Semiconductor Materials

## **Chapter 5**

## Temperature Dependence of Electrical Resistivity of ð Þ *III*, *Mn V* Diluted Magnetic Semiconductors

*Edosa Tasisa Jira*

## **Abstract**

In this work, a theory of temperature dependence of electrical resistivity is developed, with a particular emphasis on dilute magnetic semiconductors (DMSs). The approach is based on the equation of motion of the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction and considers both spin and charge disorder. The formalism is applied to the specific case of *Ga*<sup>1</sup>�*xMnxAs*.Using the RKKY exchange interaction, the relaxation time *τ* and the exchange interaction *J* are calculated. Then using spin-dependent relaxation time, electrical resistivity of the material is calculated. The electrical resistivity of *Mn*-doped III—V DMS is decreased with increasing temperature and magnetic impurity concentration.

**Keywords:** RKKY interactions, electrical resistivity, spin-dependent relaxation time

## **1. Introduction**

The motivation for this study has been to better understand semiconductors, which could be the building block for the next generation of computers. Semiconductors have been the subject of very extensive research over recent decades, because of ever more numerous and powerful applications. As conventional electronics, semiconductors are more appropriate than normal metals, since metal devices do not amplify current. Because of the complementary properties of semiconductor and ferromagnetic material systems, a growing effort is directed toward studies of semiconductor-magnetic nanostructures [1, 2].

Semiconductors and magnetic materials both play essential roles in modern electronic industry. Since the applications of semiconductors and magnetic have evolved independently, it appears logical to combine their properties for possible spinelectronic applications with increased functionalities [3]. This is known as spintronics, which utilizes both charge and spin of the electrons to process and store data. This is to control the spin degree of freedom of electrons in semiconductors. The direct method to introduce spin degrees of freedom in semiconductors is to introduce magnetic ions into semiconductors. Such semiconductors are referred to as diluted magnetic semiconductors (DMSs) [4]. The curie temperature of the DMS material and whether the ferromagnetism in the DMS material originates from free carrier mediation or purely from localized magnetic dopants limit DMS practical applications for devices [5].

For practical applications, increasing *TC* beyond room temperature is a necessity. Thus, exploring various roots of raising the value of *TC* is a great fundamental and practical significance. Carrier-mediated ferromagnetism in p-type II-VI DMS heterostructures occurs only at very low temperatures *TC* typically below 2*:*0*K*. Therefore, very significant strides have been made in developing ferromagnetic-based *III* � *V* semiconductors containing *Mn*, which remaining ferromagnetic to much higher *TC* [3, 6]. A discovery of ferromagnetic ordering in III–V DMSs with critical temperatures of annealed *Ga*1�*xMnxAs* could exceed 110 k; reaching �191 K has renewed and greatly intensified an interest in those materials, but still too low for actual applications. This was, at least partially, related to expectations that their Curie temperatures can be relatively easily brought to room temperature range through a clearly delineated path [7–9].

For the ferromagnetic transition to occur in *Ga*<sup>1</sup>�*xMnxAs*, the concentration of Mn impurities should be relatively high, x = 0.01–0.05. Such high concentration of impurities makes the study of the DMS more challenging. Moreover, the issue of solubility of the magnetic ions into the regular semiconductor was a major challenge to fabricate high-quality DMSs. In recent years, Molecular Beam Epitaxy techniques have led to the successful growth of various DMS including ð Þ *Ga*, *Mn As*. In order to prevent phase separation order, magnetic *Ga*<sup>1</sup>�*xMnxAs* should be grown at low temperatures (T; 200–300°C). These problems are not specific to *Ga*<sup>1</sup>�*xMnxAs*, in fact, all presently known DMS materials suffer from the same problems. As a result, theoretical study of DMS electrical properties such as resistivity is very difficult. Since these properties are influenced by the exchange interaction between the carriers and the localized moments, it must be taken into account non-perturbatively [10, 11]. Due to these influences, spin fluctuation scattering contributes to the resistivity. The experimentally measured dc resistivity in the DMS materials shows interesting behavior strongly depending on the concentration of the magnetic impurity and temperature. In (Ga,Mn)As, there is a metal–insulator transition between insulating samples with small manganese concentration and metallic samples with larger concentration. In and with low concentration (x < 0.03), only an insulating behavior has been observed in transport measurements. Insulating behavior is here characterized by a diverging resistivity for T ! 0, indicating localization of carriers. However, near-optimal doping x = 0.05, where the highest value of is reported, the nonmonotonic behavior of insulator–metal–insulator of the resistivity as a function of temperature is observed. A resistivity peak appears near the critical temperature, and the resistivity shows metallic behavior below and insulating behavior at higher temperatures. In metallic samples, the resistivity decreases and eventually saturates for T ! 0. The peak has been understood as the critical scattering effects of spin fluctuations [11–13]. In this paper, a Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction was used for the calculation of the resistivity. This paper focuses on the group *III* � *Mn* � *V* ferromagnetic semiconductors, more specifically *Ga*<sup>1</sup>�*xMnxAs*. Even though we focus on III–V compound based DMS such as *Ga*<sup>1</sup>�*xMnxAs*, the results presented in this paper are general for all DMS materials.

### **2. Theoretical formalism**

#### **2.1 RKKY interaction approaches to p-d exchange interaction**

The carrier-mediated spin–spin coupling is usually described in terms of the Ruderman-Kittel-Kasuya-Yosida (RKKY) model, which provides the energy *Jij* of exchange coupling. When magnetic impurity *Mn* is introduced into semiconductors, *Temperature Dependence of Electrical Resistivity of* (III, Mn) V *Diluted Magnetic Semiconductors DOI: http://dx.doi.org/10.5772/intechopen.103046*

spin flip processes become possible. The spin of the scattered electron and the spin of magnetic impurities are flipped simultaneously while the component of the total spin along the quantization axis is conserved. The interaction of the impurity with conduction electron is described as p-d or s-d interaction. The spin density of electrons determines this interaction at the localized moment. Like in the Heisenberg model, the scalar product of two spins is taking with the exchange coupling *J* [14, 15]. Writing the spin density of electrons in terms of the field operators, the interaction between the spin *S <sup>j</sup>* at *rj* and the conduction electron is

$$H\_{p-d} = -\mathfrak{Z}\mathfrak{S}\_j\mathfrak{S}\_i(r\_j) = -f\sum\_{a\beta} \mathfrak{S}\_j \mathfrak{L}\_{a\beta} \left(\hat{\boldsymbol{\psi}}^+(\boldsymbol{r}\_j)\hat{\boldsymbol{\psi}}\_{\beta}(\boldsymbol{r}\_j)\right) \tag{1}$$

