sensale-

[72] Li X, Lin L, Wu LS, et al (2017) A Bandpass Graphene Frequency Selective Surface with Tunable Polarization Rotation for THz Applications. IEEE Trans Antennas Propag 65:662-672. https://doi.org/10.1109/

[73] D.R. Smith, S. Schultz, P. Markoš, C.M. Soukoulis, determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients. Phys Rev B Condens Matter Mater Phys 65, 1-5 (2002). https ://doi.org/10.1103/

Exceptional Terahertz Wave Modulation in Graphene Enhanced by Frequency Selective Surfaces. ACS Photonics 3:315-323. https://doi.org/10.1021/

Rodriguez B, Xing HG (2016)

acsphotonics.5b00639

TAP.2016.2633163

Physrevb.65.19510 4

**32**

**Chapter 3**

**Abstract**

a Quantum Dot

excitation, fine-structure splitting

**1. Introduction**

**35**

Toward On-Demand Generation

of Entangled Photon Pairs with

*Arash Ahmadi, Andreas Fognini and Michael E. Reimer*

The generation of on-demand, optimally entangled photon pairs remains one of the most formidable challenges in the quantum optics and quantum information community. Despite the fact that recent developments in this area have opened new doors leading toward the realization of sources exhibiting either high brightness or near-unity entanglement fidelity, the challenges to achieve both together persist. Here, we will provide a historical review on the development of quantum dots (QDs) for entangled photon generation, with a focus on nanowire QDs, and address the latest research performed on nanowire QDs, including measuring entanglement fidelity, light-extraction efficiency, dephasing mechanisms, and the detrimental effects of detection systems on the measured values of entanglement fidelity. Additionally, we will discuss results recently observed pertaining to resonant excitation of a nanowire QD, revealing the potential of such sources to outperform spontaneous parametric down-conversion (SPDC) sources, providing a viable solution to the current challenges in quantum optics and quantum information.

**Keywords:** nanowire quantum dot, entanglement, dephasing, resonant two-photon

Entangled photon pairs are one of the key elements for research and in emerging quantum applications with successful results in quantum foundations [1, 2], quantum communication [3–5], and quantum information [6–8]. Thus far, nonlinear crystals exhibiting spontaneous parametric down-conversion (SPDC) [9–11] have been the main source of generating entangled photon pairs for use in these areas. This type of source results in photon pairs that exhibit near-unity entanglement fidelity, high degrees of single-photon purity and indistinguishability in each emission mode, and high temporal correlation. Moreover, these sources perform at or near room temperature. However, there are fundamental limitations to such sources, which limit their performance and scalability for use in quantum photonics; an ideal source is imperative for optimal performance. One key feature of an ideal source of entangled photons is the ability to perform on-demand, i.e., source triggering and extraction of light must be possible with near-unity efficiency. SPDC sources follow a stochastic process and therefore generate entangled photon pairs at random. Moreover, the probability of multiphoton generation follows a Poisson distribution, and thus entanglement

### **Chapter 3**

## Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot

*Arash Ahmadi, Andreas Fognini and Michael E. Reimer*

### **Abstract**

The generation of on-demand, optimally entangled photon pairs remains one of the most formidable challenges in the quantum optics and quantum information community. Despite the fact that recent developments in this area have opened new doors leading toward the realization of sources exhibiting either high brightness or near-unity entanglement fidelity, the challenges to achieve both together persist. Here, we will provide a historical review on the development of quantum dots (QDs) for entangled photon generation, with a focus on nanowire QDs, and address the latest research performed on nanowire QDs, including measuring entanglement fidelity, light-extraction efficiency, dephasing mechanisms, and the detrimental effects of detection systems on the measured values of entanglement fidelity. Additionally, we will discuss results recently observed pertaining to resonant excitation of a nanowire QD, revealing the potential of such sources to outperform spontaneous parametric down-conversion (SPDC) sources, providing a viable solution to the current challenges in quantum optics and quantum information.

**Keywords:** nanowire quantum dot, entanglement, dephasing, resonant two-photon excitation, fine-structure splitting

### **1. Introduction**

Entangled photon pairs are one of the key elements for research and in emerging quantum applications with successful results in quantum foundations [1, 2], quantum communication [3–5], and quantum information [6–8]. Thus far, nonlinear crystals exhibiting spontaneous parametric down-conversion (SPDC) [9–11] have been the main source of generating entangled photon pairs for use in these areas. This type of source results in photon pairs that exhibit near-unity entanglement fidelity, high degrees of single-photon purity and indistinguishability in each emission mode, and high temporal correlation. Moreover, these sources perform at or near room temperature. However, there are fundamental limitations to such sources, which limit their performance and scalability for use in quantum photonics; an ideal source is imperative for optimal performance. One key feature of an ideal source of entangled photons is the ability to perform on-demand, i.e., source triggering and extraction of light must be possible with near-unity efficiency. SPDC sources follow a stochastic process and therefore generate entangled photon pairs at random. Moreover, the probability of multiphoton generation follows a Poisson distribution, and thus entanglement

fidelity, single-photon purity, and photon indistinguishability [12] degrade when the pump power is increased [13]. As a result, these sources only operate at extremely low pair-production efficiencies, ϵ*<sup>p</sup>* <1%, per excitation pulse [14]. Hence, engineering and realizing an ideal source of entangled photons is necessary for the successful future of entangled photon pairs for use in quantum photonics, a future made brighter by semiconductor quantum dots (QDs).

Semiconductor quantum dots [15] are capable of generating pairs of entangled photons based on a process called the biexciton (*XX*)-exciton (*X*) cascade [16]; this cascade process is shown in **Figure 1**. The j i *XX* state is composed of two electron– hole (*e* � *h*) pairs in the QD's lowest energy level, i.e., *s*-shell. Each of these pairs possesses an angular momentum *j <sup>z</sup>* with the superposition of *j <sup>z</sup>* ¼ �1. From the j i *XX* state, there are two recombination pathways through the intermediate j i *X* state. Upon recombination, the *e* � *h* pair will emit either a left, j i *L* , or right, j i *R* , circularly polarized photon corresponding to *j <sup>z</sup>* ¼ þ1 or *j <sup>z</sup>* ¼ �1, respectively, and the QD final state at this step will be in the j i *X* state. This transition, j i *XX* ! j i *X* , is referred to as the neutral biexciton, *XX*, transition. Relaxation to the ground state occurs by the recombination of the remaining *e* � *h* pair in the j i *X* state. The latter transition, i.e., j i *X* ! j i *G* , is referred to as the neutral exciton, *X*, transition. This recombination will emit a photon with a polarization perpendicular to that of the first photon (*XX*), i.e., j i *L XX* ! j i *R <sup>X</sup>* and j i *R XX* ! j i *L <sup>X</sup>*. At the end of the process, the two emitted photons will be in the polarization entangled state [16]:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}(|RL\rangle + |LR\rangle). \tag{1}$$

refractive index, these self-assembled QDs typically suffered from isotropic emission and total internal reflection at the semiconductor-air interface and thus

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

Recent developments in micro- and nanoscale crystal growth and fabrication have resulted in structures which have improved the performance of QDs considerably. Enhancement of the spontaneous emission of QDs was first achieved by coupling an ensemble of QDs [27], and later a single QD, to a micro-cavity [28]. More recently, the coupling of QDs to micro-pillar cavities has achieved light-extraction efficiencies as high as 80% [29]. Also, such structures allow for proper control of the charge noise around the QD and thus the suppression of detrimental dephasing processes from the moving charge carriers. Excitingly, as a result, photons with >99% indistinguish-

However, such performance comes at a price. Due to Coulomb interactions [17], *XX* and *X* emission lines are separated in energy by an amount referred to as the *XX* binding energy, Δ*XX*�*<sup>X</sup>*. Within a typically used cavity with a quality factor of *Q* � 10, 000, which is tuned to photons with the wavelength *λ* � 1*μ*m, the cavity has a bandwidth of � 10*μ*eV. This small bandwidth is far less than a typical *XX* binding energy of Δ*XX*�*<sup>X</sup>* � 1meV [31]. Therefore, either the *X* or the *XX* transition lines can be coupled to the fundamental mode of the cavity, but not both as needed for a highefficiency entangled photon source. Furthermore, to avoid suboptimal entanglement, coupling to the fundamental mode of the cavity should be polarization-independent. Otherwise, one decay path in the cascade process would gain a stronger weight [32]. In an attempt to overcome these challenges, three studies have made considerable gains. Dousse et al. [32] fabricated a micropillar cavity molecule and successfully tuned both *XX* and *X* transitions to separate cavity modes in the molecule. Impressively, the fabricated device was shown to improve the pair-extraction efficiency (ϵ*e*) by three orders of magnitude, ϵ*<sup>e</sup>* ¼ 12%, as compared to a bare self-assembled QD; yet despite this, the source still generates poorly entangled photon pairs with a measured fidelity of F ¼ 67%. Other research by Chen et al. [26] showed the successful fabrication of a broadband dielectric antenna, which enables a pair-extraction efficiency of 37*:*2%, low multiphoton emission *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> <sup>≈</sup>0*:*002, and entanglement fidelity, F ¼ 90%; however, the emission profile deviates from a Gaussian, and photon indistinguishability of such sources is yet to be measured. Additionally, Wang et al. [33] have engineered a circular Bragg grating bull's-eye cavity around a single QD showing a pair-extraction efficiency, ϵ*<sup>e</sup>* ¼ 36*:*6%, and an entanglement fidelity, F ¼ 90%. Despite these impressive gains, still none exhibit the promise of quantum dots in realizing an ideal entangled photon source, showing near-unity entanglement

Another important feature of QDs affecting the measured entanglement is the finestructure splitting (FSS) of the j i *X* state, which is caused by the exchange interaction of the electron and hole in the *e* � *h* pair, together with geometrical asymmetries of the QD [34, 35]. As a result, in the *XX* � *X* cascade, there is a precession between the two recombination pathways in the j i *R =*j i *L* basis (**Figure 2(a)**), and a non-degeneracy in energy of the two pathways in the j i *H =*j i *V* basis will appear (**Figure 2(b)**). Therefore,

> *i δ* 2ℏ*t* j i *VV* � �

� �<sup>Ψ</sup> <sup>þ</sup> *<sup>i</sup>*sin *<sup>δ</sup>*

2ℏ *t* � �

2

j i Φ ,

<sup>p</sup> ð Þ j i *RR* þ j i *LL* , and *δ* ¼ FSS.

(2)

the two-photon quantum state described by Eq. (1) will change into:

*δ* 2ℏ *t*

ffiffi 2 p j i *HH* þ *e*

¼ cos

where j i <sup>Ψ</sup> is the state described by Eq. (1), j i <sup>Φ</sup> <sup>¼</sup> <sup>1</sup>ffiffi

<sup>∣</sup>Ψ~i ¼ <sup>1</sup>

exhibited a low light-extraction efficiency of � 1% [26].

*DOI: http://dx.doi.org/10.5772/intechopen.91814*

ability and single-photon purity have been reported [30].

fidelity and pair-extraction efficiency.

**37**

Over the past three decades, QDs have been extensively studied with recent advancements, as compared to other solid state quantum emitters [18–21], and have produced sources which exhibit features closest to an ideal photon source [22]. The first generation of QDs was self-assembled [23–25], which resulted in QDs with various sizes and imperfect symmetry due to the random nature of the formation process [13]. Moreover, since the bulk semiconductor material possessed a high

### **Figure 1.**

*The XX-X cascade. In the XX* j i *state, holes with jz* ¼ � <sup>3</sup> <sup>2</sup> *and electrons with jz* ¼ � <sup>1</sup> <sup>2</sup> *are paired, resulting in exciton states with jz* ¼ �1*. The two e-h pairs will then lead to two different recombination pathways, with the final state being a superposition of these two paths, i.e.,* j i *<sup>Ψ</sup>* <sup>¼</sup> <sup>1</sup>ffiffi <sup>2</sup> <sup>p</sup> ð Þ j i *RL* þ j i *LR . For a more detailed description of the QDs' electronic structure, please refer to ref. [17].*

### *Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot DOI: http://dx.doi.org/10.5772/intechopen.91814*

refractive index, these self-assembled QDs typically suffered from isotropic emission and total internal reflection at the semiconductor-air interface and thus exhibited a low light-extraction efficiency of � 1% [26].

Recent developments in micro- and nanoscale crystal growth and fabrication have resulted in structures which have improved the performance of QDs considerably. Enhancement of the spontaneous emission of QDs was first achieved by coupling an ensemble of QDs [27], and later a single QD, to a micro-cavity [28]. More recently, the coupling of QDs to micro-pillar cavities has achieved light-extraction efficiencies as high as 80% [29]. Also, such structures allow for proper control of the charge noise around the QD and thus the suppression of detrimental dephasing processes from the moving charge carriers. Excitingly, as a result, photons with >99% indistinguishability and single-photon purity have been reported [30].

However, such performance comes at a price. Due to Coulomb interactions [17], *XX* and *X* emission lines are separated in energy by an amount referred to as the *XX* binding energy, Δ*XX*�*<sup>X</sup>*. Within a typically used cavity with a quality factor of *Q* � 10, 000, which is tuned to photons with the wavelength *λ* � 1*μ*m, the cavity has a bandwidth of � 10*μ*eV. This small bandwidth is far less than a typical *XX* binding energy of Δ*XX*�*<sup>X</sup>* � 1meV [31]. Therefore, either the *X* or the *XX* transition lines can be coupled to the fundamental mode of the cavity, but not both as needed for a highefficiency entangled photon source. Furthermore, to avoid suboptimal entanglement, coupling to the fundamental mode of the cavity should be polarization-independent. Otherwise, one decay path in the cascade process would gain a stronger weight [32]. In an attempt to overcome these challenges, three studies have made considerable gains. Dousse et al. [32] fabricated a micropillar cavity molecule and successfully tuned both *XX* and *X* transitions to separate cavity modes in the molecule. Impressively, the fabricated device was shown to improve the pair-extraction efficiency (ϵ*e*) by three orders of magnitude, ϵ*<sup>e</sup>* ¼ 12%, as compared to a bare self-assembled QD; yet despite this, the source still generates poorly entangled photon pairs with a measured fidelity of F ¼ 67%. Other research by Chen et al. [26] showed the successful fabrication of a broadband dielectric antenna, which enables a pair-extraction efficiency of 37*:*2%, low multiphoton emission *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> <sup>≈</sup>0*:*002, and entanglement fidelity, F ¼ 90%; however, the emission profile deviates from a Gaussian, and photon indistinguishability of such sources is yet to be measured. Additionally, Wang et al. [33] have engineered a circular Bragg grating bull's-eye cavity around a single QD showing a pair-extraction efficiency, ϵ*<sup>e</sup>* ¼ 36*:*6%, and an entanglement fidelity, F ¼ 90%. Despite these impressive gains, still none exhibit the promise of quantum dots in realizing an ideal entangled photon source, showing near-unity entanglement fidelity and pair-extraction efficiency.

Another important feature of QDs affecting the measured entanglement is the finestructure splitting (FSS) of the j i *X* state, which is caused by the exchange interaction of the electron and hole in the *e* � *h* pair, together with geometrical asymmetries of the QD [34, 35]. As a result, in the *XX* � *X* cascade, there is a precession between the two recombination pathways in the j i *R =*j i *L* basis (**Figure 2(a)**), and a non-degeneracy in energy of the two pathways in the j i *H =*j i *V* basis will appear (**Figure 2(b)**). Therefore, the two-photon quantum state described by Eq. (1) will change into:

$$\begin{split} |\tilde{\Psi}\rangle &= \frac{1}{\sqrt{2}} \left( |HH\rangle + e^{i\frac{\delta}{2\hbar}t} |VV\rangle \right) \\ &= \cos\left(\frac{\delta}{2\hbar}t\right) \Psi + i\sin\left(\frac{\delta}{2\hbar}t\right) |\Phi\rangle, \end{split} \tag{2}$$

where j i <sup>Ψ</sup> is the state described by Eq. (1), j i <sup>Φ</sup> <sup>¼</sup> <sup>1</sup>ffiffi 2 <sup>p</sup> ð Þ j i *RR* þ j i *LL* , and *δ* ¼ FSS.

fidelity, single-photon purity, and photon indistinguishability [12] degrade when the pump power is increased [13]. As a result, these sources only operate at extremely low pair-production efficiencies, ϵ*<sup>p</sup>* <1%, per excitation pulse [14]. Hence, engineering and realizing an ideal source of entangled photons is necessary for the successful future of entangled photon pairs for use in quantum photonics, a future made

Semiconductor quantum dots [15] are capable of generating pairs of entangled photons based on a process called the biexciton (*XX*)-exciton (*X*) cascade [16]; this cascade process is shown in **Figure 1**. The j i *XX* state is composed of two electron– hole (*e* � *h*) pairs in the QD's lowest energy level, i.e., *s*-shell. Each of these pairs

j i *XX* state, there are two recombination pathways through the intermediate j i *X* state. Upon recombination, the *e* � *h* pair will emit either a left, j i *L* , or right, j i *R* ,

the two emitted photons will be in the polarization entangled state [16]:

ffiffi 2

Over the past three decades, QDs have been extensively studied with recent advancements, as compared to other solid state quantum emitters [18–21], and have produced sources which exhibit features closest to an ideal photon source [22]. The first generation of QDs was self-assembled [23–25], which resulted in QDs with various sizes and imperfect symmetry due to the random nature of the formation process [13]. Moreover, since the bulk semiconductor material possessed a high

j i <sup>Ψ</sup> <sup>¼</sup> <sup>1</sup>

the QD final state at this step will be in the j i *X* state. This transition, j i *XX* ! j i *X* , is referred to as the neutral biexciton, *XX*, transition. Relaxation to the ground state occurs by the recombination of the remaining *e* � *h* pair in the j i *X* state. The latter transition, i.e., j i *X* ! j i *G* , is referred to as the neutral exciton, *X*, transition. This recombination will emit a photon with a polarization perpendicular to that of the first photon (*XX*), i.e., j i *L XX* ! j i *R <sup>X</sup>* and j i *R XX* ! j i *L <sup>X</sup>*. At the end of the process,

*<sup>z</sup>* with the superposition of *j*

*<sup>z</sup>* ¼ þ1 or *j*

p ð Þ j i *RL* þ j i *LR :* (1)

<sup>2</sup> *and electrons with jz* ¼ � <sup>1</sup>

*exciton states with jz* ¼ �1*. The two e-h pairs will then lead to two different recombination pathways, with the*

<sup>2</sup> *are paired, resulting in*

<sup>2</sup> <sup>p</sup> ð Þ j i *RL* þ j i *LR . For a more detailed description*

*<sup>z</sup>* ¼ �1. From the

*<sup>z</sup>* ¼ �1, respectively, and

brighter by semiconductor quantum dots (QDs).

*Recent Advances in Nanophotonics - Fundamentals and Applications*

circularly polarized photon corresponding to *j*

possesses an angular momentum *j*

**Figure 1.**

**36**

*The XX-X cascade. In the XX* j i *state, holes with jz* ¼ � <sup>3</sup>

*of the QDs' electronic structure, please refer to ref. [17].*

*final state being a superposition of these two paths, i.e.,* j i *<sup>Ψ</sup>* <sup>¼</sup> <sup>1</sup>ffiffi

**Figure 2.**

*XX-X cascade in the presence of FSS. (a) R*j i *and L*j i *basis will be mixed and a precession between the two pathways will be observed. (b) In the H*j i*=*j i *V basis, the transition energies will be split by FSS* ¼ *δ.*

Due to the random nature of the growth process, self-assembled QDs have long suffered from large base asymmetries, which resulted in FSS values larger than the *X* emission linewidth. This feature will lead to the introduction of a which-path information in the *XX* � *X* cascade that will degrade the entanglement between the two emitted photons. For this reason, the early measurements on entanglement in QDs [36, 37] only led to detection of classical correlations; nonclassical correlations were only observed by improving the growth techniques and choosing QDs with FSS ≈0 [38, 39]. In the recent past, several techniques have been proposed and demonstrated in order to erase the FSS of QDs using electric fields [40–42], strain [43], and an optical approach not requiring nanofabrication [44]. However, as we show further along in the chapter, in order to reveal the effect of FSS on entanglement fidelity of the emitted photon pair, a delicate understanding of the detection system is also required. A recent study by Fognini et al. [45] shows that it is possible to measure near-unity entanglement fidelity even in the presence of finite FSS.

resonant to neither *X* nor *XX* transitions. This method can lead to near-unity population of the j i *XX* state [53] and an extreme suppression of multiphoton

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

In this review, we focus on attempts to improve the performance of entangled photon generation in by embedding them in photonic nanowires, as well as the effects of different excitation schemes in the performance of such sources. Additionally, we will also cover the improvements achieved in photon extraction efficiency, reduction of the dephasing processes, suppression of multiphoton emission, and enhancing entanglement fidelity of nanowire QD based entangled photon sources.

*Schematics of resonant TPE. A linearly polarized pulse is tuned to a virtual state halfway between X and XX transitions (the dashed blue line); and the XX* j i *is coherently populated via a two-photon absorption process.*

Embedding QDs in tapered nanowires was initially developed by using topdown approaches via reactive-ion etching [55, 56]. Such photonic structures allow for coupling of the QD emission to the waveguide's fundamental mode in a broad range of wavelengths, Δ*λ*≈ 70nm. Claudon et al. [55] managed to achieve a lightextraction efficiency of ϵ≈ 72%; however, top-down approaches are not flawless. Defects are left at the surface of the nanowire due to etching of the substrate using reactive ions, and additionally there is limited control in the positioning of the QDs at the symmetry axis of the nanowire. It is important to note that these flaws lead to suboptimal quality in the ultimate brightness of the source. As an alternative growth approach in attempt to overcome these issues, pure wurtzite InP nanowires were grown with a bottom-up growth approach and the quantum dot was placed on the nanowire axis to ensure good (�95%) coupling between the quantum dot and

A novel bottom-up approach to growing tapered nanowires was used in the work by Reimer et al. [57]. This innovative approach allowed, for the first time, the positioning of a QD on the symmetry axis of the nanowire and at a desired height with a precision of �100 nm (**Figure 4**). In this method, the growth of the

emission of *X* and *XX* transitions [54].

*DOI: http://dx.doi.org/10.5772/intechopen.91814*

fundamental mode of the nanowire waveguide.

**2.1 Bottom-up grown tapered wurtzite nanowire QDs**

**2. Nanowire QDs**

**39**

**Figure 3.**

To reveal the true potential of QDs, proper excitation schemes are needed in addition to engineering sophisticated photonic structures. Until recently, off-resonant excitation had been widely used to generate entangled and single photons from QDs in photonic structures. This scheme excites charge carriers to energy levels above the bandgap of the host semiconductor, and relaxation of the resulted *e* � *h* pairs to the QD's*s*-shell, mediated by interactions with phonons, leads to the emission of entangled photons. Admittedly, implementing this scheme is relatively straightforward, as the large difference in the frequencies of the excitation laser and the emitted photons allow for simple filtration of the reflected laser light. The excess of charge carriers and their interaction with phonons will lead to detrimental effects such as inhomogeneous broadening of emission lines [46], multiphoton emission caused by re-excitation processes [47], increased jitter in emission time [48], and dephasing [45].

Direct population of j i *XX* is forbidden due to optical selection rules. However, observation of resonant two-photon absorption in photoluminescence excitation spectroscopy of QDs [49] has recently led to development of a resonant two-photon excitation (TPE) scheme [50–52], which allows for coherent population of the j i *XX* state. In order to perform this scheme (**Figure 3**), a linearly polarized excitation pulse is tuned to a virtual state with an energy halfway between that of the ground state, *E*j i *<sup>G</sup>* , and biexciton state, *E*j i *XX* . This virtual state can also be thought of as a transition level between that of the neutral exciton and that of the neutral biexciton transitions. In other words, j i *XX* is coherently populated by a laser pulse, which is

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot DOI: http://dx.doi.org/10.5772/intechopen.91814*

### **Figure 3.**

Due to the random nature of the growth process, self-assembled QDs have long suffered from large base asymmetries, which resulted in FSS values larger than the *X* emission linewidth. This feature will lead to the introduction of a which-path information in the *XX* � *X* cascade that will degrade the entanglement between the two emitted photons. For this reason, the early measurements on entanglement in QDs [36, 37] only led to detection of classical correlations; nonclassical correlations were only observed by improving the growth techniques and choosing QDs with FSS ≈0 [38, 39]. In the recent past, several techniques have been proposed and demonstrated in order to erase the FSS of QDs using electric fields [40–42], strain [43], and an optical approach not requiring nanofabrication [44]. However, as we show further along in the chapter, in order to reveal the effect of FSS on entanglement fidelity of the emitted photon pair, a delicate understanding of the detection system is also required. A recent study by Fognini et al. [45] shows that it is possible to measure near-unity entanglement fidelity even in the presence of finite FSS. To reveal the true potential of QDs, proper excitation schemes are needed in addition to engineering sophisticated photonic structures. Until recently, off-resonant excitation had been widely used to generate entangled and single photons from QDs in photonic structures. This scheme excites charge carriers to energy levels above the bandgap of the host semiconductor, and relaxation of the resulted *e* � *h* pairs to the QD's*s*-shell, mediated by interactions with phonons, leads to the emission of entangled photons. Admittedly, implementing this scheme is relatively straightforward, as the large difference in the frequencies of the excitation laser and the emitted photons allow for simple filtration of the reflected laser light. The excess of charge carriers and their interaction with phonons will lead to detrimental effects such as inhomogeneous broadening of emission lines [46], multiphoton emission caused by re-excitation pro-

*XX-X cascade in the presence of FSS. (a) R*j i *and L*j i *basis will be mixed and a precession between the two pathways will be observed. (b) In the H*j i*=*j i *V basis, the transition energies will be split by FSS* ¼ *δ.*

*Recent Advances in Nanophotonics - Fundamentals and Applications*

**Figure 2.**

**38**

cesses [47], increased jitter in emission time [48], and dephasing [45].

Direct population of j i *XX* is forbidden due to optical selection rules. However, observation of resonant two-photon absorption in photoluminescence excitation spectroscopy of QDs [49] has recently led to development of a resonant two-photon excitation (TPE) scheme [50–52], which allows for coherent population of the j i *XX* state. In order to perform this scheme (**Figure 3**), a linearly polarized excitation pulse is tuned to a virtual state with an energy halfway between that of the ground state, *E*j i *<sup>G</sup>* , and biexciton state, *E*j i *XX* . This virtual state can also be thought of as a transition level between that of the neutral exciton and that of the neutral biexciton transitions. In other words, j i *XX* is coherently populated by a laser pulse, which is

*Schematics of resonant TPE. A linearly polarized pulse is tuned to a virtual state halfway between X and XX transitions (the dashed blue line); and the XX* j i *is coherently populated via a two-photon absorption process.*

resonant to neither *X* nor *XX* transitions. This method can lead to near-unity population of the j i *XX* state [53] and an extreme suppression of multiphoton emission of *X* and *XX* transitions [54].

In this review, we focus on attempts to improve the performance of entangled photon generation in by embedding them in photonic nanowires, as well as the effects of different excitation schemes in the performance of such sources. Additionally, we will also cover the improvements achieved in photon extraction efficiency, reduction of the dephasing processes, suppression of multiphoton emission, and enhancing entanglement fidelity of nanowire QD based entangled photon sources.

### **2. Nanowire QDs**

Embedding QDs in tapered nanowires was initially developed by using topdown approaches via reactive-ion etching [55, 56]. Such photonic structures allow for coupling of the QD emission to the waveguide's fundamental mode in a broad range of wavelengths, Δ*λ*≈ 70nm. Claudon et al. [55] managed to achieve a lightextraction efficiency of ϵ≈ 72%; however, top-down approaches are not flawless. Defects are left at the surface of the nanowire due to etching of the substrate using reactive ions, and additionally there is limited control in the positioning of the QDs at the symmetry axis of the nanowire. It is important to note that these flaws lead to suboptimal quality in the ultimate brightness of the source. As an alternative growth approach in attempt to overcome these issues, pure wurtzite InP nanowires were grown with a bottom-up growth approach and the quantum dot was placed on the nanowire axis to ensure good (�95%) coupling between the quantum dot and fundamental mode of the nanowire waveguide.