Where *ψ*^ is the field operator states expanded in terms of the block states. Using the field operators, the Fourier transform of number density operator *n r*ð Þ becomes

$$n(q) = \int n(r)e^{-q \cdot r} dr = \frac{1}{V} \sum\_{kk'\sigma} \int e^{-i\left(k+q-k'\right) \cdot r} u\_k^\*(r) u\_{k'}(r) c\_{k\sigma}^+ c\_{k'\sigma} dr \tag{2}$$

Where *ck<sup>σ</sup>* and *c* þ *<sup>k</sup><sup>σ</sup>* are the annihilation and creation operators of electrons, respectively. The integral in Eq. (2) is vanished unless *k*<sup>0</sup> ¼ *k* þ *q* þ *G* due to the lattice periodicity of the function s *uk*ð Þ*r* . Where G is the reciprocal lattice vectors. The coupling strength depends on state *k*<sup>0</sup> , the state in which the electron of wave vector *k* is scattered. If we express Eq. (2) in terms of the spin density and the system contains *Ni* localized spins, the interaction with conduction electrons is determined by

$$H\_{p-d} = -\frac{J}{V} \sum\_{l=1}^{N\_i} \sum\_{k,k'} \sum\_{k,k'} e^{i\left(k-k'\right)r} \left\{ \mathbf{S}\_l^+ c\_{k'\downarrow}^+ c\_{k\uparrow} + \mathbf{S}\_l^- c\_{k'\uparrow}^+ c\_{k\downarrow} + \mathbf{S}\_l^x \left( c\_{k'\uparrow}^+ c\_{k\uparrow} + c\_{k'\downarrow}^+ c\_{k\downarrow} \right) \right\} \tag{3}$$

The interaction of localized spin *Mn* ions with electron gives rise to the creation or annihilation of an electron–hole pair. By considering elastic transitions and the common energy *Ek* ¼ *Ek*<sup>0</sup> denoted by *E*0, the Hamiltonian is written as

$$
\langle \mathbf{k'} | H\_{\rm eff} | k \rangle = -\sum\_{j} \frac{\langle \mathbf{k'} | \lambda H\_{\rm eff} | j \rangle \langle j | \lambda H\_{\rm eff} | k \rangle}{E\_j - E\_0} \tag{4}
$$

Where j i*k* is ground state, ∣*k*<sup>0</sup> i is the state of the scattering of electron, λ is the coupling constant, and *Heff* is the effective Hamiltonian, which is equal to *Hp*�*<sup>d</sup>*. Since the flip of the impurity spin is accompanied by the flip of an electron spin, the collision term of distribution function of spin-up and spin-down electrons is

$$\begin{aligned} \left(\frac{\partial f(k)}{\partial t}\right)\_{\text{coll}} &= \sum\_{k'} W\_{k\uparrow,k'\uparrow} \left\{ f \nkern-1.5mu \left[ \begin{aligned} f \nwarrow &k' \end{aligned} \right] \left[ \mathbf{1} - f \nkern-1.5mu \left[ \begin{aligned} f \nwarrow &k \end{aligned} \right] - f \nkern-1.5mu \left[ \begin{aligned} f \nwarrow &k \end{aligned} \right] \right\} \\ &+ \sum\_{k'} W\_{k\uparrow,k'\downarrow} \left\{ f \nwarrow &k \end{aligned} \left[ \begin{aligned} f \nwarrow &k \end{aligned} \right] \left[ \mathbf{1} - f \nwarrow &k \right] \left[ \mathbf{1} - f \nwarrow &k' \end{aligned} \right] \end{aligned} \tag{5}$$

Where *Wk*,*k*<sup>0</sup> is the transition probability, *f k*ð Þ and *f k*<sup>0</sup> ð Þ are the distribution functions of the initial states and the scattered states, respectively. The transition probability between two states *α* and *β* is given by

$$\mathcal{W}\_{a\beta} = \frac{2\pi}{\hbar} \left| \langle a | H\_{p-d} | \beta \rangle \right|^2 \delta \left( E\_a - E\_\beta \right) \tag{6}$$

Where ∣*α*i is the eigen state of the initial system, and *E<sup>α</sup>* is the corresponding eigen values, ∣*β*i is the final eigen states, and *E<sup>β</sup>* is the corresponding eigen values, respectively. If the spin is flipped in the intermediate state, an electron of quantum numbers *M*00 *<sup>l</sup>* appears in the first process, and the impurity spin goes over from the initial state *<sup>S</sup><sup>z</sup>* <sup>¼</sup> *<sup>M</sup>* to *<sup>S</sup><sup>z</sup>* <sup>¼</sup> *<sup>M</sup>* <sup>þ</sup> 1. Where the electron–hole pair is composed of a spin-up electron and spin-down hole, the impurity spin has to flip down, to the state *Sz* <sup>¼</sup> *<sup>M</sup>* � 1. Then by substituting state *α* and *β* and using the distribution function for the spin of the electron–hole pair, the total contribution is

$$
\langle f|H\_{\text{eff}}|i\rangle = -\left(\frac{I}{N}\right)^2 \sum\_{\vec{\imath}\vec{\jmath}} \frac{f\_0(\mathbf{E}\_k)[\mathbf{1} - f\_0(\mathbf{E}\_{k'})]}{E\_{\mathbf{k}}\,^\prime - E\_{\mathbf{k}}} e^{i\left(\mathbf{k} - \mathbf{k}'\right) \cdot \left(r\_{\vec{\imath}} - r\_{\vec{\imath}}\right)} \left\{ 2\langle \{\mathbf{M}\_l^\dagger\}|\mathbf{S}\_l^\dagger\mathbf{S}\_j^\mathbf{r} + \frac{1}{2}\left(\mathbf{S}\_l^+\mathbf{S}\_j^- + \mathbf{S}\_l^-\mathbf{S}\_j^+\right)|\mathbf{M}\_l\rangle \right\} \tag{7}
$$

Where *M*<sup>00</sup> f g*<sup>l</sup>* j i is the state of the spins of the z-component of the intermediate states, and *M*<sup>00</sup> *<sup>l</sup>* is the corresponding eigen values, *M*<sup>0</sup> *l* �� � � � is the state of the spins of the z-component of the scattered states, and *M*<sup>0</sup> *<sup>l</sup>* is the corresponding eigen value, and f g M*<sup>l</sup>* j i is the state of the spins of initial state, and *Ml* is the corresponding eigen value.

Eq. (7) can be considered as the matrix element of the operator *H*, when we write the exchange energy between *Si* and *S <sup>j</sup>*, which is written as

$$H = -\sum\_{\vec{\eta}} J(r\_i - r\_j) \mathbf{S}\_i \mathbf{S}\_j \tag{8}$$

To determine the exchange coupling strength *J ri* � *rj* � �, we introduce *<sup>r</sup>* <sup>¼</sup> *ri* � *rj* where **r** is the displacement vector between the ions i and j, and using the following integral notation *I* as

$$I = (\mathbf{1}/\mathcal{V})^2 \sum\_{kk'} \frac{f\_0(E\_k) \left[\mathbf{1} - f\_0(E\_{k'})\right]}{E\_k - E\_k} e^{i\left(k - k'\right) \cdot r} \tag{9}$$

Replacing the sum in Eq. (9) by an integral, using the quadratic dispersion relation valid for free electrons and the notations *κ* ¼ *kr*, *κ*<sup>0</sup> ¼ *k*<sup>0</sup> *<sup>r</sup>*, the parabolic energy *Ek* <sup>¼</sup> *<sup>ћ</sup>*2*k*<sup>2</sup> <sup>2</sup>*m*<sup>∗</sup> , in which *<sup>m</sup>*<sup>∗</sup> is the effective mass, and trigonometric identity *sin <sup>κ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>iκ*<sup>0</sup> �*e*�*iκ*<sup>0</sup> <sup>2</sup>*<sup>i</sup>* and the complex variable *κ*<sup>0</sup> þ *iη* instead of *κ*<sup>0</sup> , where *η* is an infinitesimal quantity, the angular integral becomes

$$I = -\frac{m\_e k\_{F\bar{f}^2}^4}{\hbar^2 \pi^3} \frac{\sin 2k\_F r - 2k\_F r \cos 2k\_F r}{\left(2k\_F r\right)^4} \tag{10}$$

Then the effective exchange interaction or the coupling exchange interaction constant between magnetic impurities and delocalized charge carriers is given by

$$J(r) = \frac{m\_e l^2 k\_F^4}{\hbar^2 \pi^3} F(2k\_F r) \tag{11}$$

*Temperature Dependence of Electrical Resistivity of* (III, Mn) V *Diluted Magnetic Semiconductors DOI: http://dx.doi.org/10.5772/intechopen.103046*

Where *kF* is the Fermi wave vector and the function *F x*ð Þ¼ *xcosx*�*sinx <sup>x</sup>* , *x* ¼ 2*kFr:* Eq. (11) gives the well-known RKKY interaction. As the value of the oscillatory function *F*ð Þ 2*kFr* varies, the value of the exchange interaction is either positive or negative, and hence, the interaction in the system can be ferromagnetic or antiferromagnetic.

By using Eq. (11), we can determine the relaxation time *τ* of the impurity spin of electrons. For spin conserving scattering, the matrix element of Eqs. (7) and (8) is

$$
\langle k' | H\_{\text{eff}} | k \rangle = -\frac{J}{V} \mathcal{S}^{\mathfrak{c}} \tag{12}
$$

A better approximation can be obtained for the transition probability by replacing the matrix element of the scattering matrix *T* in Eq. (6). Then Eq. (6) is rewritten as

$$\mathcal{W}\_{a\beta} = \frac{2\pi}{\hbar} \left| \langle a|T|\beta \rangle \right|^2 \delta(E\_a - E\_\beta) \tag{13}$$

The scattering (transition) probability in Eq. (13) can be expressed in terms of the interaction Hamiltonian up to third order in the coupling constant. Then using the second-order correction to the *T* matrix, the Fermi-Dirac distribution function, which is expressed by *f* <sup>0</sup>ð Þ¼ *k c* þ *<sup>k</sup> ck* � � and if the z-component of the impurity spin is unchanged by the scattering, the combined contribution is

$$\left(-\frac{J}{V}\right)^2 \sum\_{k''} (\mathbf{S}^\mathbf{z})^2 \frac{\left(\mathbf{1} - f\_0\left(\mathbf{k''}\right)\right)}{E\_k - E\_{k''}} - \frac{f\_0\left(\mathbf{k''}\right)}{E\_{k''} - E\_{k'}} = \left(-\frac{J}{N}\right)^2 (\mathbf{S}^\mathbf{z})^2 \sum\_{k''} \frac{\mathbf{1}}{E\_k - E\_{k'}}\tag{14}$$

If the impurity spin goes over from the initial state *<sup>S</sup><sup>z</sup>* <sup>¼</sup> *<sup>M</sup>* to *<sup>S</sup><sup>z</sup>* <sup>¼</sup> *<sup>M</sup>* <sup>þ</sup> 1, the electron–hole pair is composed of a spin-up electron and a spin-down hole (i.e., the impurity spin has to flip down) to the state *<sup>S</sup><sup>z</sup>* <sup>¼</sup> *<sup>M</sup>* to *<sup>S</sup><sup>z</sup>* <sup>¼</sup> *<sup>M</sup>* � 1, and on the account of conservation of energy *Ek* ¼ *Ek*0, then the combined contribution of these processes gives

$$\left(-\frac{J}{V}\right)^2 \sum\_{k''} \left\{ \left[ S(\mathcal{S}+1) - M^2 \right] \frac{1}{E\_{k-}E\_{k''}} - M \frac{1 - 2f\_0(k'')}{E\_k - E\_{k''}} \right\} \tag{15}$$