### **2.1 Bottom-up grown tapered wurtzite nanowire QDs**

A novel bottom-up approach to growing tapered nanowires was used in the work by Reimer et al. [57]. This innovative approach allowed, for the first time, the positioning of a QD on the symmetry axis of the nanowire and at a desired height with a precision of �100 nm (**Figure 4**). In this method, the growth of the

nanowire core, InP, is initiated by a gold particle which defines the core of the nanowire and ultimately the size of the QD, *D* ≈20 � 30nm. After reaching the desired height, arsine is introduced to the growth chamber, and the QD, InAsP, is grown. Then, by changing the growth conditions, the nanowire is grown radially in order to create a shell *D* ≈220nm, which facilitates the waveguide effect. In the last phase of growth, the conditions are changed once again to achieve an ideal tapering at the nanowire tip with an angle of *θ* <2<sup>∘</sup> , which results in minimal internal reflection for the emitted photons leaving the nanowire.

### **2.2 Optical properties**

In terms of brightness, a value of ϵ ¼ 43 4ð Þ% for success probability of singlephoton extraction at the first lens has been reported [46]. However, theoretically these sources allow for single-photon extraction efficiencies up to ϵ ¼ 97%, which can be achieved by perfect tapering of the nanowire, *θ* ¼ 1°, placing the nanowires on top of a flawless mirror and placing the QD at the correct height in order to create perfect constructive interference [57]. In terms of multiphoton emission (**Box 1**), second-order correlation, *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> , measurements yield values <sup>&</sup>lt; 1% [45, 46], showing a true single-photon emitter. The emitted photons exhibit extremely narrow linewidths, *δω*<1GHz, with coherence lengths exceeding 1ns. Furthermore, a high level of visibility in a Hong-Ou-Mandel (HOM) measurement, *V* ¼ 85%, has been observed, indicating highly indistinguishable photons [46]. Moreover, there is a close to perfect overlap, 98*:*8% � 0*:*1%, between the far-field emission profile of these nanowires QDs and a Gaussian emission profile of a

### **1. Hanbury Brown and Twiss (HBT) setup**

In order to quantify the multiphoton emission of a source, the second-order correlation function is measured based on a setup first introduced by Hanbury Brown and Twiss [59] (**Figure a**). In this method, the light emitted from the source is sent to a beam splitter and then detected by two single-photon detectors D1 and D2. By correlating the intensities recorded by the two detectors in different time bins, one can gain information about the emission pattern of the source. Considering the particle nature of photons, if the source emits one and only one photon in each emission mode upon excitation, there will be no simultaneous detection on the two detectors; in other words, there will not be any correlation at zero time delay:

$$\mathbf{g}^{(2)}(\mathbf{0}) = \frac{\langle n\_1(t)n\_2(t) \rangle}{\langle n\_1(t) \rangle \langle n\_2(t) \rangle} = \mathbf{0},\tag{3}$$

single-mode fiber; in practice, these sources have resulted in a coupling efficiency over 93% into a single-mode fiber [58]. Impressively, this feature then allows for possibilities in long-distance fiber-based quantum communication with high

*Schematic of the bottom-up nanowire growth process and SEM image of a tapered nanowire (right). The growth process is initiated by a gold particle, which defines the dimensions of the QDf. After the quantum dot is grown the waveguide shell and the tapered tip are fabricated around the QD by controlling the growth parameters. This growth process ensures that the QD is placed on-axis of the tapered nanowire waveguide for*

*Measuring multiphoton emission and photon indistinguishability of entangled photon sources.*

photons. In this scenario, two successive photons are brought together at the beam splitter for interference. Now, at the beam splitter, four different possibilities exist (**Figure c**); photon 1 may be reflected and photon 2 transmitted (case 1), photon 1 may transmit and photon 2 be reflected (case 2), both may transmit (case 3), and, lastly, both may be reflected (case 4). With reflection from the two sides of the beam splitter differing in a *π* phase shift, the third and the fourth cases are physically identical except for a general phase. Therefore, in the interference of the two photons, these two cases will cancel out, leaving only the options with the two photons going to either D1 or D2. Building a histogram out of the correlations of the two detectors, similar to the second-order correlation measurement, exhibits zero coincidence counts at zero time delay for the case of a source generating perfectly indistinguishable photons in each mode.

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

*DOI: http://dx.doi.org/10.5772/intechopen.91814*

efficiency through low-loss communication channels.

**Box ¹.**

**Figure 4.**

**41**

*efficient light extraction.*

with *ni*ð Þ*t* being the number of photons detected by detector *i i*ð Þ ¼ 1, 2 at time *t*.

### **2. Hong-Ou-Mandel setup**

In addition to single-photon emission, for an ideal entangled photon source, the emitted photons in each mode should exhibit perfect indistinguishability. For measuring this feature, the Hong-Ou-Mandel setup is used. Using a setup similar to that the HBT (**Figure b**) and considering the wave nature of the photons, a HOM measurement enables one to test the degree of indistinguishability of the successive

photons. In this scenario, two successive photons are brought together at the beam splitter for interference. Now, at the beam splitter, four different possibilities exist (**Figure c**); photon 1 may be reflected and photon 2 transmitted (case 1), photon 1 may transmit and photon 2 be reflected (case 2), both may transmit (case 3), and, lastly, both may be reflected (case 4). With reflection from the two sides of the beam splitter differing in a *π* phase shift, the third and the fourth cases are physically identical except for a general phase. Therefore, in the interference of the two photons, these two cases will cancel out, leaving only the options with the two photons going to either D1 or D2. Building a histogram out of the correlations of the two detectors, similar to the second-order correlation measurement, exhibits zero coincidence counts at zero time delay for the case of a source generating perfectly indistinguishable photons in each mode.

**Box ¹.** *Measuring multiphoton emission and photon indistinguishability of entangled photon sources.*

single-mode fiber; in practice, these sources have resulted in a coupling efficiency over 93% into a single-mode fiber [58]. Impressively, this feature then allows for possibilities in long-distance fiber-based quantum communication with high efficiency through low-loss communication channels.

### **Figure 4.**

nanowire core, InP, is initiated by a gold particle which defines the core of the nanowire and ultimately the size of the QD, *D* ≈20 � 30nm. After reaching the desired height, arsine is introduced to the growth chamber, and the QD, InAsP, is grown. Then, by changing the growth conditions, the nanowire is grown radially in order to create a shell *D* ≈220nm, which facilitates the waveguide effect. In the last phase of growth, the conditions are changed once again to achieve an ideal tapering

In terms of brightness, a value of ϵ ¼ 43 4ð Þ% for success probability of singlephoton extraction at the first lens has been reported [46]. However, theoretically these sources allow for single-photon extraction efficiencies up to ϵ ¼ 97%, which can be achieved by perfect tapering of the nanowire, *θ* ¼ 1°, placing the nanowires on top of a flawless mirror and placing the QD at the correct height in order to create perfect constructive interference [57]. In terms of multiphoton emission (**Box 1**), second-order correlation, *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> , measurements yield values <sup>&</sup>lt; 1% [45, 46], showing a true single-photon emitter. The emitted photons exhibit extremely narrow linewidths, *δω*<1GHz, with coherence lengths exceeding 1ns. Furthermore, a high level of visibility in a Hong-Ou-Mandel (HOM) measurement, *V* ¼ 85%, has been observed, indicating highly indistinguishable photons [46]. Moreover, there is a close to perfect overlap, 98*:*8% � 0*:*1%, between the far-field emission profile of these nanowires QDs and a Gaussian emission profile of a

In order to quantify the multiphoton emission of a source, the second-order correlation function is measured based on a setup first introduced by Hanbury Brown and Twiss [59] (**Figure a**). In this method, the light emitted from the source is sent to a beam splitter and then detected by two single-photon detectors D1 and D2. By correlating the intensities recorded by the two detectors in different time bins, one can gain information about the emission pattern of the source. Considering the particle nature of photons, if the source emits one and only one photon in each emission mode upon excitation, there will be no simultaneous detection on the two detectors; in other words, there will not be any correlation at zero time delay: *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ¼ <sup>0</sup> h i *<sup>n</sup>*1ð Þ*<sup>t</sup> <sup>n</sup>*2ð Þ*<sup>t</sup>*

In addition to single-photon emission, for an ideal entangled photon source, the emitted photons in each mode should exhibit perfect indistinguishability. For measuring this feature, the Hong-Ou-Mandel setup is used. Using a setup similar to that the HBT (**Figure b**) and considering the wave nature of the photons, a HOM measurement enables one to test the degree of indistinguishability of the successive

with *ni*ð Þ*t* being the number of photons detected by detector *i i*ð Þ ¼ 1, 2 at time *t*.

, which results in minimal internal

h i *<sup>n</sup>*1ð Þ*<sup>t</sup>* h i *<sup>n</sup>*2ð Þ*<sup>t</sup>* <sup>¼</sup> 0, (3)

at the nanowire tip with an angle of *θ* <2<sup>∘</sup>

**1. Hanbury Brown and Twiss (HBT) setup**

**2. Hong-Ou-Mandel setup**

**40**

**2.2 Optical properties**

reflection for the emitted photons leaving the nanowire.

*Recent Advances in Nanophotonics - Fundamentals and Applications*

*Schematic of the bottom-up nanowire growth process and SEM image of a tapered nanowire (right). The growth process is initiated by a gold particle, which defines the dimensions of the QDf. After the quantum dot is grown the waveguide shell and the tapered tip are fabricated around the QD by controlling the growth parameters. This growth process ensures that the QD is placed on-axis of the tapered nanowire waveguide for efficient light extraction.*

### **2.3 Entanglement measurements**

Following the method introduced by James et al. [60], the first results in measuring the degree of entanglement in bottom-up grown nanowire QDs were reported in 2014 by Versteegh et al. [61]. In this work, using an above-bandgap excitation scheme, the fidelity of the emitted *XX* � *X* pairs to a maximally entangled state was found to be *F* ¼ 0*:*859ð Þ �0*:*006 , with a concurrence equal to *C* ¼ 0*:*80ð Þ �0*:*02 , under strong post-selection conditions. The fidelity is reduced to 0*:*765 � 0*:*002 with inclusion of 100% of the emitted photons. By changing the excitation conditions to excite the QD at the wurtzite InP nanowire bandgap, Jöns et al. [62] enhanced the fidelity of the same source as used by Versteegh et al. [61] to *F* ¼ 0*:*817 � 0*:*002 by including all of the collected photon pairs. This strong degree of entanglement allowed Jöns et al. [62] to perform a Bell type inequality violation test, specifically the Clauser-Horne-Shimony-Holt (CHSH) measurement [63]. The CHSH measurement yielded a violation of Bell's inequality by 25 standard deviations, clearly showing the promising features of bottom-up nanowire QDs for secure quantum communication purposes. The experimental setup can be seen in **Figure 5**. Initially, a pair of *λ=*2 and *λ=*4 waveplates corrects for the birefringence observed in the nanowire, causing the entangled state to rotate to an elliptical state instead of the expected j i <sup>Ψ</sup> <sup>¼</sup> <sup>1</sup>ffiffi 2 <sup>p</sup> ð Þ j i *HH* þ j i *VV* [61]. The stream of emitted photons are separated by a 50/50 beam splitter and sent to two separate detectors tuned to the specific wavelengths of the *X* and *XX*. In order not to be affected by the phase introduced when photons hit the reflecting surface of the beam splitter, the *λ=*2 and *λ=*4 waveplates used for projection measurements are aligned along the transmission path [62].

It is important to note that neither of the above-mentioned works addresses the ultimate entanglement fidelity achievable for nanowire QDs. In addition to the projection measurements, a more in-depth analysis is needed in order to reveal the underlying physical mechanisms such as dephasing due to nuclear spins and charge carriers through spin-flip processes. Moreover, the effect of *FSS* on the measured value of entanglement fidelity deserves more care, since, even though lifting of the degeneracy between the two decay paths in the *XX* � *X* cascade can be interpreted as an introduction of a which-path information, the effect is purely unitary, and the precession shall not destroy the entanglement alone. Here, the detection apparatus will play a major role regarding the effect of *FSS* on the entanglement fidelity.

The emission spectrum of the source is provided in **Figure 6**. Upon excitation of the sample using a green laser at *λ* ¼ 520nm (**Figure 6a**), three sets of peaks can be observed: the wurtzite InP bandgap at *λ* ¼ 830nm, levels attributed to the donors and acceptors which are formed due to the presence of impurities such as beryllium in the growth chamber, at *λ*≈870nm, and the *s*-shell of the QD at *λ*≈894nm. The *XX* � *X* cascade can be generated by exciting the sample either at the InP bandgap or the donor/acceptor levels. The charge environment around the QD is different in the two cases. Whereas excitation at the InP bandgap (**Figure 6b**) leads to appearance of a negatively charged exciton (*X*�), exciting the QD at the donor/acceptor level will lead to emission of positively charged excitons (*X*þ) and suppression of the *X*� emission line. Moreover, using the donor/acceptor levels to excite the quantum dot, this excitation scheme will fill the charge traps around the QD. As a consequence, the charge mobility will be significantly reduced; a phenomenon which we will show to be extremely effective in suppressing the dephasing caused

*QD emission spectra. (a) Emission spectrum by excitation via a green laser. Excitations at two different energy levels, wurtzite InP bandgap at 830 nm and donor/acceptor levels at* ≈*870 nm, were used for performing entanglement measurements; (b) the emission spectrum for 830 nm excitation exhibiting the exciton (X), biexciton (XX), and negatively charged exciton (X*�*) lines. (c) Excitation at to 870 nm leads to an appearance of a positively charged exciton (X*þ*) and suppression of X*�*. The spectra in (b) and (c) were taken at the saturation power of X.*

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

*DOI: http://dx.doi.org/10.5772/intechopen.91814*

Following a similar setup to the one used by Jöns et al. [62] (**Figure 5**), Fognini et al. [45] conducted two-photon quantum state tomography on the *XX* � *X* cascade in time intervals of 100 ps during the decay time of the exciton, which allowed for the construction of the density matrix of the photon pair, and gave the opportunity to observe the evolution of the two photon quantum state. The result of these

by the surrounding charge carriers.

**Figure 6.**

**43**

### *2.3.1 Dephasing-free entanglement in nanowire QDs*

In an attempt to shed light on these finer aspects of generation of entangled photons in nanowire QDs, Fognini et al. [45] studied an InAsP QD embedded in an InP photonic nanowire, revealing the effects of dephasing, *FSS*, and imperfections of the detection system on the values achieved for entanglement fidelity.

### **Figure 5.**

*Two-photon quantum state tomography setup. The setup consists of two pairs of λ=*4*–λ=*2 *wave plate sets, which combined with a pair of polarizors perform the projection measurements. A combination of λ=*2 *and λ=*4 *wave plates is used to compensate for the birefringence, if it is present in the nanowire (the image is taken from Jöns et al. [62]).*

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot DOI: http://dx.doi.org/10.5772/intechopen.91814*

### **Figure 6.**

**2.3 Entanglement measurements**

instead of the expected j i <sup>Ψ</sup> <sup>¼</sup> <sup>1</sup>ffiffi

sion path [62].

**Figure 5.**

*et al. [62]).*

**42**

2

*2.3.1 Dephasing-free entanglement in nanowire QDs*

are separated by a 50/50 beam splitter and sent to two separate detectors tuned to the specific wavelengths of the *X* and *XX*. In order not to be affected by the phase introduced when photons hit the reflecting surface of the beam splitter, the *λ=*2 and *λ=*4 waveplates used for projection measurements are aligned along the transmis-

It is important to note that neither of the above-mentioned works addresses the ultimate entanglement fidelity achievable for nanowire QDs. In addition to the projection measurements, a more in-depth analysis is needed in order to reveal the underlying physical mechanisms such as dephasing due to nuclear spins and charge carriers through spin-flip processes. Moreover, the effect of *FSS* on the measured value of entanglement fidelity deserves more care, since, even though lifting of the degeneracy between the two decay paths in the *XX* � *X* cascade can be interpreted as an introduction of a which-path information, the effect is purely unitary, and the precession shall not destroy the entanglement alone. Here, the detection apparatus will play a major role regarding the effect of *FSS* on the entanglement fidelity.

In an attempt to shed light on these finer aspects of generation of entangled photons in nanowire QDs, Fognini et al. [45] studied an InAsP QD embedded in an InP photonic nanowire, revealing the effects of dephasing, *FSS*, and imperfections

*Two-photon quantum state tomography setup. The setup consists of two pairs of λ=*4*–λ=*2 *wave plate sets, which combined with a pair of polarizors perform the projection measurements. A combination of λ=*2 *and λ=*4 *wave plates is used to compensate for the birefringence, if it is present in the nanowire (the image is taken from Jöns*

of the detection system on the values achieved for entanglement fidelity.

<sup>p</sup> ð Þ j i *HH* þ j i *VV* [61]. The stream of emitted photons

Following the method introduced by James et al. [60], the first results in measuring the degree of entanglement in bottom-up grown nanowire QDs were reported in 2014 by Versteegh et al. [61]. In this work, using an above-bandgap excitation scheme, the fidelity of the emitted *XX* � *X* pairs to a maximally entangled state was found to be *F* ¼ 0*:*859ð Þ �0*:*006 , with a concurrence equal to *C* ¼ 0*:*80ð Þ �0*:*02 , under strong post-selection conditions. The fidelity is reduced to 0*:*765 � 0*:*002 with inclusion of 100% of the emitted photons. By changing the excitation conditions to excite the QD at the wurtzite InP nanowire bandgap, Jöns et al. [62] enhanced the fidelity of the same source as used by Versteegh et al. [61] to *F* ¼ 0*:*817 � 0*:*002 by including all of the collected photon pairs. This strong degree of entanglement allowed Jöns et al. [62] to perform a Bell type inequality violation test, specifically the Clauser-Horne-Shimony-Holt (CHSH) measurement [63]. The CHSH measurement yielded a violation of Bell's inequality by 25 standard deviations, clearly showing the promising features of bottom-up nanowire QDs for secure quantum communication purposes. The experimental setup can be seen in **Figure 5**. Initially, a pair of *λ=*2 and *λ=*4 waveplates corrects for the birefringence observed in the nanowire, causing the entangled state to rotate to an elliptical state

*Recent Advances in Nanophotonics - Fundamentals and Applications*

*QD emission spectra. (a) Emission spectrum by excitation via a green laser. Excitations at two different energy levels, wurtzite InP bandgap at 830 nm and donor/acceptor levels at* ≈*870 nm, were used for performing entanglement measurements; (b) the emission spectrum for 830 nm excitation exhibiting the exciton (X), biexciton (XX), and negatively charged exciton (X*�*) lines. (c) Excitation at to 870 nm leads to an appearance of a positively charged exciton (X*þ*) and suppression of X*�*. The spectra in (b) and (c) were taken at the saturation power of X.*

The emission spectrum of the source is provided in **Figure 6**. Upon excitation of the sample using a green laser at *λ* ¼ 520nm (**Figure 6a**), three sets of peaks can be observed: the wurtzite InP bandgap at *λ* ¼ 830nm, levels attributed to the donors and acceptors which are formed due to the presence of impurities such as beryllium in the growth chamber, at *λ*≈870nm, and the *s*-shell of the QD at *λ*≈894nm. The *XX* � *X* cascade can be generated by exciting the sample either at the InP bandgap or the donor/acceptor levels. The charge environment around the QD is different in the two cases. Whereas excitation at the InP bandgap (**Figure 6b**) leads to appearance of a negatively charged exciton (*X*�), exciting the QD at the donor/acceptor level will lead to emission of positively charged excitons (*X*þ) and suppression of the *X*� emission line. Moreover, using the donor/acceptor levels to excite the quantum dot, this excitation scheme will fill the charge traps around the QD. As a consequence, the charge mobility will be significantly reduced; a phenomenon which we will show to be extremely effective in suppressing the dephasing caused by the surrounding charge carriers.

Following a similar setup to the one used by Jöns et al. [62] (**Figure 5**), Fognini et al. [45] conducted two-photon quantum state tomography on the *XX* � *X* cascade in time intervals of 100 ps during the decay time of the exciton, which allowed for the construction of the density matrix of the photon pair, and gave the opportunity to observe the evolution of the two photon quantum state. The result of these

the exciton. Fognini et al. [45] further investigated the change of the concurrence value during the exciton's emission time as compared to the measured values with a model that included the parameters of the *XX* � *<sup>X</sup>* cascade, *FSS*, *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> of *<sup>X</sup>* and *XX*, *τX*, etc., as well as the features of the detection system, including the detectors' timing

Starting with the state described by Eq. (2), the expected values for 36 possible projection correlations, *Nij* ð Þ *i*, *j*∈ f g *H*,*V*, *D*, *A*, *R*, *L* with the letters indicating the photon polarization along horizontal, vertical, diagonal, antidiagonal, right, and left, respectively, at time *t* and during a time interval Δ*t* can be written as:

where *N*<sup>0</sup> is the total number of photon pairs collected, *δ* is the value of *FSS*, *τ<sup>X</sup>* is

To construct the density matrix of the two-photon quantum state, Eq. (4) gives the correlations in all 36 bases with the effect of the detectors' timing resolution function included. However, two additional factors should be included, *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> of *<sup>X</sup>* and *XX*, and also the detectors' dark counts. The dark counts will result in detection of false correlations that are evenly distributed in time, which has to be added to the raw correlations obtained by Eq. (4). On the other hand, the system studied by

*<sup>X</sup>* ð Þ 0 is negligible in comparison, one finds that uncorrelated photons are

where *ρsim*ð Þ*t* is the density matrix of the two-photon quantum state at a particular time *t* based on the simulation and after considering all of the factors; *ρraw*ð Þ*t* is the density matrix constructed from the correlation counts of Eq. (4), together with the effect of the dark counts; and *=*4 is the density matrix of an uncorrelated pair of

**Figure 7c** shows the calculated *HH* þ *VV* correlations obtained from Eq. (4), as well as ð Þ� *RL* þ *LR* ð Þ *RR* þ *LL* correlations indicating the oscillation between jΨi and jΦi, which shows a similar trend to the experimental results as shown in **Figure 7a**. In the next step, we plot the evolution of the calculated concurrence during the exciton lifetime in **Figure 7d**, based on the density matrix constructed from Eq. (5). The measured values are plotted as light green circles, similar to **Figure 7b**, and the results from the simulation are plotted as a solid red line. Surprisingly, it is with great precision that the two data sets agree. The results from the simulation follow the same trend as the measured values, with three regimes:

ation. In other words, the density matrix constructed by considering the correlations in different bases, described in Eq. (4), and addition of the effect of the dark

> ð Þ2 *XX*ð Þ 0 *<sup>ρ</sup>raw*ðÞþ*<sup>t</sup> <sup>g</sup>*

ability of an exciton following an exponential decay, ∗ is the convolution operator,

ð Þ2

0*:*10 � 0*:*01 for *XX* by inclusion of the counts in a range of Δ*t* ¼ 100 ps in the proximity of *<sup>t</sup>* <sup>¼</sup> 0. A non-zero value of *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> for either *XX* or *<sup>X</sup>* results in the addition of uncorrelated photons in the tomography measurement and thus a reduction in the measured entanglement fidelity. Now, by only considering *g*

*n t*ð Þ , *τ<sup>X</sup>*

<sup>∗</sup> *g t*ð ÞΔ*<sup>t</sup>* (4)

*<sup>X</sup>* ð Þ¼ 0 0*:*003 � 0*:*003 for *X* and *g*

ð Þ2 *XX*ð Þ 0

ð Þ2 *XX*ð Þ 0 

fraction of the

<sup>4</sup> (5)

*XX*ð Þ 0 fraction of the times, which has to be taken into consider-

describes the emission prob-

ð Þ2 *XX*ð Þ¼ 0

> ð Þ2 *XX*ð Þ 0 ,

resolution and dark counts, but did not include any term for dephasing.

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

*Nij*ðÞ¼ *<sup>t</sup> <sup>N</sup>*<sup>0</sup> j j h i *ij*jΨð Þ *<sup>t</sup>*, *<sup>δ</sup>* <sup>2</sup>

the lifetime of the exciton state, *n t*ð Þ¼ , *<sup>τ</sup><sup>X</sup>* <sup>1</sup>*=τ<sup>X</sup> <sup>e</sup>*�*t=τ<sup>X</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.91814*

Fognini et al. [45] exhibits values of *g*

ð Þ2

since *g* ð Þ2

**45**

being detected in *g*

and *g t*ð Þ denotes the detectors' timing resolution function.

counts describe the behavior of the system only in 1 � *g*

times. Therefore, the actual density matrix is expected to be:

*ρsim*ðÞ¼ *t* 1 � *g*

photons, 1*=*4½ � j i *HH* h j *HH* þ j i *HV* h j *HV* þ j i *VH* h j *VH* þ j i *VV* h j *VV* .

### **Figure 7.**

*Dephasing-free entanglement (a) showing the correlation measurements HH* þ *VV and RL* ð Þ� þ *LR* ð Þ *RR* þ *LL . The former does not show any oscillations as H*j i *and V*j i *are the eigenstates of the Hamiltonian, whereas the latter reveals the precession of the state between* j i Ψ *and* j i Φ *, according to Eq. (2). The shaded gray bars indicate instances with the highest concurrence, (A), and instances with the lowest imaginary part in the density matrix (B-D). (b) The concurrence extracted from the correlation measurements at each instant of time, for time windows of* Δ*t* ¼ 100 *ps, along the decay time of the exciton. (c) Result of the correlations obtained from the theoretical model (Eq. (4)) with the gray shaded bars indicating similar instances as for (a). (d) Comparison of the measured values of the concurrence with that of the theoretical model, revealing the dephasing-free nature of the XX* � *X cascade.*

measurements with excitation at the donor/acceptor levels can be observed in **Figure 7a, b**. In **Figure 7a**, the results of the correlation measurements in the *H=V* basis and *R=L* basis are presented. By setting the detector tuned to the *XX* emission line as the "start," and the other detector tuned to *X* line as the "stop" in the correlation measurements, variations can be found within the coincidence counts in different bases during the exciton decay. Plotting the correlation counts ð Þ� *RL* þ *LR* ð Þ *RR* þ *LL* vs. decay time reveals the precession of the two-photon quantum state between the two entangled states j i Ψ and j i Φ , according to Eq. (2), with a frequency proportional to *FSS*. From a fit to the measured oscillation of the two-photon quantum state, we calculate the *FSS* to be *δ* ¼ 795*:*52 � 0*:*35 MHz. From the *HH* þ *VV* correlation, a fit to the data yields an *X* lifetime of *τ<sup>X</sup>* ¼ 847 � 6 ps. **Figure 7b** shows the results of calculating the concurrence [64], C, of the two-photon quantum state, with C ¼ 0 indicating no entanglement and C ¼ 1 showing a maximally entangled state, for time windows of 100 ps along the exciton decay time. The gray bars indicate the time instances when the calculated concurrence is the highest, A, and when the imaginary part of the density matrix is zero, B-D. The respective density matrices of each instance is given in the subplots on the top right of **Figure 7**. The concurrence reaches a value of C ¼ 0*:*77 � 0*:*02 at its peak, corresponding to a fidelity of F ¼ 0*:*88, with a count-weighted average of C ¼ 0*:*62 � 0*:*03.

Despite the fact that the value for concurrence does not reach near unity and that after a peak around *t* ¼ 0, it suffers a significant reduction, the observed behavior does not indicate the presence of a dephasing mechanism during the decay time of

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot DOI: http://dx.doi.org/10.5772/intechopen.91814*

the exciton. Fognini et al. [45] further investigated the change of the concurrence value during the exciton's emission time as compared to the measured values with a model that included the parameters of the *XX* � *<sup>X</sup>* cascade, *FSS*, *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> of *<sup>X</sup>* and *XX*, *τX*, etc., as well as the features of the detection system, including the detectors' timing resolution and dark counts, but did not include any term for dephasing.