The first term in Eq. (15) is negligible since it is small; however, the second term is no longer small. The second term diverges logarithmically. Using this form, the transition probability Eq. (13) in *J* by summing over the possible spin orientations is given by

$$\mathcal{W}(k\uparrow,k'\uparrow) = \mathcal{W}(k\uparrow,k'\downarrow) = \frac{N\_i 2\pi f^2 \mathcal{S}(\mathcal{S}+1)}{\mathfrak{H}} [1+4\mathfrak{J}(E\_k)] \delta(E\_k - E\_{k'}) \tag{16}$$

Where *Ni* is the concentration of magnetic impurities, which is *Ni* <sup>¼</sup> <sup>4</sup>*x=a*<sup>3</sup> *lc* where *x* is the concentrations of the impurity added to the semiconductors, *alc* is lattice constant, and the singular function in the third order correction is

$$\log(E\_k) = \frac{1}{V} \sum\_{k''} \frac{f\_0\left(k''\right)}{E\_k - E\_{k''}}\tag{17}$$

We shall investigate the dependence of Eq. (16) on the energy of the initial state *EK*, which is entirely involved in the function *g E*ð Þ*<sup>k</sup>* . At the absolute zero of temperature, *<sup>f</sup>* <sup>0</sup> *<sup>k</sup>*<sup>00</sup> � � can be replaced by a step function, which is unity when *<sup>k</sup>*<sup>00</sup> <sup>&</sup>lt; *<sup>k</sup>*<sup>0</sup> and zero when *k*<sup>00</sup> >*k*0, where *k*<sup>0</sup> is the magnitude of the Fermi momentum.

By using Fermi-Dirac distribution, *Ek* <sup>¼</sup> *<sup>ћ</sup>*2*k*<sup>2</sup> <sup>2</sup>*m*<sup>∗</sup> and for electron close to the Fermi surface at low temperature, the singular function becomes

$$\log(E\_k) = \left(\frac{\mathfrak{B}\mathfrak{z}}{2E\_F}\right) \{1 + (k/2k\_0)\log|(k - k\_0)/(k - k\_0)|\}\tag{18}$$

Where *z* is the number of conduction electrons per atom. Since at T 6¼ 0, the average of j j *k* � *k*<sup>0</sup> for thermally excited electrons is proportional to T, even at this stage of calculation we can expect a term proportional to log *T* in the expression of the resistivity. It is true that *g E*ð Þ*<sup>k</sup>* diverges when *Ek* approaches to *EF*. Eq. (18) makes the calculation easier to retain the definition of *g E*ð Þ*<sup>k</sup>* here and to carry out the summation after we get the expression for the resistivity.

Then substituting Eq. (16) into collision integral, the relaxation time *τ* is the inverse of the transition probability. Therefore, the spin-dependent relaxation time is

$$\frac{1}{\tau} = \frac{N\_i 2\pi f^2 S(S+1)}{3\hbar E\_F} \left[1 + 4\text{Jg}(E\_k)\right] \tag{19}$$

The Fermi energy *EF* is given by *EF* <sup>¼</sup> *<sup>ћ</sup>*2*k*<sup>2</sup> <sup>2</sup>*m*<sup>∗</sup> <sup>¼</sup> *<sup>ћ</sup>*<sup>2</sup> <sup>2</sup>*m*<sup>∗</sup> 3*π*<sup>2</sup> *<sup>N</sup> V* � � � � <sup>2</sup> 3 *:* In *III* � *V* semiconductors, it is found that the maximum and minimum energy locations of the Fermi level typically do not deviate by more than 1ev. In the specific case of *GaAs*, the conduction band is located at *EFS* þ 0*:*9*eV* and the valence band at *EFS* � 0*:*5*eV* [3]. Where *EFS* is the Fermi-level stabilization energy, and it is found to be located at � 4*:*9*eV* below the vacuum level and is the same for all III-V and II-VI SCs. since *Mn*-doped *GaAs* is a p-type, we use Fermi level energy of the valence band. By using this value of Fermi energy and the values of other constant, we can calculate *τ*0.

In the presence of external perturbation due to the electric field or temperature gradient, there is a variation in the function *f* � *<sup>k</sup>* . Supposing the electric field lies along the z-direction, then using Eq. (19), the rate change of the probability *f* � *<sup>k</sup>* with which the state *K*� is occupied due to the collision with the localized spins becomes

$$
\left(\frac{\partial f\_k^{\pm}}{\partial t}\right)\_{coll} = -\frac{\left(f\_k^{\pm} - f\_k^0\right)}{\tau} \tag{20}
$$

Then from standard theory, we can easily obtain the resistivity as

$$\rho\_{spin} = \mathbf{x} \rho\_0 \left\{ \mathbf{1} - \frac{\left(\hbar^2 j\right)}{\pi m\_h \, \prescript{}{}{k}\_0} \right\} \left\{ g(E\_k) \left( \frac{d f^0}{d E\_k} \right) d^3 k \right\} \tag{21}$$

Where *ρ*<sup>0</sup> is defined by

$$\rho\_0 = 3\pi m\_h \, ^\ast f^2 \text{S} (\text{S} + \text{1}) (V/\text{N}) / 2e^2 \hbar E\_F \tag{22}$$

*Temperature Dependence of Electrical Resistivity of* (III, Mn) V *Diluted Magnetic Semiconductors DOI: http://dx.doi.org/10.5772/intechopen.103046*

Where V is the volume of the crystal. Then using some techniques, the total electrical resistivity becomes

$$
\rho\_{\text{spin}} = \text{x} \rho\_0 \{ 1 - (\text{Zg} / \text{E}\_F) \log T \} \tag{23}
$$

Since substitutional *Mn* ions in *GaAs* (i.e., *Ga*1�*xMnxAs*) are a p-type semiconductor, we use the hole concentration *<sup>p</sup>* <sup>¼</sup> *<sup>N</sup> <sup>V</sup>*, instead of *n*. The hole concentration *p* would ideally to be given by *<sup>p</sup>* <sup>¼</sup> *xN*<sup>0</sup> where *<sup>N</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*:*2*x*1022*cm*�<sup>3</sup> is the concentration of *Ga* sites in *GaAs* and *x* is the concentration of impurity [3]. For ð Þ *Ga*, *Mn As*, the hole effective mass *m*<sup>∗</sup> *<sup>h</sup>* ¼ 0*:*5*m*0, where *m*<sup>0</sup> ¼ *me* is the electron mass [16].

### **3. Results and discussion**

As mentioned in Section 2, the temperature dependence of electrical resistivity is calculated. The electrical resistivity of p-type *Mn*-doped *GaAs* has been investigated theoretically. This theoretical calculation is done using RKKY interaction models. This theoretical calculation is somewhat similar to experimental work, which is done on this material. In this paper, we use RKKY to calculate the exchange interaction constant and spin-dependent relaxation time approximation. Using these exchange interaction constant and spin-dependent relaxation time, the electrical resistivity of this material is calculated*:*

Electrical resistivity is the inverse of electrical conductivity of the material. The electrical resistivity of semiconductors decreases exponentially with increasing temperature in contrast to that of pure metals. The electrical resistivity of extrinsic semiconductor is decreased with increasing both the concentration of magnetic impurity and the temperature. From this concept the electrical resistivity of DMSs, in

**Figure 1.** *Electrical resistivity of Ga*0*:*92*Mn*0*:*08*As versus temperature.*

which magnetic impurities are incorporated into standard semiconductors, is decreased with temperature and concentration of magnetic ions. The temperature dependence of resistivity in *Ga*<sup>1</sup>*xMnxAs* for 0.08≤*x*≥0*:*01 is shown in **Figure 1**. From Eq. (23) the electrical resistivity is inversely proportional to the hole density or hole concentration (i.e., *ρ*∝ <sup>1</sup> *<sup>p</sup>*). Since *Ga*<sup>1</sup>*xMnxAs* is an extrinsic semiconductor, its hole concentration is increased with adding the impurity. Then the electrical resistivity decreases with increasing concentration. From **Figure 1** the electrical resistivity of *Ga*<sup>1</sup>*xMnxAs* for x = 0.035, x = 0.05, and x = 0.08 is drawn. From the graph as the magnetic impurity concentration is increased, its resistivity decreases. As the temperature and hole concentration increase, the electrical resistivity is decreased. At x = 0.08 or at high concentration, there is low resistivity and at x = 0.035, the resistivity is high.

### **4. Conclusion**

In this paper, we studied the theoretical temperature dependence of electrical resistivity of DMS specifically *Ga*<sup>1</sup>*xMnxAs:Ga*<sup>1</sup>*xMnxAs* has become the most understood and extensively studied *III*<sup>1</sup>*xMnxV* ferromagnetic semiconductor. DMSs show interesting electrical properties, which strongly depend on the concentration of magnetic dopants and temperature. These properties of *Ga*<sup>1</sup>*xMnxAs* is affected by concentration and temperature. Since the electrical properties are dependent on the density of states, the impurity added to the semiconductor increases this density of state and decreases energy gap. The electrical resistivity of *Ga*<sup>1</sup>*xMnxAs* is exponentially decreased with an increase of temperature from 0*K* 300*K* and concentration of magnetic ions in the range of 0*:*08≤*x*≥0*:*01. In general, if *TC* of the DMSs can be increased, there is a possibility of utilizing the system under consideration for the spintronic purpose at room temperature.

### **Acknowledgements**

First of all, I would like to thank the almighty God for keeping me safe and continually blessing me in all aspects of my life. I extend my heartful thanks to my parents for their encouragement and support both morally and psychologically.

### **Conflicts of interest**

The authors declare no conflicts of interest.

## **Data availability**

No data were used to support this study.

*Temperature Dependence of Electrical Resistivity of* (III, Mn) V *Diluted Magnetic Semiconductors DOI: http://dx.doi.org/10.5772/intechopen.103046*

## **Author details**

Edosa Tasisa Jira Department of Physics, College of Natural and Computational Sciences, Mekdela Amba University, Dessie, Ethiopia

\*Address all correspondence to: edosatasisa17@gmail.com

© 2022 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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## **Chapter 6**

## Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D and D–T Fusion Neutrons

*Jean-Luc Autran and Daniela Munteanu*

## **Abstract**

This work focuses on the radiation response of Group IV (Si, Ge, SiC, diamond) and III-V (GaAs, GaN, GaP, GaSb, InAs, InP, InSb, AlAs) semiconductors subjected to D–D (2.45 MeV) and D–T (14 MeV) neutrons. The response of each material has been systematically investigated through a direct calculation using nuclear cross-section libraries, MCNP6, and Geant4 numerical simulations. For the semiconductor materials considered, we have investigated in detail the reaction rates per type of reaction (elastic, inelastic, and nonelastic) and proposed an exhaustive classification and counting of all the neutron-induced events and secondary products as a function of their nature and energy. Several metrics for quantifying the susceptibility of the related semiconductor-based electronics to neutron fusions have been finally considered and discussed.

**Keywords:** radiation effects, III-V compound semiconductors, fusion neutrons, D-D reaction, D-T reaction, numerical simulation, neutron cross section, elastic scattering, inelastic scattering, nonelastic interactions, nuclear data library, Geant4, MCNP, single-event effect

## **1. Introduction**

At the horizon of 2035 and beyond, the very large electronic equipment for command and diagnostic operations of future fusion power machines, such as ITER, the international thermonuclear experimental reactor [1], will embrace a wide variety of semiconductors materials. Indeed, new materials, such as SiC, Ge, GaN, or GaAs, are envisaged to progressively replace the traditional silicon of microelectronics for high temperature, high performance, or high-speed electronics/optoelectronics applications [2–5]. These equipment will be directly or partially (behind radiation shielding) exposed to a complex radiation field due to primary fusion neutrons created in the core of the plasma and then transported outside the machine [6]. These neutrons are produced in the deuterium–tritium (D–T) reaction that has been identified as the most efficient for fusion devices [7]:

$$\text{D} + \text{T} \rightarrow \, ^4\text{He} \,(\text{3.52MeV}) + \text{n} \,(\text{14.07MeV}) \tag{1}$$

However, one must also consider deuterium–deuterium (D–D) reaction because this less efficient fusion reaction will be used to study, during the development and commissioning of future machines, their operation before introducing tritium fuel. D–D fusion following the two main reactions with equal probability of occurrence:

$$\text{D} + \text{D} \rightarrow \, ^3\text{He} \,(0.82 \text{MeV}) + \text{n} (2.45 \text{MeV}) \tag{2}$$

$$\mathbf{D} + \mathbf{D} \rightarrow \mathbf{T} \,(\mathbf{1}.01\mathbf{MeV}) + \mathbf{p} \,(\mathbf{3}.02\mathbf{MeV})\tag{3}$$

For future fusion power machines, the radiative reliability issue for the envisaged new material-based electronics arises now, even if—(i) most of these components have not yet been manufactured and tested, and (ii) the design and the integration of machine electronics equipment will not take place for many years. In such a long-term development, numerical simulation offers a very promising solution to anticipate, in the first step at the material level, this reliability issue for semiconductors whose radiative response is little or poorly known.