Starting with the state described by Eq. (2), the expected values for 36 possible projection correlations, *Nij* ð Þ *i*, *j*∈ f g *H*,*V*, *D*, *A*, *R*, *L* with the letters indicating the photon polarization along horizontal, vertical, diagonal, antidiagonal, right, and left, respectively, at time *t* and during a time interval Δ*t* can be written as:

$$N\_{\vec{\eta}}(t) = N\_0 \Big( \left| \langle \dot{\eta} | \Psi(t, \delta) \rangle \right|^2 n(t, \tau\_X) \Big) \* \mathbf{g}(t) \Delta t \tag{4}$$

where *N*<sup>0</sup> is the total number of photon pairs collected, *δ* is the value of *FSS*, *τ<sup>X</sup>* is the lifetime of the exciton state, *n t*ð Þ¼ , *<sup>τ</sup><sup>X</sup>* <sup>1</sup>*=τ<sup>X</sup> <sup>e</sup>*�*t=τ<sup>X</sup>* describes the emission probability of an exciton following an exponential decay, ∗ is the convolution operator, and *g t*ð Þ denotes the detectors' timing resolution function.

To construct the density matrix of the two-photon quantum state, Eq. (4) gives the correlations in all 36 bases with the effect of the detectors' timing resolution function included. However, two additional factors should be included, *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> of *<sup>X</sup>* and *XX*, and also the detectors' dark counts. The dark counts will result in detection of false correlations that are evenly distributed in time, which has to be added to the raw correlations obtained by Eq. (4). On the other hand, the system studied by Fognini et al. [45] exhibits values of *g* ð Þ2 *<sup>X</sup>* ð Þ¼ 0 0*:*003 � 0*:*003 for *X* and *g* ð Þ2 *XX*ð Þ¼ 0 0*:*10 � 0*:*01 for *XX* by inclusion of the counts in a range of Δ*t* ¼ 100 ps in the proximity of *<sup>t</sup>* <sup>¼</sup> 0. A non-zero value of *<sup>g</sup>*ð Þ<sup>2</sup> ð Þ <sup>0</sup> for either *XX* or *<sup>X</sup>* results in the addition of uncorrelated photons in the tomography measurement and thus a reduction in the measured entanglement fidelity. Now, by only considering *g* ð Þ2 *XX*ð Þ 0 , since *g* ð Þ2 *<sup>X</sup>* ð Þ 0 is negligible in comparison, one finds that uncorrelated photons are being detected in *g* ð Þ2 *XX*ð Þ 0 fraction of the times, which has to be taken into consideration. In other words, the density matrix constructed by considering the correlations in different bases, described in Eq. (4), and addition of the effect of the dark counts describe the behavior of the system only in 1 � *g* ð Þ2 *XX*ð Þ 0 fraction of the times. Therefore, the actual density matrix is expected to be:

$$
\rho\_{sim}(t) = \left(\mathbf{1} - \mathbf{g}\_{XX}^{(2)}(\mathbf{0})\right) \rho\_{raw}(t) + \mathbf{g}\_{XX}^{(2)}(\mathbf{0}) \frac{\mathbf{I}}{\mathbf{4}} \tag{5}
$$

where *ρsim*ð Þ*t* is the density matrix of the two-photon quantum state at a particular time *t* based on the simulation and after considering all of the factors; *ρraw*ð Þ*t* is the density matrix constructed from the correlation counts of Eq. (4), together with the effect of the dark counts; and *=*4 is the density matrix of an uncorrelated pair of photons, 1*=*4½ � j i *HH* h j *HH* þ j i *HV* h j *HV* þ j i *VH* h j *VH* þ j i *VV* h j *VV* .

**Figure 7c** shows the calculated *HH* þ *VV* correlations obtained from Eq. (4), as well as ð Þ� *RL* þ *LR* ð Þ *RR* þ *LL* correlations indicating the oscillation between jΨi and jΦi, which shows a similar trend to the experimental results as shown in **Figure 7a**. In the next step, we plot the evolution of the calculated concurrence during the exciton lifetime in **Figure 7d**, based on the density matrix constructed from Eq. (5). The measured values are plotted as light green circles, similar to **Figure 7b**, and the results from the simulation are plotted as a solid red line. Surprisingly, it is with great precision that the two data sets agree. The results from the simulation follow the same trend as the measured values, with three regimes:

measurements with excitation at the donor/acceptor levels can be observed in **Figure 7a, b**. In **Figure 7a**, the results of the correlation measurements in the *H=V* basis and *R=L* basis are presented. By setting the detector tuned to the *XX* emission line as the "start," and the other detector tuned to *X* line as the "stop" in the correlation measurements, variations can be found within the coincidence counts in

*Dephasing-free entanglement (a) showing the correlation measurements HH* þ *VV and RL* ð Þ� þ *LR* ð Þ *RR* þ *LL . The former does not show any oscillations as H*j i *and V*j i *are the eigenstates of the Hamiltonian, whereas the latter reveals the precession of the state between* j i Ψ *and* j i Φ *, according to Eq. (2). The shaded gray bars indicate instances with the highest concurrence, (A), and instances with the lowest imaginary part in the density matrix (B-D). (b) The concurrence extracted from the correlation measurements at each instant of time, for time windows of* Δ*t* ¼ 100 *ps, along the decay time of the exciton. (c) Result of the correlations obtained from the theoretical model (Eq. (4)) with the gray shaded bars indicating similar instances as for (a). (d) Comparison of the measured values of the concurrence with that of the theoretical model, revealing the*

*Recent Advances in Nanophotonics - Fundamentals and Applications*

ð Þ� *RL* þ *LR* ð Þ *RR* þ *LL* vs. decay time reveals the precession of the two-photon quantum state between the two entangled states j i Ψ and j i Φ , according to Eq. (2), with a frequency proportional to *FSS*. From a fit to the measured oscillation of the two-photon quantum state, we calculate the *FSS* to be *δ* ¼ 795*:*52 � 0*:*35 MHz. From the *HH* þ *VV* correlation, a fit to the data yields an *X* lifetime of *τ<sup>X</sup>* ¼ 847 � 6 ps. **Figure 7b** shows the results of calculating the concurrence [64], C, of the two-photon quantum state, with C ¼ 0 indicating no entanglement and C ¼ 1 showing a maximally entangled state, for time windows of 100 ps along the exciton decay time. The gray bars indicate the time instances when the calculated concurrence is the highest, A, and when the imaginary part of the density matrix is zero, B-D. The respective density matrices of each instance is given in the subplots on the top right of **Figure 7**. The concurrence reaches a value of C ¼ 0*:*77 � 0*:*02 at its peak, corresponding to a

different bases during the exciton decay. Plotting the correlation counts

**Figure 7.**

**44**

*dephasing-free nature of the XX* � *X cascade.*

fidelity of F ¼ 0*:*88, with a count-weighted average of C ¼ 0*:*62 � 0*:*03.

Despite the fact that the value for concurrence does not reach near unity and that after a peak around *t* ¼ 0, it suffers a significant reduction, the observed behavior does not indicate the presence of a dephasing mechanism during the decay time of

(I) top, (II) flat, and (III) roll-off. Initially, as the detectors' response function *g t*ð Þ detects more photons, the plot shows an increase in the concurrence reaching a maximum. However, due to the low timing resolution of the detectors after a period of time, the phase averaging of Eq. (2) becomes significant, and the measured concurrence drops. Once the response function has been fully covered, the phase averaging becomes constant, hence reaching the "flat" part. Finally, as the probability of the exciton emission drops exponentially, the effect of dark counts dominates the actual photon counts, and the tomography system will detect uncorrelated false correlations from the dark counts, the part considered as "roll-off" whereby the concurrence drops further. In addition to this close agreement, the count-weighted average concurrence of the simulation yields C*<sup>ρ</sup>sim* ¼ 0*:*61 � 0*:*01, which agrees extremely well with the earlier mentioned value from the measurement, C*<sup>ρ</sup>* ¼ 0*:*62 � 0*:*03, within the error. Thus, the behavior of the two-photon quantum state can be explained by a model which assumes no dephasing, only considering the general features of the source, and the detection systems timing resolution. This indicates that the source at hand is not affected by dephasing during the the exciton decay time once excited at the donor/acceptor level. Hence, this excitation scheme is named "quasi-resonant," as it shows a dephasing-free two-photon quantum state, during the exciton's decay time, without being excited resonantly.

As mentioned earlier, the drop observed in the measured concurrence is the result of the low timing resolution of the detectors. Therefore, it is expected that once the detection system is improved, an enhancement in the measured concurrence will be observed. **Figure 8c** shows the result of a simulation when the features of the detection system and/or the excitation scheme have changed. The red curve

together with a regular avalanche photodiode (APD) detection system, with a timing resolution of *τ<sup>d</sup>* � 300 ps, with *τ<sup>d</sup>* being the FWHM of the response function *g t*ð Þ, and with a dark count rate of *DC* � 30 Hz. Upon resonant excitation, the yellow curve, the multiphoton emission of the source is expected to vanish,

*XX*ð Þ¼ 0 0*:*0, and the uncorrelated photons will not enter the analysis, hence, an expected increase in the concurrence value. However, the general shape of the graph does not change. In the case of conducting the experiment with a fast, lownoise detector, such as with superconducting nanowire single-photon detectors (SNSPDs), with a timing resolution of *τ<sup>d</sup>* � 30 ps and a dark count rate of

*DC* � 1 Hz, the blue curve, not only will the measured value of the concurrence be enhanced, but the overall shape of the graph will change. The drop in concurrence, observed in the case of APDs, will vanish, and the graph will only consist of the "flat" and "roll-off" parts. However the blue curve still suffers from a non-zero multiphoton emission. But, once the source is excited resonantly, and SNSPDs are used, the cyan curve, remarkably, one expects to measure near-unity concurrence. The count-weighted average concurrence in the latter case is C ¼ 0*:*996 � 0*:*008. The way in which the curve of concurrence vs. time is affected by the detectors' response function *g t*ð Þ can be analyzed in two equivalent ways. In the first approach, a low timing resolution will result in averaging of the relative phase between the two terms in Eq. (2) and turning the pure two-photon quantum state into a mixed state.

In this view, at each particular time *t*, the coincidence counts in the range

time intervals, one can write the resultant measured density matrix, *ρm*, as:

*ρm*ðÞ¼ *t*

in the measured concurrence.

**47**

½ � *t* � *τd=*2, *t* þ *τd=*2 will be included in the analysis. By dividing this range in shorter

ð*<sup>t</sup>*þ*τd=*<sup>2</sup> *t*�*τd=*2

where *ρ*ðÞ¼ *t* j i Ψð Þ*t* h j Ψð Þ*t* and *n t*ð Þ , *τ<sup>X</sup>* is the probability of the state being in the state *ρ*ð Þ*t* , based on an exponential decay. The rate of change in the density matrix is proportional to the *FSS*, since *FSS* results in precession of the state between j i Ψ and j i Φ according to Eq. (2). Therefore, in the case of a large *FSS*, and a low timing resolution, the measured density matrix will be a result of a mixture of different states in the time interval ½ � *t* � *τd=*2, *t* þ *τd=*2 . It was found that the time window for tomography analysis can be chosen to be smaller than *τd*. In the measurements presented here concerning time windows Δ*t* <100ps, the concurrence value does not show any change; however, increasing the length of time window above 100 ps reduces the calculated concurrence. This means that the effective time window is slightly less than the FWHM of *g t*ð Þ. It is straightforward to see that as the detectors' timing resolution is enhanced, *τ<sup>d</sup>* ! 0, the measured density matrix gets closer and closer to the density matrix of a pure state at each instant of time, hence, an increase

In the alternate approach, the uncertainty in timing of the arrival of the photons

can be interpreted as an uncertainty in measuring the energy of *XX* and *X*, according to Δ*τd*Δ*E* ¼ ℏ*=*2. In this picture, the timing resolution of the detectors competes with the which-path information introduced by the presence of *FSS* (**Figure 9**). Because of this, a detector with low timing resolution (*τ<sup>d</sup>* ≫ ℏ*=δ*) will

ð Þ2

*n t*ð Þ , *τ<sup>x</sup> ρ*ð Þ*t dt* (6)

*XX*ð Þ¼ 0 0*:*1,

shows the actual system at hand, quasi-resonant excitation, with *g*

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

*DOI: http://dx.doi.org/10.5772/intechopen.91814*

*g* ð Þ2

In stark contrast, under non-resonant excitation at the wurtzite InP bandgap, conducting two-photon quantum state tomography reveals the detrimental effect of the surrounding charge noise on the entangled state. By comparing **Figure 8a** and **b**, it becomes clear that shortly after the excitation laser moves to the InP bandgap, the detrimental effects of the excessive charge carriers become evident, ≈0*:*5ns, after the *XX* emission. Interestingly, these results indicate that during the exciton lifetime, interaction of the exciton state with the charge carriers is the main source of dephasing, not the presence of large nuclear spins, as was generally believed in the community; a finding which is in agreement with a previous work [65], wherein an indium-rich QD was shown not to be affected by the nuclear spins during an exciton lifetime of ≈2*:*5ns.

### **Figure 8.**

*Effect of the excitation scheme and detection system. (a) Comparison of the theoretical model and results from quasi-resonant excitation indicate suppression of dephasing during the X decay time. (b) Off-resonant excitation at the wurtzite InP bandgap, leads to the mobility of charge carriers and dephasing of the two-photon quantum state shortly after the XX's emission. (c) A combination of two different excitation schemes and detection systems were used to produce the four curves: quasi-resonant excitation and avalanche photodiodes (APDs) (red), resonant TPE and APDs (yellow), quasi-resonant excitation and superconducting nanowire*

*single-photon detectors (SNSPDs) (blue), and resonant TPE and SNSPDs (cyan). Imperfect g*ð Þ <sup>2</sup> *XX*ð Þ 0 *values in the case of quasi-resonant excitation (red and blue curves), as well as low timing resolution and relatively high noise level of APDs (red and yellow curves), result in the deterioration of the measured concurrence. Impressively, with the application of resonant TPE, and SNSPDs with a timing resolution of τ<sup>d</sup>* � 30*ps, and noise level of* � 1*Hz, the detection of perfect entanglement is expected.*

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot DOI: http://dx.doi.org/10.5772/intechopen.91814*

As mentioned earlier, the drop observed in the measured concurrence is the result of the low timing resolution of the detectors. Therefore, it is expected that once the detection system is improved, an enhancement in the measured concurrence will be observed. **Figure 8c** shows the result of a simulation when the features of the detection system and/or the excitation scheme have changed. The red curve shows the actual system at hand, quasi-resonant excitation, with *g* ð Þ2 *XX*ð Þ¼ 0 0*:*1, together with a regular avalanche photodiode (APD) detection system, with a timing resolution of *τ<sup>d</sup>* � 300 ps, with *τ<sup>d</sup>* being the FWHM of the response function *g t*ð Þ, and with a dark count rate of *DC* � 30 Hz. Upon resonant excitation, the yellow curve, the multiphoton emission of the source is expected to vanish, *g* ð Þ2 *XX*ð Þ¼ 0 0*:*0, and the uncorrelated photons will not enter the analysis, hence, an expected increase in the concurrence value. However, the general shape of the graph does not change. In the case of conducting the experiment with a fast, lownoise detector, such as with superconducting nanowire single-photon detectors (SNSPDs), with a timing resolution of *τ<sup>d</sup>* � 30 ps and a dark count rate of *DC* � 1 Hz, the blue curve, not only will the measured value of the concurrence be enhanced, but the overall shape of the graph will change. The drop in concurrence, observed in the case of APDs, will vanish, and the graph will only consist of the "flat" and "roll-off" parts. However the blue curve still suffers from a non-zero multiphoton emission. But, once the source is excited resonantly, and SNSPDs are used, the cyan curve, remarkably, one expects to measure near-unity concurrence. The count-weighted average concurrence in the latter case is C ¼ 0*:*996 � 0*:*008.

The way in which the curve of concurrence vs. time is affected by the detectors' response function *g t*ð Þ can be analyzed in two equivalent ways. In the first approach, a low timing resolution will result in averaging of the relative phase between the two terms in Eq. (2) and turning the pure two-photon quantum state into a mixed state. In this view, at each particular time *t*, the coincidence counts in the range ½ � *t* � *τd=*2, *t* þ *τd=*2 will be included in the analysis. By dividing this range in shorter time intervals, one can write the resultant measured density matrix, *ρm*, as:

$$\rho\_m(t) = \int\_{t-\tau\_d/2}^{t+\tau\_d/2} n(t, \tau\_x) \rho(t) \, dt \tag{6}$$

where *ρ*ðÞ¼ *t* j i Ψð Þ*t* h j Ψð Þ*t* and *n t*ð Þ , *τ<sup>X</sup>* is the probability of the state being in the state *ρ*ð Þ*t* , based on an exponential decay. The rate of change in the density matrix is proportional to the *FSS*, since *FSS* results in precession of the state between j i Ψ and j i Φ according to Eq. (2). Therefore, in the case of a large *FSS*, and a low timing resolution, the measured density matrix will be a result of a mixture of different states in the time interval ½ � *t* � *τd=*2, *t* þ *τd=*2 . It was found that the time window for tomography analysis can be chosen to be smaller than *τd*. In the measurements presented here concerning time windows Δ*t* <100ps, the concurrence value does not show any change; however, increasing the length of time window above 100 ps reduces the calculated concurrence. This means that the effective time window is slightly less than the FWHM of *g t*ð Þ. It is straightforward to see that as the detectors' timing resolution is enhanced, *τ<sup>d</sup>* ! 0, the measured density matrix gets closer and closer to the density matrix of a pure state at each instant of time, hence, an increase in the measured concurrence.

In the alternate approach, the uncertainty in timing of the arrival of the photons can be interpreted as an uncertainty in measuring the energy of *XX* and *X*, according to Δ*τd*Δ*E* ¼ ℏ*=*2. In this picture, the timing resolution of the detectors competes with the which-path information introduced by the presence of *FSS* (**Figure 9**). Because of this, a detector with low timing resolution (*τ<sup>d</sup>* ≫ ℏ*=δ*) will

(I) top, (II) flat, and (III) roll-off. Initially, as the detectors' response function *g t*ð Þ detects more photons, the plot shows an increase in the concurrence reaching a maximum. However, due to the low timing resolution of the detectors after a period of time, the phase averaging of Eq. (2) becomes significant, and the measured concurrence drops. Once the response function has been fully covered, the phase averaging becomes constant, hence reaching the "flat" part. Finally, as the probability of the exciton emission drops exponentially, the effect of dark counts dominates the actual photon counts, and the tomography system will detect uncorrelated false correlations from the dark counts, the part considered as "roll-off" whereby the concurrence drops further. In addition to this close agreement, the count-weighted average concurrence of the simulation yields C*<sup>ρ</sup>sim* ¼ 0*:*61 � 0*:*01, which agrees extremely well with the earlier mentioned value from the measurement, C*<sup>ρ</sup>* ¼ 0*:*62 � 0*:*03, within the error. Thus, the behavior of the two-photon quantum state can be explained by a model which assumes no dephasing, only considering the general features of the source, and the detection systems timing resolution. This indicates that the source at hand is not affected by dephasing during the the exciton decay time once excited at the donor/acceptor level. Hence, this excitation scheme is named "quasi-resonant," as it shows a dephasing-free two-photon quantum state,

*Recent Advances in Nanophotonics - Fundamentals and Applications*

during the exciton's decay time, without being excited resonantly.

exciton lifetime of ≈2*:*5ns.

**Figure 8.**

**46**

In stark contrast, under non-resonant excitation at the wurtzite InP bandgap, conducting two-photon quantum state tomography reveals the detrimental effect of the surrounding charge noise on the entangled state. By comparing **Figure 8a** and **b**, it becomes clear that shortly after the excitation laser moves to the InP bandgap, the detrimental effects of the excessive charge carriers become evident, ≈0*:*5ns, after the *XX* emission. Interestingly, these results indicate that during the exciton lifetime, interaction of the exciton state with the charge carriers is the main source of dephasing, not the presence of large nuclear spins, as was generally believed in the community; a finding which is in agreement with a previous work [65], wherein an indium-rich QD was shown not to be affected by the nuclear spins during an

*Effect of the excitation scheme and detection system. (a) Comparison of the theoretical model and results from quasi-resonant excitation indicate suppression of dephasing during the X decay time. (b) Off-resonant excitation at the wurtzite InP bandgap, leads to the mobility of charge carriers and dephasing of the two-photon quantum state shortly after the XX's emission. (c) A combination of two different excitation schemes and detection systems were used to produce the four curves: quasi-resonant excitation and avalanche photodiodes (APDs) (red), resonant TPE and APDs (yellow), quasi-resonant excitation and superconducting nanowire*

*the case of quasi-resonant excitation (red and blue curves), as well as low timing resolution and relatively high noise level of APDs (red and yellow curves), result in the deterioration of the measured concurrence. Impressively, with the application of resonant TPE, and SNSPDs with a timing resolution of τ<sup>d</sup>* � 30*ps, and*

*XX*ð Þ 0 *values in*

*single-photon detectors (SNSPDs) (blue), and resonant TPE and SNSPDs (cyan). Imperfect g*ð Þ <sup>2</sup>

*noise level of* � 1*Hz, the detection of perfect entanglement is expected.*

will reach unity when the detector is fast enough to fully erase the which-path

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

In an attempt to realize on-demand entanglement, we have performed performed resonant two-photon excitation on the same sample used by Fognini et al. [45]. The spectrum of the source under resonant TPE is given in **Figure 10a**. As it is evident from comparing this spectrum with the spectra under non-resonant excitation shown in **Figure 6**, the abundance of charge carriers surrounding the QD is significantly suppressed, leading to a lower intensity of the *X*� line, as compared to *X* and *XX*. Moreover, the PL transition rates of *XX* and *X* become closer to each other, a fact that shows an enhancement in pair-production efficiency. By integrating the area under the *X* and *XX* PL emission lines and calculating their ratio, we have achieved a pair-production efficiency of ϵ*<sup>p</sup>* ¼ 93*:*6%. Proper population of the *XX* state is affected by the center wavelength of the excitation laser, as well as its bandwidth, the length of which can be controlled via a regular 4f pulse shaper. The population of the *XX* state in resonant TPE shows a qualitatively similar Rabi oscillation as the regular resonant excitation (**Figure 10b**). The center wavelength and bandwidth of the excitation pulse is chosen so that the *π* pulse shows the highest possible count rate. Based on taking the setup efficiency and the count rate detected at the *π* pulse into consideration, the pair-extraction

Moreover, under resonant TPE, the multiphoton emission is significantly suppressed. **Figure 10c** and **d** show the results of the second-order correlation function performed on the QD once excited at the donor/acceptor levels

*<sup>X</sup>* ð Þ¼ 0 0*:*0024 � 0*:*0002, which demonstrates a two order of magnitude improvement in the case of *XX*, as compared to the values reported for

The impressive potential for nanowire QDs in detecting entangled photon pairs with near-unity entanglement fidelity is illuminated by the results of the resonant two-photon excitation. Notably, we are now at a point where we can make a comparison between SPDC sources and state-of-the-art QDs in different structures, i.e., self-assembled, micropillar cavities, nanowires, etc. As mentioned earlier, the

Poissonian nature of photon-pair emission in SPDC sources limits the performance of such sources to extremely low pair-extraction efficiencies. On the other hand, recent advances in QD growth in various photonic structures have resulted in achieving high entanglement fidelity and high pair-extraction efficiencies, simultaneously. Hüber et al. [67] have reported on measuring an entanglement fidelity of *F* ¼ 0*:*978 5ð Þ, from a self-assembled QD by strain-tuning the *FSS* down to zero. This significant result demonstrates an extensive level of improvement as compared to the results gained from the first generation of self-assembled QDs, where the entanglement fidelity was much lower [38, 68]. The results reported by Fognini et al. [45], in conjunction with the results achieved by resonant TPE, equip us with sufficient information to make such a comparison, the ultimate potential of nanowire QDs in regards to both entanglement fidelity and pair-extraction efficiency, with the values reported for other

ð Þ2

*XX*ð Þ¼ 0 0*:*0055 � 0*:*0005 and

information caused by *FSS*.

**2.4 Resonant two-photon excitation**

*DOI: http://dx.doi.org/10.5772/intechopen.91814*

efficiency is reported to be ϵ*<sup>e</sup>* ¼ 12*:*55%.

quasi-resonant excitation.

*g* ð Þ2

**49**

and under resonant TPE. For resonant TPE, *g*

**2.5 State-of-the-art entangled photon sources**

photonic structures mentioned earlier [26, 32, 33].

### **Figure 9.**

*Detectors' timing resolution and energy uncertainty. The detectors' timing resolution, τd, directly leads to an uncertainty in the energy of photons,* Δ*Eτ<sup>d</sup>* � ℏ*=*2*. For the case of a fast detector, τ<sup>d</sup>* ≪ ℏ*=δ, this uncertainty can smear out the energy difference between the two decay paths and hence retrieve the entanglement, whereas a slow detector, τ<sup>d</sup>* ≫ ℏ*=δ, will push the correlations more toward classical correlations.*

### **Figure 10.**

*Resonant two-photon excitation of a nanowire QD. (a) The spectrum of the QD under resonant TPE. The X and XX PL transition rates become more similar as compared to non-resonant excitation, indicating an enhanced pair-production efficiency; and the charged exciton is significantly suppressed, indicating a reduction of excessive charged carriers around the QD. (b) The power-dependent XX count rate exhibits a qualitatively similar Rabi oscillation as the regular direct resonant excitations, qualitatively. (c) and (d) show the comparison between results of g*ð Þ <sup>2</sup> ð Þ <sup>0</sup> *measurements in the case of quasi-resonant and resonant TPE schemes, for X and XX, respectively. Implementation of resonant TPE significantly reduces the emission time jitter of the two states, as well as multiphoton emission of the XX state.*

"notice" the energy difference between the two paths, thus degrading the entanglement fidelity. In contrast, a fast detector (*τ<sup>d</sup>* ≪ ℏ*=δ*) will render the two paths indistinguishable, since the uncertainty in energy will be more than the energy difference of the two paths. In this latter case, the entanglement will be retained and will reach unity when the detector is fast enough to fully erase the which-path information caused by *FSS*.

### **2.4 Resonant two-photon excitation**

In an attempt to realize on-demand entanglement, we have performed performed resonant two-photon excitation on the same sample used by Fognini et al. [45]. The spectrum of the source under resonant TPE is given in **Figure 10a**. As it is evident from comparing this spectrum with the spectra under non-resonant excitation shown in **Figure 6**, the abundance of charge carriers surrounding the QD is significantly suppressed, leading to a lower intensity of the *X*� line, as compared to *X* and *XX*. Moreover, the PL transition rates of *XX* and *X* become closer to each other, a fact that shows an enhancement in pair-production efficiency. By integrating the area under the *X* and *XX* PL emission lines and calculating their ratio, we have achieved a pair-production efficiency of ϵ*<sup>p</sup>* ¼ 93*:*6%. Proper population of the *XX* state is affected by the center wavelength of the excitation laser, as well as its bandwidth, the length of which can be controlled via a regular 4f pulse shaper. The population of the *XX* state in resonant TPE shows a qualitatively similar Rabi oscillation as the regular resonant excitation (**Figure 10b**). The center wavelength and bandwidth of the excitation pulse is chosen so that the *π* pulse shows the highest possible count rate. Based on taking the setup efficiency and the count rate detected at the *π* pulse into consideration, the pair-extraction efficiency is reported to be ϵ*<sup>e</sup>* ¼ 12*:*55%.

Moreover, under resonant TPE, the multiphoton emission is significantly suppressed. **Figure 10c** and **d** show the results of the second-order correlation function performed on the QD once excited at the donor/acceptor levels and under resonant TPE. For resonant TPE, *g* ð Þ2 *XX*ð Þ¼ 0 0*:*0055 � 0*:*0005 and *g* ð Þ2 *<sup>X</sup>* ð Þ¼ 0 0*:*0024 � 0*:*0002, which demonstrates a two order of magnitude improvement in the case of *XX*, as compared to the values reported for quasi-resonant excitation.

### **2.5 State-of-the-art entangled photon sources**

The impressive potential for nanowire QDs in detecting entangled photon pairs with near-unity entanglement fidelity is illuminated by the results of the resonant two-photon excitation. Notably, we are now at a point where we can make a comparison between SPDC sources and state-of-the-art QDs in different structures, i.e., self-assembled, micropillar cavities, nanowires, etc. As mentioned earlier, the Poissonian nature of photon-pair emission in SPDC sources limits the performance of such sources to extremely low pair-extraction efficiencies. On the other hand, recent advances in QD growth in various photonic structures have resulted in achieving high entanglement fidelity and high pair-extraction efficiencies, simultaneously. Hüber et al. [67] have reported on measuring an entanglement fidelity of *F* ¼ 0*:*978 5ð Þ, from a self-assembled QD by strain-tuning the *FSS* down to zero. This significant result demonstrates an extensive level of improvement as compared to the results gained from the first generation of self-assembled QDs, where the entanglement fidelity was much lower [38, 68]. The results reported by Fognini et al. [45], in conjunction with the results achieved by resonant TPE, equip us with sufficient information to make such a comparison, the ultimate potential of nanowire QDs in regards to both entanglement fidelity and pair-extraction efficiency, with the values reported for other photonic structures mentioned earlier [26, 32, 33].

"notice" the energy difference between the two paths, thus degrading the entanglement fidelity. In contrast, a fast detector (*τ<sup>d</sup>* ≪ ℏ*=δ*) will render the two paths indistinguishable, since the uncertainty in energy will be more than the energy difference of the two paths. In this latter case, the entanglement will be retained and

*Resonant two-photon excitation of a nanowire QD. (a) The spectrum of the QD under resonant TPE. The X and XX PL transition rates become more similar as compared to non-resonant excitation, indicating an enhanced pair-production efficiency; and the charged exciton is significantly suppressed, indicating a reduction of excessive charged carriers around the QD. (b) The power-dependent XX count rate exhibits a qualitatively similar Rabi oscillation as the regular direct resonant excitations, qualitatively. (c) and (d) show the comparison between results of g*ð Þ <sup>2</sup> ð Þ <sup>0</sup> *measurements in the case of quasi-resonant and resonant TPE schemes, for X and XX, respectively. Implementation of resonant TPE significantly reduces the emission time jitter of the two*

*Detectors' timing resolution and energy uncertainty. The detectors' timing resolution, τd, directly leads to an uncertainty in the energy of photons,* Δ*Eτ<sup>d</sup>* � ℏ*=*2*. For the case of a fast detector, τ<sup>d</sup>* ≪ ℏ*=δ, this uncertainty can smear out the energy difference between the two decay paths and hence retrieve the entanglement, whereas a slow*

*detector, τ<sup>d</sup>* ≫ ℏ*=δ, will push the correlations more toward classical correlations.*

*Recent Advances in Nanophotonics - Fundamentals and Applications*

**Figure 9.**

**Figure 10.**

**48**

*states, as well as multiphoton emission of the XX state.*

research and subsequent recent advances toward enhancement of the performance from such sources. Thus far, several photonic structures have been developed in order to improve the low pair-extraction efficiency of self-assembled QDs, among which bottom-up grown nanowire QDs exhibit considerable promise. Based on the detailed studies of these sources under different excitation schemes along with understanding the effects of detection systems and multiphoton emission on the measured value of entanglement fidelity, we predict nanowire QDs can undoubtedly outperform SPDC sources, once excited via resonant TPE and detected by fast,

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

*DOI: http://dx.doi.org/10.5772/intechopen.91814*

Admittedly, despite the fact that the results that indicate near-unity fidelity are achievable by nanowire QDs, the finite value of *FSS* will limit the performance of the source once a particular entangled state is required. In this case, a post selection on the collected photons is inevitable, since the two-photon quantum state precesses between the two entangled states Eq. (2). Strain fields [67], electric fields [40], and magnetic fields [68] are the most popular approaches used for addressing this issue; however, two recent proposals seem to be most compatible with a nanowire QD. The first is the method proposed by Fognini et al. [44], which uses a fast-rotating half-wave plate, realized by an electro-optical modulator, in order to change the energy of the photons after they have been emitted and correct for the energy splitting, thus making this approach a universal *FSS* eraser. The second, proposed by Zeeshan et al. [42] on the other hand, corrects the altered symmetry of the exciton's wave function, due to presence of *FSS*, via application of a quadrupole electric field, an approach which requires fabrication of electrical gates in the proximity of the nanowire. In addition to *FSS* tuning, nanowire QDs have been shown to be integrated into designs realizing performance of on-chip optical operations [71]. Such designs enhance light extraction and also allow for developing scalable quantum photonic circuits, paving the way for performing quantum computational processes on a photonic chip [72], using an on-demand entangled photon

Excitingly, this research shows that despite the challenges experienced thus far in generating on-demand and optimally entangled photon pairs, the results gained from resonant excitation of a nanowire QD have in fact revealed the enormous potential these sources have to outperform their predecessors. This research and the realization of optimally entangled photon pairs it offers have given quantum foundations, quantum communication, and quantum information a quantum

The authors gratefully acknowledge the Swiss National Science Foundation, Industry Canada, Natural Sciences and Engineering Research Council of Canada (NSERC) and Transformative Quantum Technologies (TQT), for their funding and

low-noise detectors.

source.

leap forward.

support.

**51**

**Acknowledgements**

### **Figure 11.**

*Performance of state-of-the-art entangled photon sources. Comparison between various quantum light sources in terms of entanglement fidelity and pair-extraction efficiency. Blue circles represent SPDC sources, values taken from [69] and [14]. The dashed line shows the ultimate theoretical limit of such sources, with multiphoton emission probability following a Poisson distribution. The red triangle shows results for a bare self-assembled QD, whereas the red diamonds show the results for QDs in different photonic structures. The red solid squares indicate the values reported for nanowire QDs so far. The analysis performed by Fognini et al. [45] and the results obtained by Ahmadi et al. [66] strongly suggest that the sources used for these two studies have the capacity to surpass the performance of SPDC sources once excited via resonant TPE and measured with a fast, low-noise detector. The graph is adapted and modified from [62].*

The result of such a comparison is shown in **Figure 11**. The blue circles show different values reported for entanglement fidelity vs. pair-extraction efficiency for SPDC sources. The values are taken from [69] and [14]. The dashed line shows the theoretical limit of such sources, following a Poisson distribution for the probability of multiphoton emission [70]. The two solid red squares indicate the result of two measurements performed on nanowire QDs by Jöns et al. [62] and Fognini et al. [45]. The latter work shows both an improvement in the measured entanglement fidelity and an improvement in pair-extraction efficiency. Based on the results shown by Fognini et al. [45] and the improvements gained by performing resonant TPE, we can predict measuring near-unity entanglement fidelity once two important modifications are implemented: the resonant TPE scheme is employed, and the detection system is improved to a fast and low-noise one. The final result that we predict by implementing these two changes is shown by the hollow red square. This is an extrapolation of results reported thus far on nanowire QDs based on the enhancement achieved in pair-extraction efficiency and entanglement fidelity, as well as the analysis presented in **Figure 8c**. Therefore, it is confidently predicted that nanowire QDs have the potential to surpass and outperform that of SPDC sources, revealing the significant potential of these sources for quantum communication purposes.

### **3. Conclusion and discussion**

In this chapter, we have given a historical overview of previous methods for attaining pairs of entangled photons from a QD, as well as included the latest

### *Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot DOI: http://dx.doi.org/10.5772/intechopen.91814*

research and subsequent recent advances toward enhancement of the performance from such sources. Thus far, several photonic structures have been developed in order to improve the low pair-extraction efficiency of self-assembled QDs, among which bottom-up grown nanowire QDs exhibit considerable promise. Based on the detailed studies of these sources under different excitation schemes along with understanding the effects of detection systems and multiphoton emission on the measured value of entanglement fidelity, we predict nanowire QDs can undoubtedly outperform SPDC sources, once excited via resonant TPE and detected by fast, low-noise detectors.

Admittedly, despite the fact that the results that indicate near-unity fidelity are achievable by nanowire QDs, the finite value of *FSS* will limit the performance of the source once a particular entangled state is required. In this case, a post selection on the collected photons is inevitable, since the two-photon quantum state precesses between the two entangled states Eq. (2). Strain fields [67], electric fields [40], and magnetic fields [68] are the most popular approaches used for addressing this issue; however, two recent proposals seem to be most compatible with a nanowire QD. The first is the method proposed by Fognini et al. [44], which uses a fast-rotating half-wave plate, realized by an electro-optical modulator, in order to change the energy of the photons after they have been emitted and correct for the energy splitting, thus making this approach a universal *FSS* eraser. The second, proposed by Zeeshan et al. [42] on the other hand, corrects the altered symmetry of the exciton's wave function, due to presence of *FSS*, via application of a quadrupole electric field, an approach which requires fabrication of electrical gates in the proximity of the nanowire. In addition to *FSS* tuning, nanowire QDs have been shown to be integrated into designs realizing performance of on-chip optical operations [71]. Such designs enhance light extraction and also allow for developing scalable quantum photonic circuits, paving the way for performing quantum computational processes on a photonic chip [72], using an on-demand entangled photon source.

Excitingly, this research shows that despite the challenges experienced thus far in generating on-demand and optimally entangled photon pairs, the results gained from resonant excitation of a nanowire QD have in fact revealed the enormous potential these sources have to outperform their predecessors. This research and the realization of optimally entangled photon pairs it offers have given quantum foundations, quantum communication, and quantum information a quantum leap forward.

### **Acknowledgements**

The authors gratefully acknowledge the Swiss National Science Foundation, Industry Canada, Natural Sciences and Engineering Research Council of Canada (NSERC) and Transformative Quantum Technologies (TQT), for their funding and support.

The result of such a comparison is shown in **Figure 11**. The blue circles show different values reported for entanglement fidelity vs. pair-extraction efficiency for SPDC sources. The values are taken from [69] and [14]. The dashed line shows the theoretical limit of such sources, following a Poisson distribution for the probability of multiphoton emission [70]. The two solid red squares indicate the result of two measurements performed on nanowire QDs by Jöns et al. [62] and Fognini et al. [45]. The latter work shows both an improvement in the measured entanglement fidelity and an improvement in pair-extraction efficiency. Based on the results shown by Fognini et al. [45] and the improvements gained by performing resonant TPE, we can predict measuring near-unity entanglement fidelity once two important modifications are implemented: the resonant TPE scheme is employed, and the detection system is improved to a fast and low-noise one. The final result that we predict by implementing these two changes is shown by the hollow red square. This is an extrapolation of results reported thus far on nanowire QDs based on the enhancement achieved in pair-extraction efficiency and entanglement fidelity, as well as the analysis presented in **Figure 8c**. Therefore, it is confidently predicted that nanowire QDs have the potential to surpass and outperform that of SPDC sources, revealing the

*low-noise detector. The graph is adapted and modified from [62].*

*Recent Advances in Nanophotonics - Fundamentals and Applications*

*Performance of state-of-the-art entangled photon sources. Comparison between various quantum light sources in terms of entanglement fidelity and pair-extraction efficiency. Blue circles represent SPDC sources, values taken from [69] and [14]. The dashed line shows the ultimate theoretical limit of such sources, with multiphoton emission probability following a Poisson distribution. The red triangle shows results for a bare self-assembled QD, whereas the red diamonds show the results for QDs in different photonic structures. The red solid squares indicate the values reported for nanowire QDs so far. The analysis performed by Fognini et al. [45] and the results obtained by Ahmadi et al. [66] strongly suggest that the sources used for these two studies have the capacity to surpass the performance of SPDC sources once excited via resonant TPE and measured with a fast,*

significant potential of these sources for quantum communication purposes.

In this chapter, we have given a historical overview of previous methods for attaining pairs of entangled photons from a QD, as well as included the latest

**3. Conclusion and discussion**

**50**

**Figure 11.**

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Entangling photons that never interacted. Physical Review Letters.

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### **Author details**

Arash Ahmadi1 \*, Andreas Fognini<sup>2</sup> and Michael E. Reimer<sup>3</sup>

1 Department of Physics and Astronomy, Institute for Quantum Computing, University of Waterloo, Waterloo, Canada

2 Single Quantum, Delft, Netherlands

3 Department of Electrical and Computing Engineering, Institute for Quantum Computing, Waterloo, Canada

\*Address all correspondence to: arash.ahmadi@uwaterloo.ca

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot DOI: http://dx.doi.org/10.5772/intechopen.91814*

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**Author details**

Arash Ahmadi1

**52**

University of Waterloo, Waterloo, Canada

2 Single Quantum, Delft, Netherlands

provided the original work is properly cited.

Computing, Waterloo, Canada

\*, Andreas Fognini<sup>2</sup> and Michael E. Reimer<sup>3</sup>

1 Department of Physics and Astronomy, Institute for Quantum Computing,

3 Department of Electrical and Computing Engineering, Institute for Quantum

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: arash.ahmadi@uwaterloo.ca

*Recent Advances in Nanophotonics - Fundamentals and Applications*

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*DOI: http://dx.doi.org/10.5772/intechopen.91814*

*Toward On-Demand Generation of Entangled Photon Pairs with a Quantum Dot*

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Review A. 2007;**76**:012307

**57**

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triggered entangled photon pairs. Nature. 2006;**439**(7073):179-182

Review B: Condensed Matter and Materials Physics. 2006;**73**(12):1-7

*Recent Advances in Nanophotonics - Fundamentals and Applications*

[60] James DFV, Kwiat PG, Munro WJ, White AG. Measurement of qubits. Physical Review A. 2001;**64**:052312

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nanowire quantum dot. Nature Communications. 2014;**5**:5298

[62] Jöns KD, Schweickert L,

Versteegh MAM, Dalacu D, Poole PJ, Gulinatti A, et al. Bright nanoscale source of deterministic entangled photon pairs violating Bell's inequality. Scientific Reports. 2017;**7**(1):1-11

[63] Clauser JF, Horne MA, Shimony A, Holt RA. Proposed experiment to test local hidden-variable theories. Physical Review Letters. 1969;**23**(15):880

[65] Sénés M, Liu BL, Marie X, Amand T, Gérard JM. Spin Dynamics of Neutral and Charged Excitons in InAs/GaAs Quantum Dots. Dordrecht: Springer Netherlands; 2003. pp. 79-88

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[68] Stevenson RM, Young RJ, Atkinson P, Cooper K, Ritchie DA, Shields AJ. A semiconductor source of

Submission in process

33902

[64] Bennett CH, DiVincenzo DP, Smolin JA, Wootters WK. Mixed-state entanglement and quantum error correction. Physical Review A: Atomic, Molecular, and Optical Physics. 1996;

**54**(5):3824

[52] Jayakumar H, Predojević A, Huber T, Kauten T, Solomon GS, Weihs G. Deterministic photon pairs and coherent optical control of a single quantum dot. Physical Review Letters.

[53] Müller M, Bounouar S, Jöns KD, Glässl M, Michler P. On-demand generation of indistinguishable polarization-entangled photon pairs. Nature Photonics. 2014;**8**(3):224-228

[54] Schweickert L, Jöns KD, Zeuner KD, Da Silva SFC, Huang H, Lettner T, et al. On-demand generation of backgroundfree single photons from a solid-state source. Applied Physics Letters. 2018;

[55] Claudon J, Bleuse J, Malik NS, Bazin M, Jaffrennou P, Gregersen N, et al. A highly efficient single-photon source based on a quantum dot in a photonic nanowire. Nature Photonics.

[56] Gregersen N, Nielsen TR, Claudon J, Gérard J-M, Mørk J. Controlling the emission profile of a nanowire with a conical taper. Optics Letters. 2008;

[57] Reimer ME, Bulgarini G, Akopian N, Hocevar M, Bavinck MB, Verheijen MA, et al. Bright single-photon sources in bottom-up tailored nanowires. Nature

Communications. 2012;**3**:737

[58] Bulgarini G, Reimer ME, Bavinck MB, Jons KD, Dalacu D, Poole PJ, et al. Nanowire waveguides launching single photons in a gaussian mode for ideal fiber coupling. Nano Letters. 2014;**14**(7):4102-4106

[59] Brown RH, Twiss RQ. A test of a new type of stellar interferometer on sirius. Nature. 1956;**178**(4541):

2013;**110**:135505

**112**(9):1-4

2010;**4**(3):174

**33**(15):1693-1695

1046-1048

**56**

[69] Scarani V, de Riedmatten H, Marcikic I, Zbinden H, Gisin N. Fourphoton correction in two-photon bell experiments. European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics. 2005;**32**(1):129-138

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[72] Ma X, Fung C-HF, Lo H-K. Quantum key distribution with entangled photon sources. Physical Review A. 2007;**76**:012307

**Chapter 4**

*Anand K. Bhatia*

**Abstract**

**1. Introduction**

**59**

Interactions of Positrons and

Systems, Excitation, Resonances,

There are a number of approaches to study interactions of positrons and electrons with hydrogenic targets. Among the most commonly used are the method of polarized orbital, the close-coupling approximation, and the *R*-matrix formulation. The last two approaches take into account the short-range and long-range correlations. The method of polarized orbital takes into account only long-range correlations but is not variationally correct. This method has recently been modified to take into account both types of correlations and is variationally correct. It has been applied to calculate phase

obtained using this method have lower bounds to the exact phase shifts and agree with those obtained using other approaches. This approach has also been applied to calculate resonance parameters in two-electron systems obtaining results which agree with those obtained using the Feshbach projection-operator formalism. Furthermore this method has been employed to calculate photodetachment and photoionization of twoelectron systems, obtaining very accurate cross sections which agree with the experimental results. Photodetachment cross sections are particularly useful in the study of the opacity of the sun. Recently, excitation of the atomic hydrogen by electron impact

The discovery of an electron by J.J. Thomson in 1897 led to the development of physics beyond the classical physics. Proton was discovered by Rutherford in 1909. Niels Bohr proposed a model of the structure of hydrogen atoms in 1913. Neutron was discovered by Chadwick in 1932. Other important discoveries were of X-rays and radioactivity in 1896. In 1926, Erwin Schrödinger formulated an equation to determine the wave function of quantum mechanical system. According to Max Born, the wave function can be interpreted as a probability of finding a particle at a specific point in space and time. Many processes could be studied due to such developments in physics. For example, an incoming wave behaves like a particle in

, and Li2+. The phase shifts

Electrons with Hydrogenic

and Photoabsorption in

Two-Electron Systems

shifts of scattering from hydrogenic systems like H, He+

and also by positron impact has been studied by this method.

**Keywords:** scattering, resonances, photoabsorption, excitation

### **Chapter 4**

Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances, and Photoabsorption in Two-Electron Systems

*Anand K. Bhatia*

### **Abstract**

There are a number of approaches to study interactions of positrons and electrons with hydrogenic targets. Among the most commonly used are the method of polarized orbital, the close-coupling approximation, and the *R*-matrix formulation. The last two approaches take into account the short-range and long-range correlations. The method of polarized orbital takes into account only long-range correlations but is not variationally correct. This method has recently been modified to take into account both types of correlations and is variationally correct. It has been applied to calculate phase shifts of scattering from hydrogenic systems like H, He+ , and Li2+. The phase shifts obtained using this method have lower bounds to the exact phase shifts and agree with those obtained using other approaches. This approach has also been applied to calculate resonance parameters in two-electron systems obtaining results which agree with those obtained using the Feshbach projection-operator formalism. Furthermore this method has been employed to calculate photodetachment and photoionization of twoelectron systems, obtaining very accurate cross sections which agree with the experimental results. Photodetachment cross sections are particularly useful in the study of the opacity of the sun. Recently, excitation of the atomic hydrogen by electron impact and also by positron impact has been studied by this method.

**Keywords:** scattering, resonances, photoabsorption, excitation

### **1. Introduction**

The discovery of an electron by J.J. Thomson in 1897 led to the development of physics beyond the classical physics. Proton was discovered by Rutherford in 1909. Niels Bohr proposed a model of the structure of hydrogen atoms in 1913. Neutron was discovered by Chadwick in 1932. Other important discoveries were of X-rays and radioactivity in 1896. In 1926, Erwin Schrödinger formulated an equation to determine the wave function of quantum mechanical system. According to Max Born, the wave function can be interpreted as a probability of finding a particle at a specific point in space and time. Many processes could be studied due to such developments in physics. For example, an incoming wave behaves like a particle in

processes like Compton scattering and photoabsorption. Particularly, Geiger and Bothe, using coincident counters, showed that the time between the arrival of the incident wave and the motion of the electron is of the order of 10�<sup>7</sup> second. If the incident wave acted as a wave, the time would have been much longer. Also the experiment of Compton and Simon showed that energy is conserved at every point of the scattering process.

is the target wave function. The scattering wave function of the incident electron

where *H* is the Hamiltonian and *E* is the energy of e-target system. In Rydberg units,

<sup>2</sup> � <sup>2</sup>*<sup>Z</sup>*

where *k*<sup>2</sup> is the kinetic energy of the incident electron and *Z* is the nuclear charge

*<sup>E</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> � *<sup>Z</sup>*<sup>2</sup>

<sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

*<sup>r</sup>* � <sup>2</sup>*<sup>Z</sup> r*2 � 2 *r*12

*ul*ð Þ *r*<sup>1</sup>

�*Zr*<sup>2</sup> *<sup>r</sup>*2*ul*ð Þ *<sup>r</sup>*<sup>2</sup> *dr*<sup>2</sup> � <sup>2</sup>

*l* ∞ð

*π* <sup>2</sup> <sup>þ</sup> *<sup>η</sup>*

� � � ð Þþ <sup>1</sup> \$ <sup>2</sup> <sup>Φ</sup>*<sup>L</sup> <sup>r</sup>*

*u*1*s*!*<sup>p</sup>*ð Þ *r*<sup>2</sup> *r*2

*r*

¼ 0 (6)

<sup>2</sup>*ϕ*0ð Þ *r*<sup>2</sup> *ul*ð Þ *r*<sup>2</sup> *dr*<sup>2</sup> þ *r*

Phase shift *η* (radians) of a partial wave *l* is calculated from the asymptotic value

lim*<sup>r</sup>*!∞*u r*ð Þ¼ sin *kr* � *<sup>l</sup>*

! 1, *r* ! 2

where the polarized target function is given in [7] and is defined as

! 2 � � � *<sup>χ</sup>*ð Þ *<sup>r</sup>*<sup>1</sup> *r*2 1

4 3 þ

The cutoff function, instead of *ε*ð Þ *r*1,*r*<sup>2</sup> = 1 for *r*<sup>1</sup> >*r*2, is used in the electronhydrogen scattering calculation [7], and a cutoff function *χ*ð Þ *r*<sup>1</sup> given by Shertzer

> 4ð Þ *Zr*<sup>1</sup> 3 <sup>3</sup> <sup>þ</sup> <sup>2</sup>ð Þ *Zr*<sup>1</sup>

*χ*ð Þ¼ *r*<sup>1</sup> 1 � *e*

" #

�*βr*<sup>1</sup> � �*<sup>n</sup>*

<sup>2</sup> ¼ 0 (3)

, (5)

ð Þ <sup>2</sup>*<sup>l</sup>* <sup>þ</sup> <sup>1</sup> *yl ul*, *<sup>ϕ</sup>*<sup>0</sup> ð Þ

*ϕ*0ð Þ *r*<sup>2</sup> *ul*ð Þ *r*<sup>2</sup>

� �*:* (8)

! 1, *r* ! 2 � �, (9)

*<sup>π</sup><sup>Z</sup>* <sup>p</sup> *:* (10)

(11)

cosð Þ *<sup>θ</sup>*<sup>12</sup> ffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*Zr*<sup>1</sup> <sup>þ</sup> <sup>1</sup>

, (12)

, (4)

3 5

*rl*þ<sup>1</sup> *dr*<sup>2</sup> (7)

ð

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

*ϕ*<sup>0</sup> *r* ! 2 � �j*<sup>H</sup>* � *<sup>E</sup>*j<sup>Ψ</sup> h i*d r*!

*<sup>H</sup>* ¼ �∇<sup>2</sup>

of the target, assumed fixed. The scattering equation is

1 *rl*þ<sup>1</sup>

In hybrid theory [6], we replace Eq. (1) by

ð*r*

0 *r l*

! 1 � �Φ*pol <sup>r</sup>*

�2*Zr*<sup>1</sup> ð Þ *Zr*<sup>1</sup>

is used in this calculation. It can also be of the form

�2*Zr*<sup>1</sup> <sup>1</sup> <sup>þ</sup>

þ 2*e*

2 4

*yl ϕ*<sup>0</sup> ð Þ¼ , *ul*

" #

�*Zr*<sup>1</sup> *<sup>Z</sup>*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> � �*δ<sup>l</sup>*0*r*<sup>1</sup>

<sup>1</sup> � <sup>∇</sup><sup>2</sup>

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

1 *r*1 � �

> ð ∞

0 *e*

is obtained from

*d*2 *dr*<sup>2</sup> 1

where

� <sup>4</sup>*Z*<sup>3</sup> *e*

of the scattering function.

Ψ*<sup>L</sup> r* ! 1, *r* ! 2 � � <sup>¼</sup> *uL <sup>r</sup>*

and Temkin [8] of the form

**61**

*χST*ð Þ¼ *r*<sup>1</sup> 1 � *e*

Φ*pol r* ! 1, *r* ! 2 � � <sup>¼</sup> *<sup>ϕ</sup>*<sup>0</sup> *<sup>r</sup>*

� *l l*ð Þ <sup>þ</sup> <sup>1</sup> *r*2 1

We discuss here scattering of electrons by hydrogenic systems since the wave function of the target is known exactly, and therefore we can test various theories or approximations. When the target consists of more than one electron, a reasonably accurate wave function can only be written using various configurations of the target (called configuration interaction approximation). Among the various approximations for scattering are the exchange approximation [1], the Kohn variational method [2], and the method of polarized orbitals [3] which takes into account the polarization of the target due to the incident electron. The incident electron creates an electric field which results in a change of energy of the target given by <sup>Δ</sup>*<sup>E</sup>* ¼ � <sup>1</sup> <sup>2</sup> *<sup>α</sup>E*<sup>2</sup> , where *α* is the polarizabilty of the target and is equal to 4.5 *a*<sup>3</sup> <sup>0</sup> in the case of a hydrogen target. The polarization is possible only when the incident electron is outside the target, according to the method of polarized orbitals [3]. However, this method includes only the long-range �1/*r* <sup>4</sup> potential and not the short-range correlations, and the method is not variationally correct. It is possible to use the Feshbach formalism [4] to take into account the short-range correlations via an optical potential. This method gives rigorous lower bonds on the phase shifts, i.e., they are lower than the exact phase shifts. However, it is difficult to include the long-range correlations at the same time. The close-coupling approach takes into account both the long-range and short-range correlations [5] and is variationally correct. However, a large number of target states must be included to obtain converged results and the correct polarizability. A Feshbach resonance below the *n* = 2 threshold of hydrogen atom was first discovered in a close-coupling calculation. A method which has been applied extensively to atomic, molecular, and nuclear physics is the *R*-matrix method introduced by Wigner and Eisenbud in 1947. In this method, configuration space is divided in such a way that all correlations are described within a radius *r=a*, and outside this radius simple continuum functions can be used while matching the inside and outside functions at the boundary *r=a*. This method was introduced in atomic physics by Burke. A method called the hybrid theory [6] has been introduced in which the short-range and long-range correlations are taken into account at the same time and the polarization takes place whether the incident electron is outside or inside the target. This method is variationally correct. The equations for the scattering function are very detailed and are given in [6].