In the direct continuation of our previous works [8–10] that investigated the atmospheric radiation response of a wide variety of semiconductors, the present chapter focuses on the radiation response of Group IV (Si, Ge, SiC, diamond) and III-V (GaAs, GaN, GaP, GaSb, InAs, InP, InSb, AlAs) semiconductors subjected to D–D (2.45 MeV) and D–T (14 MeV) neutrons.

In this work, the response of each material has been systematically investigated through a direct calculation using nuclear cross-section libraries, MCNP6, and Geant4 numerical simulations. In the following, for the 12 semiconductor materials considered, we will investigate in detail the reaction rates per type of reaction (elastic, inelastic, and nonelastic) and propose an exhaustive classification of all the neutroninduced secondary products as a function of their nature and distribution in the energy. Implications for quantifying the susceptibility of these related semiconductorbased electronics to neutron fusions will be finally presented and discussed.

### **2. Material properties**

**Table 1** summarizes the natural isotopic abundance of the different chemical elements entering the composition of the studied semiconductor materials. Nitrogen, phosphorus, and arsenic have only a single natural isotope which simplifies the calculations to come for these elements. On the contrary, the other elements have two (C, N, Ga, In, Sb), three (Si), and up to five (Ge) different natural isotopes. Nevertheless, for C and N, the relative weight of the minority isotope is less than or around the percent, so we can consider that the natural carbon and nitrogen consist only of the simple isotopes 12C and 14N, respectively, with a very good approximation. The other isotopes listed in **Table 1** must be considered with their respective weight in atomic composition for correctly simulating the neutron response of the natural materials to which they relate.

**Table 2** reports the value of the energy bandgap, the number of atoms per square centimeter, the density, and the value of the energy for electron-hole pair creation in bulk material for the 12 semiconductors studied in this work. Due to their atomic composition and crystallographic structure, all these materials are denser than Si (2.33 g/cm<sup>3</sup> ) and three materials exhibit a much larger number of atoms per square centimeter than Si—SiC, diamond, and GaN. This can have a direct consequence on the


*Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

#### **Table 1.**

*Natural abundance of nuclides related to the materials studied in this work. Z refers to the atomic number, A to the atomic mass.*


**Table 2.** *Main properties of the semiconductors considered in this study. From Ref. [8, 10].* number of neutron interactions per unit volume of the material, as discussed in the following. In addition, seven materials (SiC, diamond, GaAs, AlAs, InP, GaN, and GaP) exhibit a bandgap larger than Si and logically, a larger electron-hole pair creation energy (3.6 eV for Si) [10], four materials (Ge, InAs, GaSb, and InSb) are lowbandgap semiconductors (< 1eV) with lower electron-hole pair creation energy than Si. The electrical consequences of this important parameter are discussed in Section 5.

### **3. Direct evaluation of semiconductor susceptibility to fusion neutrons**

The susceptibility of the different semiconductors to neutron irradiation can be firstly evaluated via the calculation of the number of neutron-material interactions under 2.45 or 14 MeV neutrons. This can be performed via a direct analytical calculation using neutron cross-section library data; we considered here the TENDL-2021 nuclear data library [11]. The number of interactions in the target at a fixed neutron energy E is simply given by:

$$N(E) = \sum\_{i} f\_i \,\sigma\_i(E) \times \mathbf{10}^{-24} \times \mathbf{NV} \times \mathbf{e} \times \mathbf{M} \tag{4}$$

where σ<sup>i</sup> is the value at energy E of the cross section for isotope i (in barn), fi is the fraction of isotope i in the target isotopic composition, e is the target thickness, NV is the number of atoms per unit volume, and M is the number of incident monoenergetic neutrons; in this work, fixed to the arbitrary value of 5 � 108 for standardization purpose (introduced in other previous studies [8–10], M initially corresponds to the number of atmospheric neutrons impacting a surface of 1 cm<sup>2</sup> at sea level exposed to natural radiation during 25 � <sup>10</sup><sup>6</sup> h).

**Figure 1** illustrates the extraction from TENDL-2021 library data of the neutron cross-section values at 2.45 and 14 MeV for two different isotopes. Total, elastic, and inelastic data, respectively, correspond to reaction type numbers MT = 1, 2, and 4 in the standardized ENDF format [12]. When it is not available, the cross-section value for the sum of all nonelastic channels (MT = 3) is obtained by subtracting elastic + inelastic cross sections to the total cross section.

#### **Figure 1.**

*Example of neutron cross sections (total, elastic and inelastic) extracted from TENDL-2021 nuclear data library [11] for the isotopes 32-Ge-070 (left) and 51-Sb-123 (right).*

*Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

As the result of this analytical evaluation using Eq. (4), **Table 3** gives the total number of interaction events in a target (1cm<sup>2</sup> <sup>20</sup> <sup>μ</sup>m) composed of a single chemical element with an isotope distribution corresponding to the natural abundance (**Table 1**) and subjected to 2.45 or 14 MeV neutrons. Values reported in **Table 3** correspond to a normalized value NV = 1.0 <sup>10</sup><sup>22</sup> at/cm<sup>3</sup> for all targets.

From the data of **Table 3**, it is now easy to evaluate the neutron response for any material composed of these elements. **Figure 2** shows the results of this evaluation for the different semiconductor materials, considering their stoichiometry and their exact number of atoms per square centimeter, given in **Table 2**. All numerical values for **Figure 2** are detailed in the Appendix (**Table 6**). From the results of **Figure 2**, we can formulate the following observations:



#### **Table 3.**

*Total number of interaction events in a target (1cm2 <sup>20</sup> <sup>μ</sup>m) composed of a single chemical element with an isotope distribution corresponding to the natural abundance (Table 1) and subjected to 2.45 or 14 MeV neutrons. Values are calculated from Eq. (4) with a normalized value NV = 1.0 <sup>10</sup><sup>22</sup> at/cm<sup>3</sup> for all targets.*

#### **Figure 2.**

*Number of elastic, inelastic, and nonelastic interaction events in a target (1cm<sup>2</sup> <sup>20</sup> <sup>μ</sup>m) composed of natural semiconductors and subjected to 5 <sup>10</sup><sup>8</sup> neutrons of 2.45 or 14 MeV. Values are deduced from Table 3 considering the stoichiometry and the exact number of atoms per square centimeter for each material given in Table 2. Numerical values for this graph are reported in the appendix (Table 6).*

due to an elevated number of nonelastic events, the highest at 14 MeV for all materials.

• AlAs exhibits the smallest neutron response after silicon.

All these results are now investigated in detail in the next section via more complete numerical simulations.

## **4. Geant4 and MCNP6 simulations**

### **4.1 Simulation details**

The neutron susceptibility of the studied materials has been in-depth analyzed using the Monte Carlo radiation transport codes Geant4 (version 10.07 patch 02, G4NDL4.6 neutron library [13–15]) and MCNP6 (version 6.2 [16, 17]). In a similar way to previous analytical estimations, natural semiconductor targets (1 cm2 <sup>20</sup> μm) have been subjected to incoming 2.45 or 14 MeV neutrons perpendicularly to their largest surface. The same number of 5 108 incident neutrons has been considered for each simulation run.

For Geant4, a simulation run produces a single output file containing all the information related to the neutron interaction events in the target material—nature of *Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

the interaction, spatial coordinates of the reaction vertex, and exhaustive list of secondary particles produced during the interaction (energy and emission direction vector for each of these emitted particles). A post-treatment eliminates all gamma photons, neutral and light particles (e, e<sup>+</sup> , η), not able to induce significant singleevent effects in electronics.

For MCNP6 simulation, PTRAC card options were activated to obtain, for each event, the nature and the coordinates of the vertex of the interaction and the final energy of the neutron after the interaction. It is thus possible to count and to discriminate interactions as a function of the target atom and as a function of their nature (elastic, inelastic, and nonelastic).

#### **4.2 Numbers of interaction events at 2.45 and 14 MeV**

**Figures 3** and **4** show the number of interaction events for the different semiconductor targets subjected to 2.45 and 14 MeV neutrons, respectively. All the numerical values for these two figures are reported in **Table 6** of the Appendix. These results show a good agreement, on one hand, between analytical and numerical estimations, and, on the other hand, between Geant4 and MCNP6 results. If we consider the total number of interaction events (**Table 6**), analytical and numerical results agree within 6% on average at 2.45 MeV and within 2.4% on average at 14 MeV. Between Geant4 and MCNP6 results, agreements are better, within 3% on average at

2.45 MeV and within 1.6% on average at 14 MeV. The best agreement is obtained for diamond, GaN, and GaAs at 14 MeV, the largest difference is for GaP at 2.45 MeV,

**Figure 3.**

*Number of elastic, inelastic, and nonelastic events estimated by Geant4 and MCNP6 as a function of the nature of the target material for 2.45 MeV incoming neutrons.*

#### **Figure 4.**

*Number of elastic, inelastic, and nonelastic events estimated by Geant4 and MCNP6 as a function of the nature of the target material for 14 MeV incoming neutrons.*

with 11.5% of the variation between Geant4 and MCNP6, the number of elastic events being larger than 14% for Geant4 with respect to MCNP6. Globally, one can see that the different sets of results agree very satisfactorily.

In the following and because Geant4 can track all particles and, consequently, give more exhaustive information about secondaries than MCNP6, we will explore in detail the distributions and characteristics of secondaries produced in the interaction events at both 2.45 and 14 MeV from Geant4 results.

#### **4.3 Detailed analysis at 2.45 MeV**

**Figures 5**–**7** shows the energy histograms of the secondaries produced by 2.45 MeV neutrons in the targets composed of the different natural semiconductor materials under investigation. In the particular case of C (diamond), shown in **Figure 5**, all secondaries are exclusively elastic carbon recoil nuclei. They form a continuum from zero to a maximum energy Emax theoretically equal to [18]:

$$E\_{\text{max}} = 4 \times E \times \frac{A}{\left(A + 1\right)^2} \tag{5}$$

where A is the mass number of the target nucleus and E is the energy of the incident neutrons. Here, for carbon, A = 12 and with E = 2.45 MeV, we obtain a theoretical value of Emax = 0.696 MeV, in excellent agreement with the cutoff value of 0.7 MeV extracted from **Figure 5**. This figure also shows a direct comparison between *Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

#### **Figure 5.**

*Energy histograms of secondaries produced in the diamond target by 2.45 MeV neutrons and obtained from Geant4 and MCNP6 simulations. All secondaries are elastic recoil nuclei.*

MCNP6 and Geant4 simulation results because, in the case of elastic events, it is possible from MCNP6 output data to compute the energy distribution of recoil nuclei; an estimation unfortunately not possible for inelastic and nonelastic events.

**Figures 6** and **7** show Geant4 results for the other target materials. For Si, Ge, SiC, AlAs, GaAs, InAs, InSb, and GaSb (**Figure 6**), 2.45 MeV neutrons produce quasi exclusively, via elastic, inelastic, and nonelastic interactions, secondaries with the same atomic number than one of the elements of the target. The heavier the recoil nucleus or secondary product, the more the distribution is shifted toward the lower energies. This shift is more marked as the difference in atomic number is greater, for example for InP (Z = 15 and 49). Only three materials, i.e., GaN, GaP, and InP, show (**Figure 7**) a significant production of secondaries at 2.45 MeV that differ from target elements:


In addition to the previous analysis, **Figure 8** (left) gives, for each material, the sum of the initial kinetic energy for all produced secondaries (we recall here that neutrons, gamma rays, and light particles, such as electrons or positrons are not considered, only charged particles with Z ≥ 1 are considered). Such a quantity represents the energy susceptible to be deposited by all these secondaries in the considered target if they are totally stopped (after transferring the totality of their kinetic energy, essentially by ionization process). This energy amount is the highest for C (about 105 MeV), followed by SiC, GaN, Si, and GaP (all above 104 MeV); it is minimum for InSb (around 3 103

#### **Figure 6.**

*Energy histograms of secondaries produced by 2.45 MeV neutrons in targets of Si, Ge, GaAs, SiC, AlAs, InAs, InSb, and GaSb semiconductor materials.*

MeV). **Figure 8** (right) gives the average energy conveyed per particle for the four categories of secondaries. The consequence of this result on the electrical point-of-view and in the framework of single-event effect occurrence is be discussed in Section 5.