### **2. Scattering function calculations**

In the exchange approximation [1], we write the wave function of incident electron and the target as

$$
\Psi\left(\vec{r}\_1, \vec{r}\_2\right) = \mathfrak{u}\left(\vec{r}\_1\right)\mathfrak{\phi}\_0\left(\vec{r}\_2\right) \pm (\mathfrak{1} \leftrightarrow \mathfrak{2}).\tag{1}
$$

In the above equation, the plus sign refers to the singlet state, and the minus sign refers to the triplet state, *u r*! 1 � � is the scattering function, and

$$
\phi\_0 \left( \stackrel{\rightarrow}{r}\_2 \right) = 2 \sqrt{Z^3} e^{-Zr\_2} Y\_{00}(\hat{r}\_2) \tag{2}
$$

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

is the target wave function. The scattering wave function of the incident electron is obtained from

$$\int \left[\phi\_0 \left(\overrightarrow{r\_2}\right) |H - E|\Psi\right] d\overrightarrow{r\_2} = \mathbf{0} \tag{3}$$

where *H* is the Hamiltonian and *E* is the energy of e-target system. In Rydberg units,

$$H = -\nabla\_1^2 - \nabla\_2^2 - \frac{2Z}{r} - \frac{2Z}{r\_2} - \frac{2}{r\_{12}},\tag{4}$$

$$E = k^2 - Z^2,\tag{5}$$

where *k*<sup>2</sup> is the kinetic energy of the incident electron and *Z* is the nuclear charge of the target, assumed fixed. The scattering equation is

$$\begin{aligned} & \left[ \frac{d^2}{dr\_1^2} - \frac{l(l+1)}{r\_1^2} + 2e^{-2Zr\_1} \left( 1 + \frac{1}{r\_1} \right) + k^2 \right] u\_l(r\_1) \\ & \pm 4Z^3 e^{-Zr\_1} \left[ \left( Z^2 + k^2 \right) \delta\_{l0} r\_1 \right] e^{-Zr\_2} r\_2 u\_l(r\_2) dr\_2 - \frac{2}{(2l+1)} y\_l(u\_l, \phi\_0) \right] \\ & = 0 \end{aligned} \tag{6}$$

where

processes like Compton scattering and photoabsorption. Particularly, Geiger and Bothe, using coincident counters, showed that the time between the arrival of the incident wave and the motion of the electron is of the order of 10�<sup>7</sup> second. If the incident wave acted as a wave, the time would have been much longer. Also the experiment of Compton and Simon showed that energy is conserved at every point

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We discuss here scattering of electrons by hydrogenic systems since the wave function of the target is known exactly, and therefore we can test various theories or approximations. When the target consists of more than one electron, a reasonably accurate wave function can only be written using various configurations of the target (called configuration interaction approximation). Among the various approximations for scattering are the exchange approximation [1], the Kohn variational method [2], and the method of polarized orbitals [3] which takes into account the polarization of the target due to the incident electron. The incident electron creates an electric field which results in a change of energy of the target given by

, where *α* is the polarizabilty of the target and is equal to 4.5 *a*<sup>3</sup>

case of a hydrogen target. The polarization is possible only when the incident electron is outside the target, according to the method of polarized orbitals [3].

short-range correlations, and the method is not variationally correct. It is possible to use the Feshbach formalism [4] to take into account the short-range correlations via an optical potential. This method gives rigorous lower bonds on the phase shifts, i.e., they are lower than the exact phase shifts. However, it is difficult to include the long-range correlations at the same time. The close-coupling approach takes into account both the long-range and short-range correlations [5] and is variationally correct. However, a large number of target states must be included to obtain converged results and the correct polarizability. A Feshbach resonance below the *n* = 2 threshold of hydrogen atom was first discovered in a close-coupling calculation. A method which has been applied extensively to atomic, molecular, and nuclear physics is the *R*-matrix method introduced by Wigner and Eisenbud in 1947. In this method, configuration space is divided in such a way that all correlations are described within a radius *r=a*, and outside this radius simple continuum functions can be used while matching the inside and outside functions at the boundary *r=a*. This method was introduced in atomic physics by Burke. A method called the hybrid theory [6] has been introduced in which the short-range and long-range correlations are taken into account at the same time and the polarization takes place whether the incident electron is outside or inside the target. This method is variationally correct. The equations for the scattering function are very detailed and

In the exchange approximation [1], we write the wave function of incident

*ϕ*<sup>0</sup> *r* ! 2 � �

is the scattering function, and

In the above equation, the plus sign refers to the singlet state, and the minus sign

ffiffiffiffiffi *<sup>Z</sup>*<sup>3</sup> <sup>p</sup> *e*

¼ *u r*! 1 � �

¼ 2

However, this method includes only the long-range �1/*r*

<sup>0</sup> in the

<sup>4</sup> potential and not the

� ð Þ 1 \$ 2 *:* (1)

�*Zr*2*Y*00ð Þ ^*r*<sup>2</sup> (2)

of the scattering process.

<sup>Δ</sup>*<sup>E</sup>* ¼ � <sup>1</sup>

<sup>2</sup> *<sup>α</sup>E*<sup>2</sup>

are given in [6].

**60**

electron and the target as

refers to the triplet state, *u r*!

**2. Scattering function calculations**

Ψ *r* ! 1, *r* ! 2 � �

> 1 � �

> > *ϕ*<sup>0</sup> *r* ! 2 � �

$$\mathcal{Y}\_l(\phi\_0, \boldsymbol{u}\_l) = \frac{1}{r^{l+1}} \int\_0^r r\_2^l \phi\_0(r\_2) \boldsymbol{u}\_l(r\_2) dr\_2 + r^l \int\_r^\infty \frac{\phi\_0(r\_2) \boldsymbol{u}\_l(r\_2)}{r^{l+1}} dr\_2 \tag{7}$$

Phase shift *η* (radians) of a partial wave *l* is calculated from the asymptotic value of the scattering function.

$$\lim\_{r \to \infty} u(r) = \sin\left(kr - l\frac{\pi}{2} + \eta\right). \tag{8}$$

In hybrid theory [6], we replace Eq. (1) by

$$
\Psi\_L(\overrightarrow{r}\_1, \overrightarrow{r}\_2) = u\_L(\overrightarrow{r}\_1) \Phi^{pol} \left(\overrightarrow{r}\_1, \overrightarrow{r}\_2\right) \pm (\mathbf{1} \leftrightarrow \mathbf{2}) + \Phi\_L \left(\overrightarrow{r}\_1, \overrightarrow{r}\_2\right), \tag{9}
$$

where the polarized target function is given in [7] and is defined as

$$\Phi^{pol}\left(\overrightarrow{r}\_1, \overrightarrow{r}\_2\right) = \phi\_0\left(\overrightarrow{r}\_2\right) - \frac{\chi(r\_1)}{r\_1^2} \frac{u\_{1\gets p}(r\_2)}{r\_2} \frac{\cos\left(\theta\_{12}\right)}{\sqrt{\pi Z}}.\tag{10}$$

The cutoff function, instead of *ε*ð Þ *r*1,*r*<sup>2</sup> = 1 for *r*<sup>1</sup> >*r*2, is used in the electronhydrogen scattering calculation [7], and a cutoff function *χ*ð Þ *r*<sup>1</sup> given by Shertzer and Temkin [8] of the form

$$\chi\_{ST}(r\_1) = 1 - e^{-2Zr\_1} \left[ \frac{(Zr\_1)^4}{3} + \frac{4(Zr\_1)^3}{3} + 2(Zr\_1)^2 + 2Zr\_1 + 1 \right] \tag{11}$$

is used in this calculation. It can also be of the form

$$\chi(r\_1) = \left(1 - e^{-\beta r\_1}\right)^n,\tag{12}$$

where the exponent *n*≥3 and *β*, another parameter which is a function of *k,* can be used to optimize the phase shifts*.* Both cutoffs (11) and (12) are used in calculations on scattering. Φ*L*ð Þ *r*1,*r*<sup>2</sup> is a short-range correlation function [6] which can be written using the Euler angle decomposition [9] for all partial waves. This formulism allows the separation of the angular parts and the radial parts consisting of radial coordinates *r*1, *r*2, and *r*12. This facilitates writing the equation for radial functions [9]. The function *u*1*s*!*p*ð Þ *r*<sup>1</sup> in Eq. (10) is given by

$$u\_{1s \to p}(r\_2) = e^{-Zr\_2} \left(\frac{Z}{2}r\_2^3 + r\_2^2\right) \tag{13}$$

A comparison of results for singlet and triplet phase shifts obtained using different methods is given in **Tables 1** and **2**. Results from most methods agree. A comparison of the singlet and triplet phase shifts obtained by the *R*-matrix and the hybrid theory is given in **Figure 1**. The curves for the two methods cannot be distinguished, showing that accurate result can be obtained using the hybrid theory [6].

The scattering length *a* is defined by.

$$\lim\_{k \to 0} k \cot \eta = -\mathbf{1}/\mathfrak{a}.\tag{14}$$

The scattering length is calculated at a distance *R*, and there is a correction for the long-range polarization [12].

$$a = a(R) - a\left(\frac{\mathbf{1}}{R} - \frac{a}{R^2} + \frac{a^2}{3R^3}\right),\tag{15}$$

1.76815, which agree with those calculated by Schwartz [2], using the Kohn variational method. In this method, scattering lengths have upper bounds, while phase shifts do not have any bounds in this method (Kohn variational method). Similar calculations

*The upper represents singlet phase shifts obtained in the hybrid and* R*-matrix theories. The two curves cannot be*

*k* **EA<sup>a</sup> POb Kohn<sup>c</sup> Close-coupling<sup>d</sup> R-matrixe Feshbach method<sup>f</sup> Hybrid theory<sup>g</sup>** 0.1 2.908 2.949 2.9388 2.9355 2.939 2.93853 2.93856 0.2 2.679 2.732 2.7171 2.715 2.717 2.71741 2.71751 0.3 2.461 2.519 2.4996 2.461 2.500 2.49975 2.49987 0.4 2.257 2.320 2.2938 2.2575 2.294 2.29408 2.29465 0.5 2.070 2.133 2.1046 2.0956 2.105 2.10454 2.10544 0.6 1.901 1.9329 1.933 1.93272 1.93322 0.7 1.749 1.815 1.7797 1.780 1.77950 1.77998 0.8 1.614 1.682 1.643 1.616 1.64379 1.64425

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

Resonance parameters in two-electron systems have been calculated using various approaches. Among them are the stabilization method, the complex-rotation method, the close-coupling method, and the Feshbach projection-operator formalism. In the hybrid theory, phase shifts have been calculated in the resonance region

for the phase shifts for e-He+ and e-Li2+ have been carried out in Ref. [13].

*distinguished. Similarly, the lower curves represent triplet phase shifts obtained in the two theories.*

S *phase shifts obtained in different methods.*

[13] and are fitted to the Breit-Wigner form

*a Ref. [1]. b Ref. [7]. c Ref. [2]. d Ref. [5]. e Ref. [10]. f Ref. [11]. g Ref. [6].*

**Table 2.** *Comparison of <sup>3</sup>*

**Figure 1.**

**63**

where *a* is the true scattering length and *α* is the polarizability of the hydrogen atom. At *R* = 117.3088, <sup>1</sup> *S* scattering length *a*(*R*) is 5.96554, obtained using *χβ*. The corrected <sup>1</sup> *S* scattering length is 5.96611. At *R* = 349.0831, the <sup>3</sup> *S* scattering length is 1.781542, obtained using *χβ*, and it is 1.76815, after correction for the long-range polarization given by [12]. The <sup>1</sup> *S* and <sup>3</sup> *S* scattering lengths are therefore 5.96611 and



*Ref. [6].*

**Table 1.** *Comparison of <sup>1</sup>* S *phase shifts obtained in different methods.*

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*


*a Ref. [1]. b Ref. [7].*

where the exponent *n*≥3 and *β*, another parameter which is a function of *k,* can be used to optimize the phase shifts*.* Both cutoffs (11) and (12) are used in calculations on scattering. Φ*L*ð Þ *r*1,*r*<sup>2</sup> is a short-range correlation function [6] which can be

formulism allows the separation of the angular parts and the radial parts consisting of radial coordinates *r*1, *r*2, and *r*12. This facilitates writing the equation for radial

> �*Zr*<sup>2</sup> *Z* 2 *r* 3 <sup>2</sup> þ *r* 2 2

A comparison of results for singlet and triplet phase shifts obtained using different methods is given in **Tables 1** and **2**. Results from most methods agree. A comparison of the singlet and triplet phase shifts obtained by the *R*-matrix and the hybrid theory is given in **Figure 1**. The curves for the two methods cannot be distinguished, showing that accurate result can be obtained using the hybrid theory [6].

The scattering length is calculated at a distance *R*, and there is a correction for

*<sup>R</sup>* � *<sup>α</sup>*

where *a* is the true scattering length and *α* is the polarizability of the hydrogen

*k* **EAa POb Kohn<sup>c</sup> Close-coupling<sup>d</sup> R-matrix<sup>e</sup> Feshbach method<sup>f</sup> Hybrid theory<sup>g</sup>** 0.1 2.396 2.583 2.553 2.491 2.550 2.55358 2.55372 0.2 1.870 2.144 2.673 1.9742 2.062 2.06678 2.06699 0.3 1.508 1.750 1.6964 1.519 1.691 1.09816 1.69853 0.4 1.239 1.469 1.4146 1.257 1.410 1.41540 1.41561 0.5 1.031 1.251 1.202 1.082 1.196 1.20094 1.20112 0.6 0.869 1.041 1.035 1.04083 1.04110 0.7 0.744 0.947 0.930 0.925 0.93111 0.93094 0.8 0.651 0.854 0.886 0.608 0.88718 0.88768

*<sup>R</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> 3*R*<sup>3</sup>

*S* scattering length *a*(*R*) is 5.96554, obtained using *χβ*. The

*k* cot *η* ¼ �1*=a:* (14)

*S* scattering lengths are therefore 5.96611 and

, (15)

*S* scattering length is

(13)

written using the Euler angle decomposition [9] for all partial waves. This

functions [9]. The function *u*1*s*!*p*ð Þ *r*<sup>1</sup> in Eq. (10) is given by

*Recent Advances in Nanophotonics - Fundamentals and Applications*

The scattering length *a* is defined by.

the long-range polarization [12].

polarization given by [12]. The <sup>1</sup>

atom. At *R* = 117.3088, <sup>1</sup>

corrected <sup>1</sup>

*a Ref. [1]. b Ref. [7]. c Ref. [2]. d Ref. [5]. e Ref. [10]. f Ref. [11]. g Ref. [6].*

**Table 1.** *Comparison of <sup>1</sup>*

**62**

*u*1*s*!*p*ð Þ¼ *r*<sup>2</sup> *e*

lim *k*!0

*<sup>a</sup>* <sup>¼</sup> *a R*ð Þ� *<sup>α</sup>* <sup>1</sup>

*S* scattering length is 5.96611. At *R* = 349.0831, the <sup>3</sup>

*S* and <sup>3</sup>

S *phase shifts obtained in different methods.*

1.781542, obtained using *χβ*, and it is 1.76815, after correction for the long-range

*c Ref. [2]. d*

*Ref. [5]. e Ref. [10].*

*f Ref. [11].*

*g Ref. [6].*

### **Table 2.**

*Comparison of <sup>3</sup>* S *phase shifts obtained in different methods.*

### **Figure 1.**

*The upper represents singlet phase shifts obtained in the hybrid and* R*-matrix theories. The two curves cannot be distinguished. Similarly, the lower curves represent triplet phase shifts obtained in the two theories.*

1.76815, which agree with those calculated by Schwartz [2], using the Kohn variational method. In this method, scattering lengths have upper bounds, while phase shifts do not have any bounds in this method (Kohn variational method). Similar calculations for the phase shifts for e-He+ and e-Li2+ have been carried out in Ref. [13].

Resonance parameters in two-electron systems have been calculated using various approaches. Among them are the stabilization method, the complex-rotation method, the close-coupling method, and the Feshbach projection-operator formalism. In the hybrid theory, phase shifts have been calculated in the resonance region [13] and are fitted to the Breit-Wigner form

$$
\eta\_{calc.} (E) = \eta\_0 + AE + \tan^{-1} \frac{0.5 \Gamma}{(E\_R - E)}.\tag{16}
$$

*hν* þ *e* þ *H* ! *e* þ *H*, (18)

*<sup>H</sup>*� <sup>þ</sup> *<sup>H</sup>* ! *<sup>H</sup>*<sup>2</sup> <sup>þ</sup> *<sup>e</sup>:* (19)

<sup>2</sup> *<sup>r</sup>*<sup>12</sup> � ð Þ <sup>1</sup> \$ <sup>2</sup> � �*:* (21)

<sup>2</sup>∣. Ohmura and Ohmura [20] have calculated the

*.*

*<sup>o</sup>* for the

*:* (20)

In the first process, after absorption of the radiation by the bound electron, it becomes a free electron in the final state, while in the free-free transition, the electron is in the continuum state in the initial state as well as in the final state. It is

*e* þ *H* ! *H*� þ *hν*

� � <sup>¼</sup> <sup>4</sup>*παω*∣<sup>&</sup>lt; <sup>Ψ</sup> *<sup>f</sup>* <sup>∣</sup>*z*<sup>1</sup> <sup>þ</sup> *<sup>z</sup>*2j j <sup>Φ</sup>*<sup>i</sup>* <sup>&</sup>gt; <sup>2</sup>

outgoing electron, *<sup>ω</sup>* <sup>=</sup> *<sup>I</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> is the energy of the incident photon, and *<sup>I</sup>* is the ionization potential of the system absorbing the photon. Photoabsorption cross sections for H�, He, and Li<sup>+</sup> are given in [16] and in **Table 4**. The correlated wave functions for H�, He, and Li<sup>+</sup> are of Hylleraas form having terms 364, 220, and 165,

In Eq. (21), *C*'s are the linear parameters, while *γ* and *δ* are nonlinear

photodetachment cross section of H�, using the effective range theory and the loosely bound structure of hydrogen ion. These cross sections are shown in

*k* **H**� **He Li<sup>+</sup>**

0.2 38.5443 7.1544 2.5677 0.3 35.2318 6.8716 2.5231 0.4 24.4774 6.4951 2.4373 0.5 16.0858 6.0461 2.3870 0.6 10.7410 5.5925 2.2988 0.7 7.4862 5.0120 2.0005 0.8 5.6512 4.4740 2.0925 0.9 3.9296 1.9792

**Figure 2**. It should be noticed that with the outgoing energy of the electron going to zero, the cross section goes to zero when the remaining system is neutral and is

A comparison of the cross sections of the ground state of He with those obtained in the *R*-matrix [21] is shown in **Figure 3**. The agreement is very good except at *k* = 0.6 where the *R*-matrix result is slightly lower. It seems that the cross section has not been calculated accurately at this *k*. The precision measurements of the cross sections of photoionization of He by Samson et al. [22] are also shown in **Figure 3**,

In the above equation, *α* is the fine-structure constant, *k* is the momentum of the

�*γr*1�*δr*<sup>2</sup> *r l* 1*r m*

possible to have the following reactions which help molecular formation:

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

The photoabsorption cross section in length form and in units of *a*<sup>2</sup>

transition from the initial states *I* to the final state *f* is given by

*σ a*<sup>2</sup> 0

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

respectively. A Hyllerraas wave function is given by

! <sup>1</sup> � *r* !

finite when the remaining system is an ion [16].

parameters, and *r*<sup>12</sup> ¼ ∣*r*

**Table 4.**

**65**

<sup>Ψ</sup>ð Þ¼ *<sup>r</sup>*1,*r*2,*r*<sup>12</sup> <sup>X</sup>*Clmn <sup>e</sup>*

0.1 15.3024 7.3300

*Photoabsorption cross (Mb) for the ground state of H*�*, He, and Li+*

In the above equation, *E=k*<sup>2</sup> is the incident energy, *ηcalc:* are the calculated phase shifts, and *η*0, *A*, Γ, and *ER* are the fitting parameters. *ER* and Γ represent the resonance position and resonance width, respectively. We find that in the hybrid theory, the He singlet resonance is at *ER* = 57.8481 eV with respect to the ground state of He and Γ = 0.1233 eV. They agree well with *ER* = 57.8435 eV and Γ = 0.125 eV obtained using the Feshbach projection-operator formalism [14]. A similar calculation [13] has been carried out for Li2+ resonance parameters. The resonance parameters agree with those obtained in [14].

P-wave phase shifts have been calculated for scattering of electrons from He<sup>+</sup> and Li2+ in Ref. [15] and in Ref. [16], respectively. Singlet P and triplet P phase shifts are shown in **Table 3** for e + He<sup>+</sup> and for e + Li2+ scattering. Phase shifts for e + He<sup>+</sup> agree well with those obtained by Oza [17] using the close-coupling approximation. Phase shifts for e + Li2+ agree with those obtained by Gien [18] using the Harris-Nesbet method.


**Table 3.**

*Phase shifts (radians) for e + He+ and e + Li2+ scattering.*

### **3. Photoabsorption**

Photodetachment and photoionization are required to calculate radiativeattachment cross sections. The recombination rates are required to calculate the ionization balance in astrophysical plasmas. Cross sections for bound-free transitions of H� are required to account for the absorption in the solar atmosphere [19]. The opacity in the sun is due to photodetachment and free-free absorption of the radiation:

$$h\nu + H^- \to H + \varepsilon,\tag{17}$$

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

$$h\nu + \mathfrak{e} + H \to \mathfrak{e} + H,\tag{18}$$

In the first process, after absorption of the radiation by the bound electron, it becomes a free electron in the final state, while in the free-free transition, the electron is in the continuum state in the initial state as well as in the final state. It is possible to have the following reactions which help molecular formation:

$$\begin{aligned} e + H &\to H^- + h\nu \\ H^- + H &\to H\_2 + e. \end{aligned} \tag{19}$$

The photoabsorption cross section in length form and in units of *a*<sup>2</sup> *<sup>o</sup>* for the transition from the initial states *I* to the final state *f* is given by

$$
\sigma(a\_0^2) = 4\pi a a \nu | < \Psi\_f | \mathbf{z}\_1 + \mathbf{z}\_2 | \Phi\_i > |^2. \tag{20}
$$

In the above equation, *α* is the fine-structure constant, *k* is the momentum of the outgoing electron, *<sup>ω</sup>* <sup>=</sup> *<sup>I</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> is the energy of the incident photon, and *<sup>I</sup>* is the ionization potential of the system absorbing the photon. Photoabsorption cross sections for H�, He, and Li<sup>+</sup> are given in [16] and in **Table 4**. The correlated wave functions for H�, He, and Li<sup>+</sup> are of Hylleraas form having terms 364, 220, and 165, respectively. A Hyllerraas wave function is given by

$$\Psi(r\_1, r\_2, r\_{12}) = \sum C\_{lmn} \left[ e^{-\gamma r\_1 - \delta r\_2} r\_1^l r\_2^m r\_{12} \pm (\mathbf{1} \leftrightarrow \mathbf{2}) \right]. \tag{21}$$

In Eq. (21), *C*'s are the linear parameters, while *γ* and *δ* are nonlinear parameters, and *r*<sup>12</sup> ¼ ∣*r* ! <sup>1</sup> � *r* ! <sup>2</sup>∣. Ohmura and Ohmura [20] have calculated the photodetachment cross section of H�, using the effective range theory and the loosely bound structure of hydrogen ion. These cross sections are shown in **Figure 2**. It should be noticed that with the outgoing energy of the electron going to zero, the cross section goes to zero when the remaining system is neutral and is finite when the remaining system is an ion [16].

A comparison of the cross sections of the ground state of He with those obtained in the *R*-matrix [21] is shown in **Figure 3**. The agreement is very good except at *k* = 0.6 where the *R*-matrix result is slightly lower. It seems that the cross section has not been calculated accurately at this *k*. The precision measurements of the cross sections of photoionization of He by Samson et al. [22] are also shown in **Figure 3**,


*.*

### **Table 4.** *Photoabsorption cross (Mb) for the ground state of H*�*, He, and Li+*

*<sup>η</sup>calc:*ð Þ¼ *<sup>E</sup> <sup>η</sup>*<sup>0</sup> <sup>þ</sup> *AE* <sup>þ</sup> tan �<sup>1</sup> <sup>0</sup>*:*5<sup>Γ</sup>

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shifts, and *η*0, *A*, Γ, and *ER* are the fitting parameters. *ER* and Γ represent the resonance position and resonance width, respectively. We find that in the hybrid theory, the He singlet resonance is at *ER* = 57.8481 eV with respect to the ground state of He and Γ = 0.1233 eV. They agree well with *ER* = 57.8435 eV and Γ = 0.125 eV obtained using the Feshbach projection-operator formalism [14]. A similar calculation [13] has been carried out for Li2+ resonance parameters. The resonance param-

eters agree with those obtained in [14].

using the Harris-Nesbet method.

**P <sup>3</sup>**

*k* **<sup>1</sup>**

**3. Photoabsorption**

*Phase shifts (radians) for e + He+ and e + Li2+ scattering.*

radiation:

**64**

**Table 3.**

In the above equation, *E=k*<sup>2</sup> is the incident energy, *ηcalc:* are the calculated phase

P-wave phase shifts have been calculated for scattering of electrons from He<sup>+</sup> and Li2+ in Ref. [15] and in Ref. [16], respectively. Singlet P and triplet P phase shifts are shown in **Table 3** for e + He<sup>+</sup> and for e + Li2+ scattering. Phase shifts for e + He<sup>+</sup> agree well with those obtained by Oza [17] using the close-coupling approximation. Phase shifts for e + Li2+ agree with those obtained by Gien [18]

0.1 �0.038308 0.21516 �0.049083 0.16323 0.2 �0.038956 0.21683 �0.048990 0.16334 0.3 �0.039873 0.21945 �0.048934 0.16341 0.4 �0.040902 0.22283 �0.48823 0.16351 0.5 �0.041469 0.22662 �0.048565 0.16360 0.6 �0.041641 0.23088 �0.048306 0.16379 0.7 �0.041438 0.23417 �0.047972 0.16382 0.8 �0.039927 0.23753 �0.047547 0.16374 1.0 �0.037132 0.24205 �0.045966 0.16409 1.1 �0.035430 0.24323 �0.045029 0.16399 1.3 �0.026419 0.24370 �0.042670 0.16345 1.4 �0.020773 �0.041251 0.16299 1.6 �0.037973 0.16158

**P <sup>1</sup>**

**e + He<sup>+</sup> [15] e + Li2+ [16]**

Photodetachment and photoionization are required to calculate radiativeattachment cross sections. The recombination rates are required to calculate the ionization balance in astrophysical plasmas. Cross sections for bound-free transitions of H� are required to account for the absorption in the solar atmosphere [19]. The opacity in the sun is due to photodetachment and free-free absorption of the

*hν* þ *H*� ! *H* þ *e*, (17)

ð Þ *ER* � *<sup>E</sup> :* (16)

**P <sup>3</sup>**

**P**

**Figure 2.**

*Photodetachment of a hydrogen ion. The lowest curve is obtained when only the long-range correlations are included; the middle curve is obtained when the short-range and long-range correlations are also included. The top curve is obtained using Ohmura and Ohmura formulation.*

sections of the ground state, metastable states singlet *S*, and triplet *S* is shown in **Figure 4**. The cross section for the singlet (1s2s) state is highest at *k* = 0.1, and then

*Photoionization cross section for the ground state (middle curve from the right), for the (1s2s) singlet* S *state*

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

Excitation of the 1*S* state of atomic hydrogen to the 2*S* state has been calculated

using the hybrid theory in the distorted-wave approximation [23]. The total

*Cross section (Mb) for exciting the 1*S *state to 2*S *state of atomic hydrogen by electron impact.*

the cross sections decrease rapidly.

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

excitation cross section is written in the form

*(top curve), and for the (1s2s) triplet* S *state (lower curve) of the He atom.*

**4. Excitation**

**Figure 5.**

**67**

**Figure 4.**

**Figure 3.**

*The upper curve represents photoionization cross sections (Mb) for the ground state of He in the hybrid theory, while the lower represents cross sections obtained in* R*-matrix calculations. The curve starting at the top left represents experimental results of Samson et al. [22].*

showing that the agreement with the cross sections obtained using the hybrid theory is very good.

Similarly, cross sections have been calculated in [16] for the (1s2s) <sup>1</sup> *S* and (1s2s) 3 *S* metastable states of He and Li<sup>+</sup> . The number of Hylleraas terms used are 455 and 364 for He singlet and triplet states, respectively. For Li<sup>+</sup> , 120 and 220 terms are used for the singlet and triplet states, respectively. These cross sections are comparable to those obtained for the ground state. A comparison of photoionization cross

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

### **Figure 4.**

*Photoionization cross section for the ground state (middle curve from the right), for the (1s2s) singlet* S *state (top curve), and for the (1s2s) triplet* S *state (lower curve) of the He atom.*

sections of the ground state, metastable states singlet *S*, and triplet *S* is shown in **Figure 4**. The cross section for the singlet (1s2s) state is highest at *k* = 0.1, and then the cross sections decrease rapidly.

### **4. Excitation**

Excitation of the 1*S* state of atomic hydrogen to the 2*S* state has been calculated using the hybrid theory in the distorted-wave approximation [23]. The total excitation cross section is written in the form

**Figure 5.** *Cross section (Mb) for exciting the 1*S *state to 2*S *state of atomic hydrogen by electron impact.*

showing that the agreement with the cross sections obtained using the hybrid

*The upper curve represents photoionization cross sections (Mb) for the ground state of He in the hybrid theory, while the lower represents cross sections obtained in* R*-matrix calculations. The curve starting at the top left*

*Photodetachment of a hydrogen ion. The lowest curve is obtained when only the long-range correlations are included; the middle curve is obtained when the short-range and long-range correlations are also included. The*

*top curve is obtained using Ohmura and Ohmura formulation.*

*Recent Advances in Nanophotonics - Fundamentals and Applications*

used for the singlet and triplet states, respectively. These cross sections are comparable to those obtained for the ground state. A comparison of photoionization cross

*S* and (1s2s)

, 120 and 220 terms are

. The number of Hylleraas terms used are 455 and

Similarly, cross sections have been calculated in [16] for the (1s2s) <sup>1</sup>

364 for He singlet and triplet states, respectively. For Li<sup>+</sup>

theory is very good.

*S* metastable states of He and Li<sup>+</sup>

*represents experimental results of Samson et al. [22].*

3

**66**

**Figure 3.**

**Figure 2.**


### **Table 5.**

*Photoionization cross sections (Mb) for the metastable states of He and Li<sup>+</sup> with the outgoing electron with momentum* k.

$$
\sigma = \frac{k\_f}{k\_i} \int |T\_{fi}|^2 d\Omega. \tag{22}
$$

In the above equation, *ki* and *kf* are the incident and final momenta, and *Tfi* is the matrix element for the excitation of the initial state *i* to the final state *f* and is given by

$$T\_{fi} = -(\mathbf{1}/4\pi) < \Psi\_f |V| \Psi\_i > . \tag{23}$$

$$\mathbf{V} = -\frac{2\mathbf{Z}}{r\_1} + \frac{2}{r\_{12}}.\tag{24}$$

*Z* is the nuclear charge, and *r*<sup>1</sup> and *r*<sup>2</sup> indicate the position of the incident and target electron, respectively. The excitation cross sections at *k*(Ry) = 0.8–2 are shown in **Figure 5**. There is a maximum at *k* = 0.907 and another at *k* = 1.50 (**Table 5**).

### **5. Recombination**

Recombination rate coefficients for a process like that indicated in Eq. (17) have been calculated in Ref. [16] for the ground states as well as for the metastable states. The attachment cross section *σ<sup>a</sup>* is given by

$$
\sigma\_a = \left(\frac{h\nu}{c p\_e}\right)^2 \frac{\mathbf{g}(f)}{\mathbf{g}(i)} \sigma. \tag{25}
$$

calculate photoionization cross sections from *P*-state of an atom, both *S*-wave and

*, and e-Li2+.*

*<sup>T</sup> <sup>α</sup>R*ð Þ� *<sup>T</sup>* **1015, H**� *<sup>α</sup>*ð Þ *<sup>R</sup>* ð Þ� *<sup>T</sup>* **1013, He** *<sup>α</sup>R*ð Þ� *<sup>T</sup>* **1013, Li**<sup>þ</sup> 1000 0.99 2.50 0.12 2000 1.28 2.39 1.04 5000 2.40 1.87 2.62 7000 2.82 1.66 2.92 10,000 3.20 1.45 3.03 12,000 3.37 1.35 3.02 15,000 3.56 1.23 2.95 17,000 3.65 1.17 2.89 20,000 3.75 1.10 2.79 22,000 3.79 1.05 2.73 25,000 3.83 0.99 2.63 30,000 3.83 0.92 2.49 35,000 3.77 0.87 2.36 40,000 3.63 0.82 2.25

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

, and Li2+.

Scattering cross sections have also been calculated in the presence of laser fields [25]. A strong suppression in the laser-assisted cross sections is noted when compared to cross sections in the field-free situation. Further, scattering cross sections have also been calculated in the presence of Debye potential [26], in addition to the

*D* and <sup>3</sup>

*D*

*D*-wave continuum functions are needed. We give a few phase shifts for <sup>1</sup>

*/s) for (1s2*

*D***, e-H <sup>1</sup>**

*) 1*

0.1 1.3193 (�3) 1.3217 (�3) 5.9268 (�3) 8.5133 (�3) 3.0363 (�3) 8.2703 (�3) 0.2 5.0217 (�3) 5.0835 (�3) 6.1299 (�3) 9.0331 (�3) 3.0585 (�3) 8.4642 (�3) 0.3 1.0531 (�2) 1.0898 (�2) 6.4446 (�3) 9.8834 (�3) 3.0508 (�3) 8.7011 (�3) 0.4 1.7250 (�2) 1.8401 (�2) 6.8511 (�3) 1.1044 (�2) 3.0776 (�3) 9.0700 (�2) 0.5 2.4675 (�2) 2.7204 (�2) 7.3028 (�3) 1.2473 (–2) 3.0782 (�3) 9.5041 (�2) 0.6 3.2495 (�2) 3.6934 (�2) 7.7904 (�3) 1.4152 (�2) 3.0608 (�3) 1.0009 (�2) 0.7 4.0544 (�2) 4.7286 (�2) 8.3087 (�3) 1.6066 (�2) 3.0831 (�3) 1.0622 (�2) 0.8 4.8620 (�2) 5.7990 (�2) 8.8420 (�3) 1.8172 (�2) 3.1396 (�3) 1.1380 (�2) 0.9 5.6532 (�2) 6.8791 (�2) 9.3860 (�3) 2.0439 (�2) 3.1537 (�3) 1.2151 (�2)

S *states of H*�*, He, and Li+*

*.*

*D***, e-He+ 3***D***, e-He+ 1***D***, e-Li2+ 3***D***, e-Li2+**

in **Table 7** for scattering from H, He+

*D-wave phase shifts (radians) for e-H, e-He<sup>+</sup>*

**6. Laser fields**

**Table 7.**

**Table 6.**

*k* **<sup>1</sup>**

*Recombination rate coefficients (cm3*

*D***, e-H <sup>3</sup>**

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

laser field.

**69**

The above relation between the photodetachment and photoionization follows from the principle of detailed balance, where *g*(*i*) and *g*(*f*) are the weight factors for the initial and final states, and *p*<sup>e</sup> = *k* is the electron momentum. The attachment cross sections are in general smaller than the photoabsorption cross sections. Recombination rate coefficients, averaged over the Maxwellian velocity distribution, are given in **Table 6** for H�, He, and Li<sup>+</sup> for a temperature range from 1000 to 40,000 K.

Elastic *P*-wave scattering from e- Be3+, C5+, and O7+ and photoionization in twoelectron systems have been carried out. Phase shifts and photoionization cross sections are given in [24] for the ground as well as metastable states. In order to

*<sup>T</sup> <sup>α</sup>R*ð Þ� *<sup>T</sup>* **1015, H**� *<sup>α</sup>*ð Þ *<sup>R</sup>* ð Þ� *<sup>T</sup>* **1013, He** *<sup>α</sup>R*ð Þ� *<sup>T</sup>* **1013, Li**<sup>þ</sup> 1000 0.99 2.50 0.12 2000 1.28 2.39 1.04 5000 2.40 1.87 2.62 7000 2.82 1.66 2.92 10,000 3.20 1.45 3.03 12,000 3.37 1.35 3.02 15,000 3.56 1.23 2.95 17,000 3.65 1.17 2.89 20,000 3.75 1.10 2.79 22,000 3.79 1.05 2.73 25,000 3.83 0.99 2.63 30,000 3.83 0.92 2.49

35,000 3.77 0.87 2.36 40,000 3.63 0.82 2.25

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

### **Table 6.**

*<sup>σ</sup>* <sup>¼</sup> *<sup>k</sup> <sup>f</sup> ki* ð *Tfi* � � � � 2

*k* **He Li<sup>+</sup>**

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*S* **(1s2s)3**

0.1 8.7724 5.2629 2.4335 2.9889 0.2 7.5894 5.0795 2.3742 2.8570 0.3 6.0523 4.2004 2.2287 2.6434 0.4 4.5403 3.4403 2.3733 0.5 3.2766 2.7189 2.0865 0.6 2.2123 2.1531 1.7962 0.7 1.6047 1.4564 1.5182 0.8 1.1230 1.3539 1.2627

*S* **(1s2s)<sup>1</sup>**

**(1s2s)1**

*<sup>V</sup>* ¼ � <sup>2</sup>*<sup>Z</sup> r*1 þ 2 *r*12

**5. Recombination**

**Table 5.**

*momentum* k.

40,000 K.

**68**

The attachment cross section *σ<sup>a</sup>* is given by

*Z* is the nuclear charge, and *r*<sup>1</sup> and *r*<sup>2</sup> indicate the position of the incident and target electron, respectively. The excitation cross sections at *k*(Ry) = 0.8–2 are shown in **Figure 5**. There is a maximum at *k* = 0.907 and another at *k* = 1.50 (**Table 5**).

Recombination rate coefficients for a process like that indicated in Eq. (17) have been calculated in Ref. [16] for the ground states as well as for the metastable states.

The above relation between the photodetachment and photoionization follows from the principle of detailed balance, where *g*(*i*) and *g*(*f*) are the weight factors for the initial and final states, and *p*<sup>e</sup> = *k* is the electron momentum. The attachment cross sections are in general smaller than the photoabsorption cross sections. Recombination rate coefficients, averaged over the Maxwellian velocity distribution, are given in **Table 6** for H�, He, and Li<sup>+</sup> for a temperature range from 1000 to

Elastic *P*-wave scattering from e- Be3+, C5+, and O7+ and photoionization in two-

electron systems have been carried out. Phase shifts and photoionization cross sections are given in [24] for the ground as well as metastable states. In order to

*<sup>σ</sup><sup>a</sup>* <sup>¼</sup> *<sup>h</sup><sup>ν</sup> cpe* � �<sup>2</sup> *g f* ð Þ

In the above equation, *ki* and *kf* are the incident and final momenta, and *Tfi* is the matrix element for the excitation of the initial state *i* to the final state *f* and is given by

*Photoionization cross sections (Mb) for the metastable states of He and Li<sup>+</sup> with the outgoing electron with*

*d*Ω*:* (22)

*S* **(1s2s)3**

*S*

*:* (24)

*g i*ð Þ *<sup>σ</sup>:* (25)

*Tfi* ¼ �ð Þ 1*=*4*π* < Ψ *<sup>f</sup>* ∣*V*∣Ψ*<sup>i</sup>* >*:* (23)

*Recombination rate coefficients (cm3 /s) for (1s2 ) 1* S *states of H*�*, He, and Li+ .*


**Table 7.**

*D-wave phase shifts (radians) for e-H, e-He<sup>+</sup> , and e-Li2+.*

calculate photoionization cross sections from *P*-state of an atom, both *S*-wave and *D*-wave continuum functions are needed. We give a few phase shifts for <sup>1</sup> *D* and <sup>3</sup> *D* in **Table 7** for scattering from H, He+ , and Li2+.

### **6. Laser fields**

Scattering cross sections have also been calculated in the presence of laser fields [25]. A strong suppression in the laser-assisted cross sections is noted when compared to cross sections in the field-free situation. Further, scattering cross sections have also been calculated in the presence of Debye potential [26], in addition to the laser field.

### **7. Positron-hydrogen scattering**

Dirac in 1928, combining the ideas of relativity and quantum mechanics, formulated the well-known relativistic wave equation and predicted an antiparticle of the electron of spin ħ/2. At that time only protons and electrons were known. He thought that the antiparticle must be proton. Hermann Weyl showed from symmetry considerations that the antiparticle must have the same mass as an electron. There are many other examples where symmetry played an important role, e.g., Newton's third law of motion (for every action there is an equal and opposite reaction) and Faraday's laws of electricity and magnetism (electric currents generate magnetic fields, and magnetic fields generate electric currents). Symmetry laws have some profound implications as shown by Emmy Noether in 1918 that every symmetry in the action is related to a conservation law [27].

Positrons, produced by cosmic rays in a cloud chamber, were detected by Anderson [28] in 1932. Positrons can form positronium atoms which annihilate, giving 511 KeV line with a width of 1.6 keV. This line has been observed from the center of the galaxy. Positrons have become very useful to scan the human brain (PET scans). They have been used to probe the Fermi surfaces, and the annihilation of positronium atoms in metals has been used to detect defects in metals.

Calculations of positron-hydrogen scattering should be simpler than the electron-hydrogen scattering because of the absence of the exchange between electrons and positrons. However, the complications arise due to the possibility of positronium atom formation. In electron-hydrogen system, the two electrons are on either side of the proton because of the repulsion between two electrons. However, because of the attraction between a positron and an electron, both the positron and the bound electron tend to be on the same side of the proton. This configuration shows that the correlations are more important in the case of a positron incident on a hydrogen atom. In 1971, we [29] carried out calculations using the projectionoperator formulism of Feshbach [4] and using generalized Hylleraas-type functions:

$$
\Psi(r\_1, r\_2, r\_{12}) = e^{-\gamma r\_1 - \delta r\_2 - ar\_{12}} \sum\_{lmn} \mathbf{C}\_{lmn} r\_1^l r\_2^m r\_{12}^n. \tag{26}
$$

bounds. In the hybrid theory, as indicated earlier, both short-range and long-range correlations can be taken into account at the same time. It should be noted that in the case of positrons, the sign before the second term in Eq. (10) is plus instead of minus, as in the case of electrons. The phase shifts obtained in two approaches are given in **Table 8**. We used 84 terms in the Hylleraas wave function previously, and improved results are obtained now with shorter expansions [30] as indicated in the table. A comparison of the results obtained with different approaches is shown in **Figure 6**. Using a fewer number of terms, higher phase shifts have been obtained in the hybrid theory [30]. P-wave shifts have been calculated in the hybrid theory [31] and are compared with those using the Feshbach projection-operator formalism

*The upper curve represents phase shifts obtained using the hybrid theory, and the lower curve represents phase*

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

The incident positron can annihilate the atomic electron with the emission of two gamma rays. The cross section for this process has been given by Ferrell [34]:

> <sup>2</sup> Ψ *r* ! 1, *r* ! 2 � � � �

*Zeff* for partial waves *l* = 0 and 1, obtained in [31, 32], is given in **Table 9** along

1

� � � 2 *δ r* ! 1, *r* ! 2

*<sup>k</sup>*<sup>2</sup> ln 1 <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> � � � <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*5*k*<sup>2</sup>

" #

<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

�

with the contribution from *l* > 1. For higher partial waves, *Zeff* is given by

6 *k*2 *k*�<sup>1</sup> (27)

� �*:* (28)

*:* (29)

*σ<sup>a</sup> πa*<sup>2</sup> 0 � � <sup>¼</sup> *Zeff <sup>α</sup>*<sup>3</sup>

[32] and with those obtained by Armstrong [33].

where *α* is the fine-structure constant and

*Zeff* ¼

*Zeff*ð Þ¼ *l* >1

ð *d r*! 1*d r*!

*k*2 <sup>1</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> <sup>þ</sup>

**8.** *Zeff*

**71**

**Figure 6.**

*shifts obtained using the Feshbach formalism.*

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

Nonlinear parameters are *γ*, *δ*, and *α* and *C*'s are the linear coefficients. The results obtained agree with those obtained by Schwartz [2]. However, the longrange correlations could not be taken into account in [29] at the same time and had to be added separately, with the result that the final phase shifts ceased to have any


**Table 8.**

*Comparison of* S*-wave and* P*-wave phase shifts obtained in the hybrid theory with results obtained earlier.*

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

### **Figure 6.**

**7. Positron-hydrogen scattering**

Dirac in 1928, combining the ideas of relativity and quantum mechanics, formulated the well-known relativistic wave equation and predicted an antiparticle of the electron of spin ħ/2. At that time only protons and electrons were known. He thought that the antiparticle must be proton. Hermann Weyl showed from symmetry considerations that the antiparticle must have the same mass as an electron. There are many other examples where symmetry played an important role, e.g., Newton's third law of motion (for every action there is an equal and opposite reaction) and Faraday's laws of electricity and magnetism (electric currents generate magnetic fields, and magnetic fields generate electric currents). Symmetry laws have some profound implications as shown by Emmy Noether in 1918 that every

Positrons, produced by cosmic rays in a cloud chamber, were detected by Anderson [28] in 1932. Positrons can form positronium atoms which annihilate, giving 511 KeV line with a width of 1.6 keV. This line has been observed from the center of the galaxy. Positrons have become very useful to scan the human brain (PET scans). They have been used to probe the Fermi surfaces, and the annihilation

�*γr*1�*δr*2�*αr*<sup>12</sup>

Nonlinear parameters are *γ*, *δ*, and *α* and *C*'s are the linear coefficients. The results obtained agree with those obtained by Schwartz [2]. However, the longrange correlations could not be taken into account in [29] at the same time and had to be added separately, with the result that the final phase shifts ceased to have any

> **Schwarz [2]**

*S***-wave** *P***-wave** 0.1 0.14918 0.1483 0.151 0.008871 0.00876 0.008 0.2 0.18803 0.1877 0.188 0.032778 0.03251 0.032 0.3 0.16831 0.1677 0.168 0.06964 0.6556 0.064 0.4 0.12083 0.1201 0.120 0.10047 0.10005 0.099 0.5 0.06278 0.0624 0.062 0.13064 0.13027 0.130 0.6 0.00903 0.0039 0.007 0.15458 0.15410 0.153 0.7 0–0.04253 �0.0512 �0.54 0.17806 0.17742 0.175

*Comparison of* S*-wave and* P*-wave phase shifts obtained in the hybrid theory with results obtained earlier.*

X *lmn*

**Hybrid theory [31]**

*Clmnr l* 1*r m* 2 *r n*

<sup>12</sup>*:* (26)

**Armstrong [33]**

**Bhatia et al. [32]**

of positronium atoms in metals has been used to detect defects in metals. Calculations of positron-hydrogen scattering should be simpler than the electron-hydrogen scattering because of the absence of the exchange between electrons and positrons. However, the complications arise due to the possibility of positronium atom formation. In electron-hydrogen system, the two electrons are on either side of the proton because of the repulsion between two electrons. However, because of the attraction between a positron and an electron, both the positron and the bound electron tend to be on the same side of the proton. This configuration shows that the correlations are more important in the case of a positron incident on a hydrogen atom. In 1971, we [29] carried out calculations using the projectionoperator formulism of Feshbach [4] and using generalized Hylleraas-type functions:

symmetry in the action is related to a conservation law [27].

*Recent Advances in Nanophotonics - Fundamentals and Applications*

Ψð Þ¼ *r*1,*r*2,*r*<sup>12</sup> *e*

**Bhatia et al. [29]**

*k* **Hybrid theory [30]**

**Table 8.**

**70**

*The upper curve represents phase shifts obtained using the hybrid theory, and the lower curve represents phase shifts obtained using the Feshbach formalism.*

bounds. In the hybrid theory, as indicated earlier, both short-range and long-range correlations can be taken into account at the same time. It should be noted that in the case of positrons, the sign before the second term in Eq. (10) is plus instead of minus, as in the case of electrons. The phase shifts obtained in two approaches are given in **Table 8**. We used 84 terms in the Hylleraas wave function previously, and improved results are obtained now with shorter expansions [30] as indicated in the table. A comparison of the results obtained with different approaches is shown in **Figure 6**. Using a fewer number of terms, higher phase shifts have been obtained in the hybrid theory [30]. P-wave shifts have been calculated in the hybrid theory [31] and are compared with those using the Feshbach projection-operator formalism [32] and with those obtained by Armstrong [33].

### **8.** *Zeff*

The incident positron can annihilate the atomic electron with the emission of two gamma rays. The cross section for this process has been given by Ferrell [34]:

$$
\sigma\_{\mathfrak{a}} \left( \mathfrak{a} \mathfrak{a}\_0^2 \right) = Z\_{\mathfrak{eff}} \mathfrak{a}^3 \mathfrak{k}^{-1} \tag{27}
$$

where *α* is the fine-structure constant and

$$Z\_{\rm eff} = \int d\vec{r}\_1 d\vec{r}\_2 \left| \Psi(\vec{r}\_1, \vec{r}\_2) \right|^2 \delta(\vec{r}\_1, \vec{r}\_2) . \tag{28}$$

*Zeff* for partial waves *l* = 0 and 1, obtained in [31, 32], is given in **Table 9** along with the contribution from *l* > 1. For higher partial waves, *Zeff* is given by

$$Z\_{\rm eff}(l>1) = \frac{k^2}{1+k^2} + \frac{6}{k^2} \left[ \frac{1}{k^2} \ln\left(1+k^2\right) - \frac{1+0.5k^2}{1+k^2} \right]. \tag{29}$$


**Table 9.**

Zeff *for positron-hydrogen scattering.*

### **9. Positronium formation**

Positronium, the bound state of an electron and a positron, was predicted by Mohorovicic [35] in connection with the spectra of nebulae. Positronium (Ps) formation takes place when the incident positron captures the bound electron of the hydrogen atom:

$$\text{Per}^+ + H(\text{1s}) \to \text{Ps} + P.\tag{30}$$

*<sup>σ</sup>* <sup>¼</sup> <sup>1</sup>*:*<sup>32</sup> � <sup>10</sup>�18cm2 *<sup>k</sup>*<sup>3</sup>

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

the outgoing electron. These cross sections are much larger than that for the

Lyman-α radiation (2*P* ! 1*S*) at 1216 Å has been seen from astrophysical sources and the sun. It has been observed from Voyager measurements [40]. Similarly, Lyman-α radiation (2*P* ! 1*S*) at 2416 Å is expected when in the photodetachment of Ps�, the remaining positronium is left in the 2*P* state. Following Ohmura and Ohmura [20], photodetachment cross sections have been calculated when the

> *ο*ð Þ¼ 2*p* 164*:*492*C k*ð Þ *σ*ð Þ¼ 3*p* 26*:*3782*C k*ð Þ *σ*ð Þ¼ 4*p* 9*:*1664*C k*ð Þ *σ*ð Þ¼ 5*p* 4*:*3038*C k*ð Þ *σ*ð Þ¼ 6*p* 2*:*3764*C k*ð Þ *σ*ð Þ¼ 7*p* 0*:*2675*C k*ð Þ

These cross sections are given in **Figure 7** for various photon energies.

Positrons do not bind with hydrogen atoms. However, they do bind with various atoms as has been shown byMitroy et al. [42]. The binding energies are given in**Table 11**.

*The upper most curve represents photodetachment cross sections for* n *= 2 on log10 scale. The curves below it are*

The electron affinity is <sup>3</sup>*γ*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

**11. Binding energies**

*for n = 3, 4, 5, 6, and 7.*

**Figure 7.**

**73**

photodetachment of the negative hydrogen ion.

remaining atom is in 2*p*, 3*p*, 4*p*, 5*p*, 6*p*, and 7*p* states [41].

*<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup> <sup>3</sup> *:* (31)

(32)

<sup>2</sup> , where *γ* = 0.12651775 [39] and *k* is the momentum of

Cross sections for the positronium formation are given in **Table 10** and are compared with those obtained by Khan and Ghosh [36] and Humberston [37].


**Table 10.**

*Cross sections* πa<sup>2</sup> 0 *for positronium formation obtained in hybrid theory and in comparison with those obtained by Khan and Ghosh [36] and Humberston [37].*

### **10. Photodetachment of positronium ion (Ps**�**)**

Photodetachment has been discussed above already. Following the work of Ohmura and Ohmura [20], Bhatia and Drachman [38] calculated cross sections (in the length and velocity form) for photodetachment of Ps�. Their result in length and velocity form is

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

$$
\sigma = \left( \mathbf{1}.32 \times 10^{-18} \text{cm}^2 \right) \frac{\mathbf{k}^3}{\left(\mathbf{k}^2 + \mathbf{y}^2\right)^3}. \tag{31}
$$

The electron affinity is <sup>3</sup>*γ*<sup>2</sup> <sup>2</sup> , where *γ* = 0.12651775 [39] and *k* is the momentum of the outgoing electron. These cross sections are much larger than that for the photodetachment of the negative hydrogen ion.

Lyman-α radiation (2*P* ! 1*S*) at 1216 Å has been seen from astrophysical sources and the sun. It has been observed from Voyager measurements [40]. Similarly, Lyman-α radiation (2*P* ! 1*S*) at 2416 Å is expected when in the photodetachment of Ps�, the remaining positronium is left in the 2*P* state. Following Ohmura and Ohmura [20], photodetachment cross sections have been calculated when the remaining atom is in 2*p*, 3*p*, 4*p*, 5*p*, 6*p*, and 7*p* states [41].

$$\begin{aligned} \sigma(2p) &= 164.492 \text{C}(k) \\ \sigma(3p) &= 26.3782 \text{C}(k) \\ \sigma(4p) &= 9.1664 \text{C}(k) \\ \sigma(5p) &= 4.3038 \text{C}(k) \\ \sigma(6p) &= 2.3764 \text{C}(k) \\ \sigma(7p) &= 0.2675 \text{C}(k) \end{aligned} \tag{32}$$

These cross sections are given in **Figure 7** for various photon energies.

### **Figure 7.**

**9. Positronium formation**

0.5476 0.018783

0.6724 0.022350

0.81 0.19256 0.9025 0.016760 1.00 0.014327

0

Zeff *for positron-hydrogen scattering.*

hydrogen atom:

**Table 9.**

and velocity form is

**Table 10.** *Cross sections* πa<sup>2</sup>

**72**

Positronium, the bound state of an electron and a positron, was predicted by Mohorovicic [35] in connection with the spectra of nebulae. Positronium (Ps) formation takes place when the incident positron captures the bound electron of the

*k Zeff* **(***l* **= 0)** *Zeff* **(***l* **= 1)** *Zeff* **(***l* **> 1) Total** 0.1 7.363 0.022 <0.001 7.385 0.2 5.538 0.90 0.001 5.629 0.3 4.184 0.187 0.004 4.375 0.4 3.327 0.294 0.010 3.631 0.5 2.730 0.390 0.022 3.142 0.6 2.279 0.464 0.039 2.782 0.7 1.850 0.528 0.063 2.541

*Recent Advances in Nanophotonics - Fundamentals and Applications*

Cross sections for the positronium formation are given in **Table 10** and are compared with those obtained by Khan and Ghosh [36] and Humberston [37].

*k***<sup>2</sup> Hybrid theory Khan and Ghosh [36] Humberston [37]** 0.5041 0.0066228 0.009037 0.0041

0.5625 0.0.20249 0.024795 0.0044 0.64 0.022566 0.0248 0.0049

0.7225 0.21456 0.021164 0.0058

Photodetachment has been discussed above already. Following the work of Ohmura and Ohmura [20], Bhatia and Drachman [38] calculated cross sections (in the length and velocity form) for photodetachment of Ps�. Their result in length

*for positronium formation obtained in hybrid theory and in comparison with those*

<sup>þ</sup> þ *H*ð Þ! 1*s Ps* þ *P:* (30)

*e*

**10. Photodetachment of positronium ion (Ps**�**)**

*obtained by Khan and Ghosh [36] and Humberston [37].*

0.75 0.020835 0.019707

*The upper most curve represents photodetachment cross sections for* n *= 2 on log10 scale. The curves below it are for n = 3, 4, 5, 6, and 7.*

### **11. Binding energies**

Positrons do not bind with hydrogen atoms. However, they do bind with various atoms as has been shown byMitroy et al. [42]. The binding energies are given in**Table 11**.


*e* <sup>þ</sup> þ *e*

Born approximation, total cross sections for e—-He and e+

**15. Total positron-hydrogen cross sections**

*Cross sections have been interpolated from those given in Ref. [50].*

*Cross sections measured by Kauppila et al. [47] for e—-He and e+*

**14. High-energy cross sections**

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

tend to be equal.

experimental results.

**Table 12.**

*a*

**75**

**Table 13.**

*Positron-hydrogen total cross sections.*

antimatter. However, there is no experimental confirmation up to now.

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

<sup>þ</sup> þ *p* ! *H* þ *e*

If the antihydrogen is formed in the excited state, then it can decay to the lower states. It would then be possible to verify if the quantum mechanics principles are the same in the antimatter universe. It is thought that gravitational interactions should be the same between matter and antimatter and between antimatter and

At high energies, only static potential remains. Therefore, according to the first

This fact has been verified experimentally by Kauppila et al. [47], and their results are shown in **Table 12**. We see that as the incident energy increases, cross sections

**Energy (eV) e—-He e<sup>+</sup>**

 1.27 1.97 1.16 1.26 0.967 0.987 0.796 0.812 0.614 0.612 0.437 0.434 0.371 0.381

Total cross sections for positron scattering from hydrogen atoms have been measured by Zhou et al. [48] and have been calculated by Walters [49] using the close-coupling approximation and also by Gien [50] using the modified Glauber approximation. Their results are given in **Table 13**. They are fairly close to the

*E* **(eV) Ref. [49] Ref. [50]<sup>a</sup>** 54.4 3.02 2.85 100 2.24 2.00 200 1.33 1.24 300 9.69 (�1) 8.73 (�1)

*-He scattering.*

<sup>þ</sup>*:* (38)


**-He**

**Table 11.**

*Binding energies (Ry) of positrons with various atoms.*

### **12. Resonances**

Resonances formed in the scattering of electrons from atoms are very common. However, they are not that common in positron-target systems. The first successful prediction of *S*-wave Feshbach resonance in positron-hydrogen system is by Doolen et al. [43], who, using the complex-rotation method, obtained the position �0.2573741 and for width 0.0000677 Ry. In this method

$$r \to r e^{-i\theta}, T \to T e^{-2i\theta}, \text{and } V \to V e^{-i\theta} \text{ and } H = T + V. \tag{33}$$

Eigenvalues are complex now. The real part gives position of the resonance, and the imaginary part gives its half width.

A number of Feshbach and shape resonances in Ps� have been calculated by using the complex-rotation method. Parameters of a <sup>1</sup> *P* shape resonance above *n* = 2 have been calculated by Bhatia and Ho [44]. They obtained �0.06217 and 0.000225 Ry for the position and width of the resonance. These results have been confirmed experimentally by Michishio et al. [45].

### **13. Antihydrogen formation**

Antihydrogen can be formed in the collision of Ps with antiproton

$$\text{Ps} + \overline{p} \to \overline{H} + e. \tag{34}$$

In the above equation, *p* represents antiproton and *H* represents antihydrogen. According to the time reversal invariance, the above reaction is related to

$$\text{Ps} + \text{p} \to \text{H} + \text{e}^+.\tag{35}$$

From this reaction, the positronium formation cross sections are known from positron-hydrogen scattering. This implies that the cross section for antihydrogen is related to the cross section for Ps formation:

$$
\sigma\_{\overline{H}} = \left(\frac{k\_{\varepsilon^+}}{k\_{\mathbb{P}s}}\right)^2 \sigma\_{\mathbb{P}s}.\tag{36}
$$

*ke*<sup>þ</sup> and *k*Ps are momenta of positron and positronium. Humberston et al.

[46] have calculated cross sections for the formation of antihydrogen in reaction (34). It is possible to form antihydrogen by radiative recombination or three-body recombination:

$$e^{+} + \overline{p} \to \overline{H} + h\nu. \tag{37}$$

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

$$
\epsilon^+ + \epsilon^+ + \overline{p} \to \overline{H} + \epsilon^+. \tag{38}
$$

If the antihydrogen is formed in the excited state, then it can decay to the lower states. It would then be possible to verify if the quantum mechanics principles are the same in the antimatter universe. It is thought that gravitational interactions should be the same between matter and antimatter and between antimatter and antimatter. However, there is no experimental confirmation up to now.

### **14. High-energy cross sections**

At high energies, only static potential remains. Therefore, according to the first Born approximation, total cross sections for e—-He and e+ -He should be the same. This fact has been verified experimentally by Kauppila et al. [47], and their results are shown in **Table 12**. We see that as the incident energy increases, cross sections tend to be equal.


**Table 12.**

**12. Resonances**

**He (<sup>3</sup>**

**Table 11.**

Resonances formed in the scattering of electrons from atoms are very common. However, they are not that common in positron-target systems. The first successful prediction of *S*-wave Feshbach resonance in positron-hydrogen system is by Doolen

*S***) Li Be Na Mg** 0.0011848 0.004954 0.006294 0.000946 0.031224 **Ca Sr Cu Au Cd** 0.03300 0.0201 0.011194 0.011664 0.01220

Eigenvalues are complex now. The real part gives position of the resonance, and

A number of Feshbach and shape resonances in Ps� have been calculated by

have been calculated by Bhatia and Ho [44]. They obtained �0.06217 and 0.000225 Ry for the position and width of the resonance. These results have been confirmed

In the above equation, *p* represents antiproton and *H* represents antihydrogen.

*Ps* þ *p* ! *H* þ *e*

*<sup>σ</sup><sup>H</sup>* <sup>¼</sup> *ke*<sup>þ</sup> *kPs* <sup>2</sup>

*ke*<sup>þ</sup> and *k*Ps are momenta of positron and positronium. Humberston et al. [46] have calculated cross sections for the formation of antihydrogen in reaction (34). It is possible to form antihydrogen by radiative recombination or three-body

*e*

From this reaction, the positronium formation cross sections are known from positron-hydrogen scattering. This implies that the cross section for antihydrogen is

Antihydrogen can be formed in the collision of Ps with antiproton

According to the time reversal invariance, the above reaction is related to

, and *<sup>V</sup>* ! *Ve*�*i<sup>θ</sup>* and *<sup>H</sup>* <sup>¼</sup> *<sup>T</sup>* <sup>þ</sup> *<sup>V</sup>:* (33)

*Ps* þ *p* ! *H* þ *e:* (34)

<sup>þ</sup>*:* (35)

*σPs:* (36)

<sup>þ</sup> þ *p* ! *H* þ *hν:* (37)

*P* shape resonance above *n* = 2

et al. [43], who, using the complex-rotation method, obtained the position

�0.2573741 and for width 0.0000677 Ry. In this method

*Recent Advances in Nanophotonics - Fundamentals and Applications*

using the complex-rotation method. Parameters of a <sup>1</sup>

, *<sup>T</sup>* ! *Te*�2*i<sup>θ</sup>*

*<sup>r</sup>* ! *re*�*i<sup>θ</sup>*

*Binding energies (Ry) of positrons with various atoms.*

the imaginary part gives its half width.

experimentally by Michishio et al. [45].

related to the cross section for Ps formation:

recombination:

**74**

**13. Antihydrogen formation**

*Cross sections measured by Kauppila et al. [47] for e—-He and e+ -He scattering.*

### **15. Total positron-hydrogen cross sections**

Total cross sections for positron scattering from hydrogen atoms have been measured by Zhou et al. [48] and have been calculated by Walters [49] using the close-coupling approximation and also by Gien [50] using the modified Glauber approximation. Their results are given in **Table 13**. They are fairly close to the experimental results.


**Table 13.** *Positron-hydrogen total cross sections.*

### **16. Threshold laws**

When the incident electron or positron has just enough energy to ionize the hydrogen atom, how do the cross sections behave? Wannier [51], using classical methods and supposing the two electrons emerge opposite to each other, showed that *σ* ∝ *E*1*:*127, where *E i*s the excess energy. As indicated earlier in the positronhydrogen scattering, positron and electron tend to be on the same side of the nucleus. This shows the cross section at threshold cannot be the same for positrons and electrons. The threshold behavior of the positron impact on hydrogen has been analyzed by Klar [52]. He finds that at threshold, *σ* ∝*E*2*:*650.

Wigner [53] has emphasized the importance of long-range forces near the threshold which have been included in these calculations (hybrid theory). At the threshold, the cross section for exciting the 1*S* state of hydrogen atom to the 2*S* state is proportional to ln *k <sup>f</sup>* �<sup>2</sup> [54].

### **17. Conclusions**

In this chapter, we have discussed various interactions of electrons and positrons with atoms, ions, and radiation fields. There are various approximations and theories to calculate scattering functions. Theories which provide variational bounds on the calculated phase shifts are preferable because improved results can be obtained when the number of functions in the closed channels is increased. Such theories are the close-coupling, *R*-matrix, and hybrid theory. It should be possible to formulate the hybrid theory for more complicated systems.

The continuum functions obtained using the hybrid theory have been used to calculate photoabsorption cross sections, obtaining results which agree with definitive results obtained using other methods and experiments. Such cross sections are needed to study the opacity in the sun. The resonances play an important role when they are included in the calculations of excitation cross sections, which are important to infer temperatures and densities of solar and astrophysical plasmas.

When Feshbach projection-operator formalism [4] is used to calculate resonance position, *E* ¼ *ε* þ Δ, where Δ, the shift in the resonance position due to its coupling with the continuum, has to be calculated separately [14]. However, in the hybrid theory, the calculated position includes this correction. This is an advantage since the calculation of the shift is nontrivial.

We have indicated that in addition to obtaining accurate phase shifts for positron scattering from a hydrogen atom, we have described calculations of annihilation, positronium, and antihydrogen formation. We have discussed resonances in a positron-hydrogen system. We have discussed photodetachment of a positronium ion and a possibility of observing Lyman-α radiation from a positronium atom when the final state is the 2*p* state.

**Author details**

Anand K. Bhatia

**77**

NASA/Goddard Space Flight Center, United States

provided the original work is properly cited.

\*Address all correspondence to: anand.k.bhatia@nasa.gov

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances…*

*DOI: http://dx.doi.org/10.5772/intechopen.91763*

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

### **Author details**

**16. Threshold laws**

is proportional to ln *k <sup>f</sup>*

**17. Conclusions**

**76**

When the incident electron or positron has just enough energy to ionize the hydrogen atom, how do the cross sections behave? Wannier [51], using classical methods and supposing the two electrons emerge opposite to each other, showed that *σ* ∝ *E*1*:*127, where *E i*s the excess energy. As indicated earlier in the positronhydrogen scattering, positron and electron tend to be on the same side of the nucleus. This shows the cross section at threshold cannot be the same for positrons and electrons. The threshold behavior of the positron impact on hydrogen has been

Wigner [53] has emphasized the importance of long-range forces near the threshold which have been included in these calculations (hybrid theory). At the threshold, the cross section for exciting the 1*S* state of hydrogen atom to the 2*S* state

In this chapter, we have discussed various interactions of electrons and positrons with atoms, ions, and radiation fields. There are various approximations and theories to calculate scattering functions. Theories which provide variational bounds on the calculated phase shifts are preferable because improved results can be obtained when the number of functions in the closed channels is increased. Such theories are the close-coupling, *R*-matrix, and hybrid theory. It should be possible to formulate

The continuum functions obtained using the hybrid theory have been used to calculate photoabsorption cross sections, obtaining results which agree with definitive results obtained using other methods and experiments. Such cross sections are needed to study the opacity in the sun. The resonances play an important role when they are included in the calculations of excitation cross sections, which are important to infer temperatures and densities of solar and astrophysical plasmas.

When Feshbach projection-operator formalism [4] is used to calculate resonance position, *E* ¼ *ε* þ Δ, where Δ, the shift in the resonance position due to its coupling with the continuum, has to be calculated separately [14]. However, in the hybrid theory, the calculated position includes this correction. This is an advantage since

We have indicated that in addition to obtaining accurate phase shifts for positron scattering from a hydrogen atom, we have described calculations of annihilation, positronium, and antihydrogen formation. We have discussed resonances in a positron-hydrogen system. We have discussed photodetachment of a

positronium ion and a possibility of observing Lyman-α radiation from a

analyzed by Klar [52]. He finds that at threshold, *σ* ∝*E*2*:*650.

*Recent Advances in Nanophotonics - Fundamentals and Applications*

�<sup>2</sup> [54].

the hybrid theory for more complicated systems.

the calculation of the shift is nontrivial.

positronium atom when the final state is the 2*p* state.

Anand K. Bhatia NASA/Goddard Space Flight Center, United States

\*Address all correspondence to: anand.k.bhatia@nasa.gov

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[19] Wildt R. Electron affinity in astrophysics. The Astrophysical Journal. 1939;**89**:295

[20] Ohmura T, Ohmura H. Electronhydrogen scattering at low energies. Physics Review. 1960;**118**:154

[21] Nahar SN. In: Chavez M, Bertone E, Rosa-Gonzalez D, Rodriguez-Merino

*Interactions of Positrons and Electrons with Hydrogenic Systems, Excitation, Resonances… DOI: http://dx.doi.org/10.5772/intechopen.91763*

LR, editors. New Quests Stellar Astrophysics. II. The Ultraviolet Properties of Evolved Stellar Populations. New York: Springer; 2009. p. 245

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[12] Temkin A. Polarization and triplet electron-hydrogen scattering length. Physical Review Letters. 1961;**6**:354

[13] Bhatia AK. Applications of the hybrid theory to the scattering of electrons from He+ and Li2+ and resonances in two-electron systems. Physical Review A. 2008;**77**:052707

[14] Bhatia AK, Temkin A. Calculation of autoionization of He and H using the projection-operator formalism. Physical

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[22] Samson JAR, He ZX, Yin L, Haddad GN. Precision measurements of the absolute photoionization cross sections of He. Journal of Physics B. 1994;**27**:887

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[43] Doolen GD, Nuttal J, Wherry CJ. Evidence of a resonance in e+ -H S-wave scattering. Physical Review Letters. 1978;**40**:313

[44] Bhatia AK, Ho YK. Complexcoordinate calculation of 1,3P resonances in Ps- using hylleraas functions. Physical Review A. 1990;**42**:1119

[45] Michishio K, Kanai T, Azuma T, Wade K, Mochizuki I, Hyodo T, et al. Photodetachment of positronium negative ions. Nature Communications. 2016;**7**:11060

[46] Humberston JW, Charlton M, Jacobson FM, Deutch BI. On antihydrogen formation in collisions of antiprotons with positronium. Journal of Physics B. 1987;**20**:347

[47] Kauppila WE, Stein TS, Smart JH, Debabneh MS, Ho YK, Dawning JP, et al. Meaurements of total scattering cross sections for intermediate-energy positrons and electrons colliding with helium, neon, and argon. Physical Review A. 1981;**24**:308

[48] Zhou S, Kappila WE, Kwan CK, Stein TS. Measurements of total cross sections for positrons and electron colliding with atomic hydrogen. Physical Review Letters. 1994;**72**:1443

[49] Walters HRJ. Positron scattering by atomic hydrogen at intermediate energies. Journal of Physics B. 1988;**21**: 1893

[50] Gien TT. Total cross sections for positron-hydrogen scattering. Journal of Physics B. 1995;**28**:L321

[51] Wannier GH. The threshold law of single ionization of atoms or ions by electrons. Physical Review. 1953;**90**:817

[52] Klar H. Threshold ionization of atoms by positrons. Journal of Physics B. 1981;**14**:4165

[53] Wigner EP. On the behavior of cross sections near thresholds. Physics Review. 1948;**73**:1002

[54] Sadeghpour HR, Bohn JL, Cavagnero MJ, Esry BD, Fabrikant II, Macek JH, et al. Collisions near threshold in atomic and molecular physics. Journal of Physics B. 2000;**33**: R93

**81**

A = Cs+

, CH3NH3

+

; B = Pb2+ or Sn2+ and X = I−

**Chapter 5**

**Abstract**

Origin and Fundamentals of

In the last few decades, the energy demand has been increased dramatically. Different forms of energy have utilized to fulfill the energy requirements. Solar energy has been proven an effective and highly efficient energy source which has the potential to fulfill the energy requirements in the future. Previously, various kind of solar cells have been developed. In 2013, organic–inorganic metal halide perovskite materials have emerged as a rising star in the field of photovoltaics. The methyl ammonium lead halide perovskite structures were employed as visible light sensitizer for the development of highly efficient perovskite solar cells (PSCs). In 2018, the highest power conversion efficiency of 23.7% was achieved for methyl ammonium lead halide based PSCs. This obtained highest power conversion efficiency makes them superior over other solar cells. The PSCs can be employed for practical uses, if their long term stability improved by utilizing some novel strategies. In this chapter, we have discussed the optoelectronic properties of the perovskite materials, construction of PSCs and recent advances in the electron

**Keywords:** methyl ammonium lead halide, perovskites light absorbers, photovoltaics,

The researchers believe that the solar energy have the potential to fulfill the energy requirements [1]. The earth receive enormous amount of solar energy in the form of sunlight [2]. This sunlight can be converted to the electrical energy to fulfill our energy requirements [3]. The photovoltaic device (solar cells) can directly converted the sunlight to the electricity [4]. In previous decades, different kinds of photovoltaic devices (dye-sensitized solar cells = DSSCs, organic solar cells = OSCs, polymer solar cells, quantum-dot sensitized solar cells and perovskite solar cells) were developed [5–9]. These kinds of solar cells have attracted the scientific community due to their simple manufacturing procedure and cost-effectiveness [10]. The PSCs gained huge attention because of their excellent photovoltaic performance and low-cost [11–16]. The perovskite solar cells (PSCs) involve a perovskite light absorber layer. The perovskite is a material which has a molecular formula of ABO3. The perovskite term was given to the calcium titanate (CaTiO3). There is also another class of perovskite materials exists with molecular formula of ABX3 (where

, Br−

materials possesses excellent absorption properties, charge carrier properties and suitable band gap. In 2009, Kojima et al. [17] prepared the methyl ammonium lead

or Cl−

). This class of perovskite

Perovskite Solar Cells

transport layers for the fabrication of PSCs.

perovskite solar cells

**1. Introduction**

*Mohd Quasim Khan and Khursheed Ahmad*

### **Chapter 5**

[43] Doolen GD, Nuttal J, Wherry CJ. Evidence of a resonance in e+

scattering. Physical Review Letters.

[44] Bhatia AK, Ho YK. Complexcoordinate calculation of 1,3P resonances in Ps- using hylleraas functions. Physical

[45] Michishio K, Kanai T, Azuma T, Wade K, Mochizuki I, Hyodo T, et al. Photodetachment of positronium negative ions. Nature Communications.

[46] Humberston JW, Charlton M, Jacobson FM, Deutch BI. On

antihydrogen formation in collisions of antiprotons with positronium. Journal of

[47] Kauppila WE, Stein TS, Smart JH, Debabneh MS, Ho YK, Dawning JP, et al. Meaurements of total scattering cross sections for intermediate-energy positrons and electrons colliding with helium, neon, and argon. Physical

[48] Zhou S, Kappila WE, Kwan CK, Stein TS. Measurements of total cross sections for positrons and electron colliding with atomic hydrogen. Physical Review Letters. 1994;**72**:1443

[49] Walters HRJ. Positron scattering by atomic hydrogen at intermediate energies. Journal of Physics B. 1988;**21**:

[50] Gien TT. Total cross sections for positron-hydrogen scattering. Journal of

[51] Wannier GH. The threshold law of single ionization of atoms or ions by electrons. Physical Review. 1953;**90**:817

[52] Klar H. Threshold ionization of atoms by positrons. Journal of Physics B.

Physics B. 1995;**28**:L321

1981;**14**:4165

**80**

Review A. 1990;**42**:1119

Physics B. 1987;**20**:347

Review A. 1981;**24**:308

1893

1978;**40**:313

2016;**7**:11060


*Recent Advances in Nanophotonics - Fundamentals and Applications*

[53] Wigner EP. On the behavior of cross

Cavagnero MJ, Esry BD, Fabrikant II, Macek JH, et al. Collisions near threshold in atomic and molecular physics. Journal of Physics B. 2000;**33**:

sections near thresholds. Physics

[54] Sadeghpour HR, Bohn JL,

Review. 1948;**73**:1002

R93

## Origin and Fundamentals of Perovskite Solar Cells

*Mohd Quasim Khan and Khursheed Ahmad*

### **Abstract**

In the last few decades, the energy demand has been increased dramatically. Different forms of energy have utilized to fulfill the energy requirements. Solar energy has been proven an effective and highly efficient energy source which has the potential to fulfill the energy requirements in the future. Previously, various kind of solar cells have been developed. In 2013, organic–inorganic metal halide perovskite materials have emerged as a rising star in the field of photovoltaics. The methyl ammonium lead halide perovskite structures were employed as visible light sensitizer for the development of highly efficient perovskite solar cells (PSCs). In 2018, the highest power conversion efficiency of 23.7% was achieved for methyl ammonium lead halide based PSCs. This obtained highest power conversion efficiency makes them superior over other solar cells. The PSCs can be employed for practical uses, if their long term stability improved by utilizing some novel strategies. In this chapter, we have discussed the optoelectronic properties of the perovskite materials, construction of PSCs and recent advances in the electron transport layers for the fabrication of PSCs.

**Keywords:** methyl ammonium lead halide, perovskites light absorbers, photovoltaics, perovskite solar cells

### **1. Introduction**

The researchers believe that the solar energy have the potential to fulfill the energy requirements [1]. The earth receive enormous amount of solar energy in the form of sunlight [2]. This sunlight can be converted to the electrical energy to fulfill our energy requirements [3]. The photovoltaic device (solar cells) can directly converted the sunlight to the electricity [4]. In previous decades, different kinds of photovoltaic devices (dye-sensitized solar cells = DSSCs, organic solar cells = OSCs, polymer solar cells, quantum-dot sensitized solar cells and perovskite solar cells) were developed [5–9]. These kinds of solar cells have attracted the scientific community due to their simple manufacturing procedure and cost-effectiveness [10].

The PSCs gained huge attention because of their excellent photovoltaic performance and low-cost [11–16]. The perovskite solar cells (PSCs) involve a perovskite light absorber layer. The perovskite is a material which has a molecular formula of ABO3. The perovskite term was given to the calcium titanate (CaTiO3). There is also another class of perovskite materials exists with molecular formula of ABX3 (where A = Cs+ , CH3NH3 + ; B = Pb2+ or Sn2+ and X = I− , Br− or Cl− ). This class of perovskite materials possesses excellent absorption properties, charge carrier properties and suitable band gap. In 2009, Kojima et al. [17] prepared the methyl ammonium lead

halide (MAPbX3; where MA = CH3NH3 + , X = halide anion) perovskite materials and investigated their optoelectronic properties. Further, authors fabricated the dye sensitized solar cells using MAPbX3 visible light sensitizer [17]. The performance of the developed dye sensitized solar cells was evaluated and the device exhibited good power conversion efficiency (PCE) and open circuit voltage. The fabricated dye sensitized solar cells with MAPbX3 visible light sensitizer exhibited the good PCE of ~3.8% [17]. Although, this power conversion efficiency was quite interesting but the presence of liquid electrolyte vanished this performance. Hence, it was observed that the use of alternative solid state electrolyte/hole transport material would be of great significance. In this regard, numerous strategies were developed to overcome the issue of liquid electrolyte. In this regard, a solid state electrolyte was employed by Lee et al. [18] to develop the PSCs. The developed PSCs device showed the good power conversion efficiency of 10.9%. In last few years, various novel approaches were advanced to improve the photovoltaic performance of the PSCs [19–30] and recently the best PCE of 23.3% was achieved for PSCs [31].

In this chapter, we have discussed the construction of PSCs. Recent advances in PSCs with respect to the charge collection layer/electron transport layer and future prospective have also been discussed.

### **2. Construction of PSCs**

The fabrication of PSCs required different layers such as transparent conductive oxide coated fluorine doped tin oxide = FTO, electron transport layer (generally consists of semiconducting metal oxides), light absorber layer (perovskite), hole transport material (HTM) layer and metal contact (Au) layer. In the first step, FTO glass substrate etched with the help of zinc powder and HCl followed by the washing of the etched FTO glass substrate with acetone, DI water and 2-propanol. The compact layer of the TiO2 deposited on to the FTO glass substrate using spin coater and annealed at ~500°C for 30–40 min. Further electron transport layer also deposited on the electrode using spin coater and annealed at ~500°C for 30–40 min. The perovskite light absorber layer deposited using spin coater and annealed at ~80–120°C for 20–60 min. Further, hole transport material (HTM) also deposited using spin coater. Finally the metal contact layer (Au) deposited using thermal evaporation approach. The construction of PSCs has been presented in **Figure 1**.

**83**

thin film.

*Origin and Fundamentals of Perovskite Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.94376*

PSCs devices worked as light absorber.

and Ag have been displayed in **Figure 2b**.

**3. Origin of PSCs**

The photovoltaic performance of the constructed PSCs can be determined by different techniques like external quantum efficiency (EQE), photocurrent-voltage (I-V), photoluminescence spectroscopy and incident-photo-to-current-conversion efficiency = IPCE etc. In general, the performance of any solar cell device can be determined in terms of fill factor, PCE, photocurrent density and open circuit voltage. Photoluminescence spectroscopy can also be employed to check the lifetime of the generated electrons inside the perovskite materials which is related to the recombination processes. It has also been known that the PSCs devices with high electron lifetime and lower recombination reaction rates may provide better

The PSCs device absorbs the sunlight which created the electron–hole pairs inside the perovskite light absorber layer. This electron has to be transferred to the conductive electrode surface. Thus, electron transport layer consists of transition metal oxides (generally TiO2 or SnO2) transport this generated electron to the conductive electrode (FTO). The hole can be collected by the hole transport material layers. In some cases, the transferred electron can recombine and influence the performance of the PSCs device. Thus, some buffer or compact layers have been used to reduce the recombination process. The perovskite materials used in such

The PSCs were originated in 2009 by Kojima et al. [17] and reported an interesting PCE of 3.1%. Further different approaches were utilized to improve the PCE of the PSCs. Recently, Tang et al. [32] prepared the low temperature processed zinc oxide nanowalls (ZnO NWs). Authors employed these prepared ZnO NWs as electron collection layer for the development of PSCs [32]. The morphological features of the prepared ZnO NWs were determined by scanning electron microscopy = SEM and transmission electron microscopy = TEM. The SEM and TEM results confirmed the formation of ZnO NWs. Further PSCs were constructed using ZnO NWs as electron collection layer whereas MAPbI3 as light absorber. The device architecture of the PSCs has been presented in **Figure 2a**. The energy level values of the perovskite light absorber, ZnO, indium doped tin oxide (ITO), spiro-OMeTAD

The performance of the PSCs devices with ZnO NWs and ZnO thin films were evaluated by J-V analysis. The J-V curves of the PSCs developed with ZnO NWs and ZnO thin films have been depicted in **Figure 2c**. The constructed PSCs device with ZnO NWs exhibited the highest PCE of 13.6% whereas the PSCs developed with ZnO thin films showed the PCE of 11.3%. This showed that ZnO NWs plays crucial role in charge collection compare to the ZnO thin films. The NWs of ZnO collect the generated electron more efficiently compare to the ZnO thin films. Further, incident IPCE of the constructed PSCs was also checked. The IPCE curves of the PSCs developed with ZnO NWs and ZnO thin films have been presented in **Figure 2d**. The PSCs developed with ZnO NWs showed the highest open circuit voltage of 1000 mV whereas the PSCs device fabricated with ZnO thin films showed the open circuit voltage of 980 mV. The constructed PSCs with ZnO NWs exhibited the improved IPCE compared to the PSCs device developed with ZnO thin films.

In other work, Mahmud et al. [33] synthesized low-temperature processed ZnO

The optical properties of the prepared ZnO thin film were investigated by employing ultraviolet–visible (UV–vis) absorption spectroscopy. The Tauc plot of the ZnO has been presented in **Figure 3A**. The synthesized ZnO possess an optical

photovoltaic performance in terms of power conversion efficiency.

**Figure 1.** *Fabrication process for the PSCs.*

*Origin and Fundamentals of Perovskite Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.94376*

*Recent Advances in Nanophotonics - Fundamentals and Applications*

recently the best PCE of 23.3% was achieved for PSCs [31].

+

investigated their optoelectronic properties. Further, authors fabricated the dye sensitized solar cells using MAPbX3 visible light sensitizer [17]. The performance of the developed dye sensitized solar cells was evaluated and the device exhibited good power conversion efficiency (PCE) and open circuit voltage. The fabricated dye sensitized solar cells with MAPbX3 visible light sensitizer exhibited the good PCE of ~3.8% [17]. Although, this power conversion efficiency was quite interesting but the presence of liquid electrolyte vanished this performance. Hence, it was observed that the use of alternative solid state electrolyte/hole transport material would be of great significance. In this regard, numerous strategies were developed to overcome the issue of liquid electrolyte. In this regard, a solid state electrolyte was employed by Lee et al. [18] to develop the PSCs. The developed PSCs device showed the good power conversion efficiency of 10.9%. In last few years, various novel approaches were advanced to improve the photovoltaic performance of the PSCs [19–30] and

In this chapter, we have discussed the construction of PSCs. Recent advances in PSCs with respect to the charge collection layer/electron transport layer and future

The fabrication of PSCs required different layers such as transparent conductive oxide coated fluorine doped tin oxide = FTO, electron transport layer (generally consists of semiconducting metal oxides), light absorber layer (perovskite), hole transport material (HTM) layer and metal contact (Au) layer. In the first step, FTO glass substrate etched with the help of zinc powder and HCl followed by the washing of the etched FTO glass substrate with acetone, DI water and 2-propanol. The compact layer of the TiO2 deposited on to the FTO glass substrate using spin coater and annealed at ~500°C for 30–40 min. Further electron transport layer also deposited on the electrode using spin coater and annealed at ~500°C for 30–40 min. The perovskite light absorber layer deposited using spin coater and annealed at ~80–120°C for 20–60 min. Further, hole transport material (HTM) also deposited using spin coater. Finally the metal contact layer (Au) deposited using thermal evaporation approach. The construction of PSCs has

, X = halide anion) perovskite materials and

halide (MAPbX3; where MA = CH3NH3

prospective have also been discussed.

**2. Construction of PSCs**

been presented in **Figure 1**.

**82**

**Figure 1.**

*Fabrication process for the PSCs.*

The photovoltaic performance of the constructed PSCs can be determined by different techniques like external quantum efficiency (EQE), photocurrent-voltage (I-V), photoluminescence spectroscopy and incident-photo-to-current-conversion efficiency = IPCE etc. In general, the performance of any solar cell device can be determined in terms of fill factor, PCE, photocurrent density and open circuit voltage. Photoluminescence spectroscopy can also be employed to check the lifetime of the generated electrons inside the perovskite materials which is related to the recombination processes. It has also been known that the PSCs devices with high electron lifetime and lower recombination reaction rates may provide better photovoltaic performance in terms of power conversion efficiency.

The PSCs device absorbs the sunlight which created the electron–hole pairs inside the perovskite light absorber layer. This electron has to be transferred to the conductive electrode surface. Thus, electron transport layer consists of transition metal oxides (generally TiO2 or SnO2) transport this generated electron to the conductive electrode (FTO). The hole can be collected by the hole transport material layers. In some cases, the transferred electron can recombine and influence the performance of the PSCs device. Thus, some buffer or compact layers have been used to reduce the recombination process. The perovskite materials used in such PSCs devices worked as light absorber.

### **3. Origin of PSCs**

The PSCs were originated in 2009 by Kojima et al. [17] and reported an interesting PCE of 3.1%. Further different approaches were utilized to improve the PCE of the PSCs. Recently, Tang et al. [32] prepared the low temperature processed zinc oxide nanowalls (ZnO NWs). Authors employed these prepared ZnO NWs as electron collection layer for the development of PSCs [32]. The morphological features of the prepared ZnO NWs were determined by scanning electron microscopy = SEM and transmission electron microscopy = TEM. The SEM and TEM results confirmed the formation of ZnO NWs. Further PSCs were constructed using ZnO NWs as electron collection layer whereas MAPbI3 as light absorber. The device architecture of the PSCs has been presented in **Figure 2a**. The energy level values of the perovskite light absorber, ZnO, indium doped tin oxide (ITO), spiro-OMeTAD and Ag have been displayed in **Figure 2b**.

The performance of the PSCs devices with ZnO NWs and ZnO thin films were evaluated by J-V analysis. The J-V curves of the PSCs developed with ZnO NWs and ZnO thin films have been depicted in **Figure 2c**. The constructed PSCs device with ZnO NWs exhibited the highest PCE of 13.6% whereas the PSCs developed with ZnO thin films showed the PCE of 11.3%. This showed that ZnO NWs plays crucial role in charge collection compare to the ZnO thin films. The NWs of ZnO collect the generated electron more efficiently compare to the ZnO thin films. Further, incident IPCE of the constructed PSCs was also checked. The IPCE curves of the PSCs developed with ZnO NWs and ZnO thin films have been presented in **Figure 2d**. The PSCs developed with ZnO NWs showed the highest open circuit voltage of 1000 mV whereas the PSCs device fabricated with ZnO thin films showed the open circuit voltage of 980 mV. The constructed PSCs with ZnO NWs exhibited the improved IPCE compared to the PSCs device developed with ZnO thin films.

In other work, Mahmud et al. [33] synthesized low-temperature processed ZnO thin film.

The optical properties of the prepared ZnO thin film were investigated by employing ultraviolet–visible (UV–vis) absorption spectroscopy. The Tauc plot of the ZnO has been presented in **Figure 3A**. The synthesized ZnO possess an optical

### *Recent Advances in Nanophotonics - Fundamentals and Applications*

### **Figure 2.**

*Schematic picture of PSCs device architecture (a). Energy level diagram of PSCs components (b). J-V graphs of the PSCs constructed with ZnO NWs and ZnO thin films (c). IPCE of the PSCs constructed with ZnO NWs and ZnO thin films (d). Reprinted with permission [32].*

band gap of 3.53 eV as confirmed by Tauc relation. The formation of ZnO on ITO glass substrate was confirmed by employing X-ray diffraction = XRD method. The XRD pattern of the ZnO has been presented in **Figure 3B**. The XRD pattern of the ZnO showed the crystalline nature with strong diffraction peaks. Authors employed ZnO thin film as electro transport layer for the construction of PSCs [33]. The MAPbI3 was utilized as light absorber layer. Authors also investigated the morphological features of the MAPbI3 films prepared on ZnO. The SEM results showed the presence of uniform surface morphology of the MAPbI3 perovskite [33]. Further, PSCs were fabricated and the device architecture of the fabricated PSCs has been depicted in **Figure 4A**.

The energy level diagram of the fabricated PSCs device has been presented in **Figure 4B**. The photovoltaic performance of the constructed PSCs device was evaluated by recording J-V curve. The obtained results showed that the fabricated PSCs device with ZnO thin film possess a highest PCE of 8.77% with open circuit voltage of 932 mV.

In 2017, Li et al. [34] synthesized ZnO/Zn2SnO4 under facile conditions. The synthesized ZnO/Zn2SnO4 was utilized as compact layer for the fabrication of MAPbI3 based PSCs. Authors recorded the XRD pattern of the MAPbI3 perovskite layer [34]. The XRD pattern of the MAPbI3 perovskite layer has been presented in **Figure 5a**.

The XRD pattern of the MAPbI3 perovskite layer showed the well-defined diffraction planes which suggested the successful formation of MAPbI3 perovskite as shown in **Figure 5a**. The formation of the ZnO/Zn2SnO4 was checked by XRD and X-ray photoelectron spectroscopy (XPS). The recorded XRD pattern of the ZnO/ Zn2SnO4 has been presented in **Figure 5b**. The XRD pattern showed the diffraction planes for the ZnO, Zn2SnO4 and SnO2.

**85**

**Figure 3.**

This confirmed the formation of ZnO/Zn2SnO4**.** Further, authors also investigated

the morphological characteristics of the ZnO/Zn2SnO4 using SEM analysis [34]. Authors employed ZnO/Zn2SnO4 as compact layer and developed the PSCs devices [34]. Authors also developed the PSCs using TiO2 with different thickness [34]. The performance of the developed PSCs devices were evaluated by J-V approach. The recorded J-V curves of the developed PSCs with different thicknesses (40 nm, 60 nm, 80 nm, 100 nm and 120 nm) of TiO2 have been presented in **Figure 6**. The PSCs device fabricated with TiO2 (thickness = 100 nm) exhibited the highest performance

*Tauc plot of the ZnO (A). XRD pattern of the ZnO/ITO (B). Reprinted with permission [33].*

compared to the PSCs device fabricated with TiO2 of different thicknesses.

*Origin and Fundamentals of Perovskite Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.94376*

*Recent Advances in Nanophotonics - Fundamentals and Applications*

band gap of 3.53 eV as confirmed by Tauc relation. The formation of ZnO on ITO glass substrate was confirmed by employing X-ray diffraction = XRD method. The XRD pattern of the ZnO has been presented in **Figure 3B**. The XRD pattern of the ZnO showed the crystalline nature with strong diffraction peaks. Authors employed ZnO thin film as electro transport layer for the construction of PSCs [33]. The MAPbI3 was utilized as light absorber layer. Authors also investigated the morphological features of the MAPbI3 films prepared on ZnO. The SEM results showed the presence of uniform surface morphology of the MAPbI3 perovskite [33]. Further, PSCs were fabricated and the device architecture of the fabricated PSCs has been

*Schematic picture of PSCs device architecture (a). Energy level diagram of PSCs components (b). J-V graphs of the PSCs constructed with ZnO NWs and ZnO thin films (c). IPCE of the PSCs constructed with ZnO NWs* 

The energy level diagram of the fabricated PSCs device has been presented in **Figure 4B**. The photovoltaic performance of the constructed PSCs device was evaluated by recording J-V curve. The obtained results showed that the fabricated PSCs device with ZnO thin film possess a highest PCE of 8.77% with open circuit

In 2017, Li et al. [34] synthesized ZnO/Zn2SnO4 under facile conditions. The synthesized ZnO/Zn2SnO4 was utilized as compact layer for the fabrication of MAPbI3 based PSCs. Authors recorded the XRD pattern of the MAPbI3 perovskite layer [34]. The XRD pattern of the MAPbI3 perovskite layer has been presented in **Figure 5a**. The XRD pattern of the MAPbI3 perovskite layer showed the well-defined diffraction planes which suggested the successful formation of MAPbI3 perovskite as shown in **Figure 5a**. The formation of the ZnO/Zn2SnO4 was checked by XRD and X-ray photoelectron spectroscopy (XPS). The recorded XRD pattern of the ZnO/ Zn2SnO4 has been presented in **Figure 5b**. The XRD pattern showed the diffraction

**84**

depicted in **Figure 4A**.

*and ZnO thin films (d). Reprinted with permission [32].*

**Figure 2.**

voltage of 932 mV.

planes for the ZnO, Zn2SnO4 and SnO2.

**Figure 3.** *Tauc plot of the ZnO (A). XRD pattern of the ZnO/ITO (B). Reprinted with permission [33].*

This confirmed the formation of ZnO/Zn2SnO4**.** Further, authors also investigated the morphological characteristics of the ZnO/Zn2SnO4 using SEM analysis [34]. Authors employed ZnO/Zn2SnO4 as compact layer and developed the PSCs devices [34]. Authors also developed the PSCs using TiO2 with different thickness [34]. The performance of the developed PSCs devices were evaluated by J-V approach. The recorded J-V curves of the developed PSCs with different thicknesses (40 nm, 60 nm, 80 nm, 100 nm and 120 nm) of TiO2 have been presented in **Figure 6**. The PSCs device fabricated with TiO2 (thickness = 100 nm) exhibited the highest performance compared to the PSCs device fabricated with TiO2 of different thicknesses.

**Figure 4.**

*Schematic diagram of the PSCs device (A). Energy level diagram of the PSCs (B). Reprinted with permission [33].*

**Figure 5.**

*XRD patterns of the MAPbI3 (a) and ZnO/ZSO CL (b). Reprinted with permission [34].*

Furthermore, the photovoltaic performance of the PSCs developed using ZnO/ Zn2SnO4 as compact layer with different thickness (15 nm, 35 nm, 55 nm, 75 nm and 95 nm) were also evaluated. The J-V curves of the PSCs developed using ZnO/

**87**

*Origin and Fundamentals of Perovskite Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.94376*

the poor performance [34].

**Figure 6.**

effective charge compact layer compared to the TiO2.

CH3NH3PbI3 perovskite light absorber layer.

Zn2SnO4 with different thickness (15 nm, 35 nm, 55 nm, 75 nm and 95 nm) has been presented in **Figure 7**. Authors found that the PSCs device fabricated with ZnO/ Zn2SnO4 (thickness = 15 nm) has poor photovoltaic parameters which resulted to

*J-V curves of PSCs based on different thickness of TiO2 CLs. Reprinted with permission [34].*

The PSCs device fabricated with ZnO/Zn2SnO4 (thickness = 75 nm) showed enhanced photovoltaic parameters which resulted to the improved photovoltaic performance (**Figure 7**). This showed that ZnO/Zn2SnO4 (thickness = 75 nm) is more

In another recent work, Chang et al. [35] developed the PSCs using Ce doped

In this work, Chang et al. [35] prepared the thin films of Ce doped CH3NH3PbI3 perovskite light absorber layer using a post treatment approach. Authors used CsI to promote the morphological features and crystallization of the thin films of Ce doped CH3NH3PbI3 perovskite light absorber layer. The use of Cs helps to obtain the large grain size of the CH3NH3PbI3 perovskite. The grain size of the CH3NH3PbI3 perovskite absorber layers were ranges 270 nm–650 nm. The formation of the perovskite light absorber layers were confirmed by XRD analysis. The optical band gap of the perovskite light absorber layer was also calculated by using Tauc relation. The Cs doped CH3NH3PbI3 perovskite light absorber has a band gap of 1.59 eV whereas this band gap slightly increases with increasing CsI concentrations. The increase in the optical band gap of the CH3NH3PbI3 perovskite absorber layer also

confirmed the insertion of Cs in to the perovskite light absorber layer.

The SEM pictures of the CH3NH3PbI3 perovskite light absorber layers were also recorded. The recorded SEM pictures of the CH3NH3PbI3 perovskite light absorber layers without and with CsI treatment have been presented in **Figure 8a**–**f**. The SEM picture of the CH3NH3PbI3 perovskite light absorber layer without CsI treatment showed the small grain size (**Figure 8a**). However, the insertion of CsI to the CH3NH3PbI3 perovskite light absorber layer increases the grain size as confirmed by the SEM investigations. The highly uniform surface morphology was observed in case of CH3NH3PbI3 perovskite absorber layer treated with 6mg mL−1 CsI (**Figure 8d**). Furthermore, PSCs devices were fabricated using CH3NH3PbI3 perovskite light absorber layers. The schematic picture of the developed PSCs device has been presented in **Figure 9**. The

### **Figure 6.**

*Recent Advances in Nanophotonics - Fundamentals and Applications*

Furthermore, the photovoltaic performance of the PSCs developed using ZnO/ Zn2SnO4 as compact layer with different thickness (15 nm, 35 nm, 55 nm, 75 nm and 95 nm) were also evaluated. The J-V curves of the PSCs developed using ZnO/

*XRD patterns of the MAPbI3 (a) and ZnO/ZSO CL (b). Reprinted with permission [34].*

*Schematic diagram of the PSCs device (A). Energy level diagram of the PSCs (B). Reprinted with* 

**86**

**Figure 5.**

**Figure 4.**

*permission [33].*

*J-V curves of PSCs based on different thickness of TiO2 CLs. Reprinted with permission [34].*

Zn2SnO4 with different thickness (15 nm, 35 nm, 55 nm, 75 nm and 95 nm) has been presented in **Figure 7**. Authors found that the PSCs device fabricated with ZnO/ Zn2SnO4 (thickness = 15 nm) has poor photovoltaic parameters which resulted to the poor performance [34].

The PSCs device fabricated with ZnO/Zn2SnO4 (thickness = 75 nm) showed enhanced photovoltaic parameters which resulted to the improved photovoltaic performance (**Figure 7**). This showed that ZnO/Zn2SnO4 (thickness = 75 nm) is more effective charge compact layer compared to the TiO2.

In another recent work, Chang et al. [35] developed the PSCs using Ce doped CH3NH3PbI3 perovskite light absorber layer.

In this work, Chang et al. [35] prepared the thin films of Ce doped CH3NH3PbI3 perovskite light absorber layer using a post treatment approach. Authors used CsI to promote the morphological features and crystallization of the thin films of Ce doped CH3NH3PbI3 perovskite light absorber layer. The use of Cs helps to obtain the large grain size of the CH3NH3PbI3 perovskite. The grain size of the CH3NH3PbI3 perovskite absorber layers were ranges 270 nm–650 nm. The formation of the perovskite light absorber layers were confirmed by XRD analysis. The optical band gap of the perovskite light absorber layer was also calculated by using Tauc relation.

The Cs doped CH3NH3PbI3 perovskite light absorber has a band gap of 1.59 eV whereas this band gap slightly increases with increasing CsI concentrations. The increase in the optical band gap of the CH3NH3PbI3 perovskite absorber layer also confirmed the insertion of Cs in to the perovskite light absorber layer.

The SEM pictures of the CH3NH3PbI3 perovskite light absorber layers were also recorded. The recorded SEM pictures of the CH3NH3PbI3 perovskite light absorber layers without and with CsI treatment have been presented in **Figure 8a**–**f**. The SEM picture of the CH3NH3PbI3 perovskite light absorber layer without CsI treatment showed the small grain size (**Figure 8a**). However, the insertion of CsI to the CH3NH3PbI3 perovskite light absorber layer increases the grain size as confirmed by the SEM investigations. The highly uniform surface morphology was observed in case of CH3NH3PbI3 perovskite absorber layer treated with 6mg mL−1 CsI (**Figure 8d**). Furthermore, PSCs devices were fabricated using CH3NH3PbI3 perovskite light absorber layers. The schematic picture of the developed PSCs device has been presented in **Figure 9**. The

### **Figure 8.**

*SEM pictures of the CH3NH3PbI3 thin films: untreated (a) and treated with 2.5 mg mL−1 (b), 5 mg mL−1 (c), 6 mg mL−1 (d), 7 mg mL−1 (e), 9 mg mL−1 CsI (f). Reprinted with permission [35].*

**89**

*Origin and Fundamentals of Perovskite Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.94376*

> **cm2 )**

**VOC (mV)**

MAPbI3 19.2 720 10.2 PSCs [36] perovskite 22.7 240 2.02 PSCs [37] MASnI3 16.8 880 6.4 PSCs [38] FASnI3 17.53 600 6.7 PSCs [39] FASnI3 24.1 520 9 PSCs [40] MASnI3 11.1 970 7.6 PSCs [41] FASn0.5Pb0.5I3 21.9 700 10.2 PSCs [42] MASn0.25Pb0.75 15.8 730 7.37 PSCs [43] Al3+doped CH3NH3PbI3 22.4 1001 19.1 PSCs [44] Dye 13.2 570 4.63 DSSCs [45] Dye 15.46 821 8.20 DSSCs [46] Polymer light absorber 20.65 946 14.45 Polymer [47] Polymer light absorber 19.1 990 11.5 Polymer [48] Perovskite quantum dot 15.1 1220 13.8 Quantum dot PSCs [49]

**PCE (%)**

**Type of solar cells References**

cells

21.8 940 14.8 Organic solar cells [51]

[50]

**Absorber layer JSC (mA/**

constructed PSCs device with CH3NH3PbI3 perovskite absorber layer (with 6 mg mL−1 CsI treatment) exhibited the best PCE of 14.4% with open circuit voltage of 1.05 V. However, the PSCs developed without CsI treatment showed the relatively lower PCE of 10.5%. There are different kinds of solar cells existed and each type of solar cell has different light absorbing materials. The photovoltaic performance of the PSCs has been

Quantum dot light absorber 26.70 780 13.84 Quantum dot solar

Since the origin, PSCs have received enormous attention due to their simple solution-processed fabrication, high performance and cost-effectiveness. This is because of the excellent optoelectronic properties of the organic–inorganic lead halide perovskite light absorbers. The PSCs achieved a highest power conversion efficiency of more than 24%. The PSCs can be employed for practical applications due to their high performance and cost-effectiveness. However, the poor aerobic stability and moisture sensitivity of the perovskite light absorbers restricts their practical applications. Thus, it is of great importance to overcome the issue of moisture sensitivity and poor stability of the PSCs. In previous years numerous strategies and methods were developed to enhance the stability of the PSCs. However, further

We believe that the following points/strategies would be beneficial to further

1.New device architectures are required to develop the highly efficient PSCs.

2.The photovoltaic parameters/performance of the PSCs can be further improved by utilizing/developing new electron transport/charge extraction layers.

improvements are necessary to commercialize the PSCs at large scale.

enhance the stability and photovoltaic performance of the PSCs:

compared with other reported solar cells as shown in **Table 1**.

*Comparison of the photovoltaic parameters of the PSCs with other reported solar cells.*

**4. Future prospective**

Organic light absorbing

material

**Table 1.**

**Figure 9.** *Schematic picture of the constructed PSCs device. Reprinted with permission [35].*


*Origin and Fundamentals of Perovskite Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.94376*

### **Table 1.**

*Recent Advances in Nanophotonics - Fundamentals and Applications*

*J-V curves of PSCs based on different thickness of ZnO/ZSO CLs. Reprinted with permission [34].*

*SEM pictures of the CH3NH3PbI3 thin films: untreated (a) and treated with 2.5 mg mL−1 (b), 5 mg mL−1 (c), 6* 

**88**

**Figure 9.**

**Figure 8.**

**Figure 7.**

*Schematic picture of the constructed PSCs device. Reprinted with permission [35].*

*mg mL−1 (d), 7 mg mL−1 (e), 9 mg mL−1 CsI (f). Reprinted with permission [35].*

*Comparison of the photovoltaic parameters of the PSCs with other reported solar cells.*

constructed PSCs device with CH3NH3PbI3 perovskite absorber layer (with 6 mg mL−1 CsI treatment) exhibited the best PCE of 14.4% with open circuit voltage of 1.05 V. However, the PSCs developed without CsI treatment showed the relatively lower PCE of 10.5%. There are different kinds of solar cells existed and each type of solar cell has different light absorbing materials. The photovoltaic performance of the PSCs has been compared with other reported solar cells as shown in **Table 1**.

### **4. Future prospective**

Since the origin, PSCs have received enormous attention due to their simple solution-processed fabrication, high performance and cost-effectiveness. This is because of the excellent optoelectronic properties of the organic–inorganic lead halide perovskite light absorbers. The PSCs achieved a highest power conversion efficiency of more than 24%. The PSCs can be employed for practical applications due to their high performance and cost-effectiveness. However, the poor aerobic stability and moisture sensitivity of the perovskite light absorbers restricts their practical applications. Thus, it is of great importance to overcome the issue of moisture sensitivity and poor stability of the PSCs. In previous years numerous strategies and methods were developed to enhance the stability of the PSCs. However, further improvements are necessary to commercialize the PSCs at large scale.

We believe that the following points/strategies would be beneficial to further enhance the stability and photovoltaic performance of the PSCs:


3.Some novel hydrophobic cationic groups should be introduced to the perovskite light absorbers to improve the aerobic stability and moisture sensitivity.

### **5. Conclusions**

In present scenario, energy crisis is the major challenge for today's world. Solar cells have the potential to overcome the issue of energy crisis. In last 10 years, PSCs have attracted the materials scientists due to its excellent photovoltaic performance and easy fabrication procedures. The highly efficient PSCs involve MAPbX3 as light absorber layer. The photovoltaic performance of the PSCs can be influenced by the presence of absorber layer or electron transport layer. Previously different kinds of electron transport layers have been widely studied to enhance the performance of the MAPbX3 based PSCs. The highest PCE of more than 24% has been certified by NREL for MAPbX3 based PSCs device. This excellent PCE is close to the commercialized silicon based solar cells. Thus, it can be said that PSCs can fulfill our energy requirements in the future. In this chapter, the fabrication of PSCs has been discussed. The recent advances in the development of PSCs with different compact layers, electron transport layers and charge collection layers have been reviewed.

### **Acknowledgements**

K.A. would like to acknowledge Discipline of Chemistry, IIT Indore. M.Q.K. acknowledged Department of Chemistry, Faculty of Applied Science and Humanities, Invertis University.

### **Conflict of interest**

"The authors declare no conflict of interest."

### **Author details**

Mohd Quasim Khan1 and Khursheed Ahmad2 \*

1 Department of Chemistry, Faculty of Applied Science and Humanities, Invertis University, Bareilly, 243123, U.P., India

2 Discipline of Chemistry, Indian Institute of Technology Indore, Simrol, Khandwa Road, 453552, M.P., India

\*Address all correspondence to: khursheed.energy@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**91**

*Origin and Fundamentals of Perovskite Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.94376*

[1] Reddy VS, Kaushik SC, Ranjan KR, Tyagi SK. State-of-the-art of solar thermal power plants. Renewable and Sustainable Energy Reviews.

[9] Tran VA, Truong TT, Phan TAP, Nguyen TN, Huynh TV, Agrestic A, et al. Application of nitrogen-doped TiO2 nano-tubes in dye-sensitized solar cells. Applied Surface Science.

[10] Deng J, Wang M, Fang J, Song X, Yang Z, Yuan Z. Synthesis of Zn-doped TiO2 nano-particles using metal Ti and Zn as raw materials and application in quantum dot sensitized solar cells. J. Alloy Compound. 2019;**791**:371-379

Natarajan K, Mobin SM. A two-step modified deposition method based (CH3NH3)3Bi2I9 perovskite: Lead free, highly stable and enhanced photovoltaic performance. ChemElectroChem.

Natarajan K, Mobin SM. Design and synthesis of 1D-polymeric chain based [(CH3NH3)3Bi2Cl9]n perovskite: A new light absorber material for lead free perovskite solar cells. ACS Applied Energy Materials. 2018;**01**:2405-2409

[13] Ahmad K, Mobin SM. Graphene oxide based planar heterojunction perovskite solar cell under ambient condition. New Journal of Chemistry.

2017;**399**:515-522

[11] Ahmad K, Ansari SN,

[12] Ahmad K, Ansari SN,

2019;**6**:1192-1198

2017;**41**:14253-14258

2019;**7**:6659-6664

[14] Ahmad K, Mohammad A, Mobin SM. Hydrothermally grown α-MnO2 nanorods as highly efficient low cost counter-electrode material for dye-sensitized solar cells and electrochemical sensing applications. Electrochimica Acta. 2017;**252**:549-557

[15] Zhong M, Liang Y, Zhang J, Wei Z, Li Q, Xu D. Highly efficient flexible MAPbI3 solar cells with a fullerene derivative-modified SnO2layer as the electron transport layer. Journal of Materials Chemistry A.

[2] Chen GY, Seo J, Yang CH, Prasad PN. Nanochemistry and nanomaterials for photovoltaics. Chemical Society

Hammed MG, Barakat NAM. Cd-doped TiO2 nanofibers as effective working electrode for the dye sensitized solar cells. Materials Letters.

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*Origin and Fundamentals of Perovskite Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.94376*

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*Recent Advances in Nanophotonics - Fundamentals and Applications*

3.Some novel hydrophobic cationic groups should be introduced to the perovskite light absorbers to improve the aerobic stability and moisture sensitivity.

In present scenario, energy crisis is the major challenge for today's world. Solar cells have the potential to overcome the issue of energy crisis. In last 10 years, PSCs have attracted the materials scientists due to its excellent photovoltaic performance and easy fabrication procedures. The highly efficient PSCs involve MAPbX3 as light absorber layer. The photovoltaic performance of the PSCs can be influenced by the presence of absorber layer or electron transport layer. Previously different kinds of electron transport layers have been widely studied to enhance the performance of the MAPbX3 based PSCs. The highest PCE of more than 24% has been certified by NREL for MAPbX3 based PSCs device. This excellent PCE is close to the commercialized silicon based solar cells. Thus, it can be said that PSCs can fulfill our energy requirements in the future. In this chapter, the fabrication of PSCs has been discussed. The recent advances in the development of PSCs with different compact layers, electron transport layers and charge collection layers have been reviewed.

**90**

**Author details**

**5. Conclusions**

Mohd Quasim Khan1

**Acknowledgements**

**Conflict of interest**

Humanities, Invertis University.

Road, 453552, M.P., India

University, Bareilly, 243123, U.P., India

provided the original work is properly cited.

and Khursheed Ahmad2

"The authors declare no conflict of interest."

\*Address all correspondence to: khursheed.energy@gmail.com

\*

1 Department of Chemistry, Faculty of Applied Science and Humanities, Invertis

K.A. would like to acknowledge Discipline of Chemistry, IIT Indore. M.Q.K.

acknowledged Department of Chemistry, Faculty of Applied Science and

2 Discipline of Chemistry, Indian Institute of Technology Indore, Simrol, Khandwa

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

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**95**

Section 3

Semiconductor Devices

Section 3