#### **4.4 Detailed analysis at 14 MeV**

**Figures 9** and **10** show the energy histograms of the secondaries produced by 14 MeV neutrons in the different targets. Contrary to the previous case at 2.45 MeV, a

*Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

**Figure 7.** *Energy histograms of secondaries produced by 2.45 MeV neutrons in targets of GaP, InP, and GaN semiconductor materials.*

#### **Figure 8.**

*Left: Sum of the kinetic energy of all secondaries produced by 2.45 MeV in the different targets (charged particle with Z* ≥ *1). Right: Average particle energy per particle type (same legend as for the left figure).*

significant number of nonelastic reaction channels is opened at this higher neutron energy. The ratio of the number of nonelastic reactions with respect to the total number of events varies from about 5% in the case of C up to 38% for InP; it is around 33% for Si. These nonelastic reactions produce several types of ions, protons, and alpha particles—their respective numbers are given in **Table 4** which also indicates the number of secondary ions produced characterized by the same atomic number as target nuclei. This last category includes, of course, all recoil nuclei produced in elastic and inelastic reactions but also ions produced in certain nonelastic reactions, for

#### **Figure 9.**

*Energy histograms of secondaries produced by 2.45 MeV neutrons in targets of GaN, GaP, GaAs, GaSb, InSb, InP, InAs, and AlAs III-V semiconductor materials.*

example, in the (n,n'γ) or (n,2n'γ) reactions that are increasingly frequent when increasing the atomic number of the target element. These nonelastic reactions give a final nucleus with a lighter mass with respect to the initial impacted nucleus (for example, 74Ge(n,2n)73Ge or 115In(n,2n)113In, etc.).

For the good understanding of **Figures 9** and **10**, **Table 5** indicates the main nonelastic reactions leading to the production of protons and alpha particles. The energy ranges of the ejected particles are also reported. Protons, deuterons, and tritons (Z=1) are produced in large numbers in six materials, by order of importance—GaP, Si, SiC, InP, AlAs, and GaN. The main reactions for these targets are the following (we indicated in parenthesis the energy threshold of the reactions): 31P(n,n'p)30Si (7.5 MeV), 31P(n,p)31Si (0.7 MeV), 31P(n,d)30Si (5.2 MeV), 28Si(n,p)28Al (4 MeV), 27Al(n,p)27Mg (1.9 MeV), 14N(n,d)13C (5.7 MeV). Protons and deuterons are also observed in smaller numbers in the other materials, i.e., GaAs, Ge, GaSb, InAs, and InSb. Only two tritons are observed for C, they are produced in the rare reaction

*Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

#### **Figure 10.**

*Energy histograms of secondaries produced by 2.45 MeV neutrons in targets of Si, SiC-4H, Ge, and C (diamond) Group-IV semiconductor materials.*


#### **Table 4.**

*Number of secondaries produced by 14 MeV neutrons in the different targets.*

13C(n,t)11B. Concerning alpha particles (Z=2), they are produced in large or significant numbers in seven materials—SiC, C, Si, GaN, GaP, InP, and AlAs. The main reactions for these targets are the following—28Si(n,α) 25Mg (2.7 MeV), 12C(n,α) 9 Be (6.2 MeV), 14N(n,α) 11B (0.17 MeV), 31P(n,α) 28Al (2 MeV), 27Al(n,α) 24Na (3.2 MeV). Traces of alpha particles are also observed in Ge, GaAs, GaSb, and InSb.


#### **Table 5.**

*Main reactions for proton, deuteron, or alpha particle production in the nonelastic interactions of 14 MeV neutrons with target elements. The energy ranges of the ejected particles are also indicated (see Figures 9 and 10 for visualization).*

**Figure 11** (left) shows the sum of the kinetic energy of all secondaries produced by 14 MeV in the different targets. **Figure 11** (right) gives the average energy conveyed per particle for the four categories of secondaries. Protons and alphas are of prime importance in the radiation response of these materials insofar as they convey a very important part of the total kinetic energy; this part is predominant in the case of five materials—Si, SiC, AlAs, GaN, and GaP. As compared to **Figure 8** (left) obtained for 2.45 MeV neutrons, the amplification factor of the total kinetic energy at 14 MeV for

#### **Figure 11.**

*Left: Sum of the kinetic energy of all secondaries produced by 14 MeV in the different targets (charged particle with Z* ≥ *1). Right: Average particle energy per particle type (same legend as for the left figure).*

all secondaries is minimal for C (2.5) and InSb ((3.5) and maximal for InP, GaP, AlAs (around 10), and Si (14.3).

### **5. Implication for electronics subjected to fusion neutrons**

In this last section, we examine the previous results in light of the mechanisms of creation of single-event effects (SEEs) in electronics subjected to fusion neutrons. We recall here that these SEEs are initiated by neutron-matter interactions in four main steps—(1) interactions produce secondary fragments or recoil nuclei from target nuclei. (2) The produced ion(s) interact(s) with material to generate free charge carriers via the production of electron and hole pairs. (3) Electrons and holes are transported by drift-diffusion mechanisms through the circuit materials (oxides and semiconductors) up to a sensitive node (reversely biased junctions); free carriers also recombine during their transport. (4) The collected charge on a sensitive node can significantly alter its voltage that leads to a change in device or circuit operation. This voltage glitch may propagate through the circuit. The present work allows us to provide quantitative information on steps 1 and 2 in so far as from step 3, it would be necessary to consider the circuit architecture, the applied polarizations, and the transport and recombination properties of the materials. In the following, we examine the different materials in light of the number of interactions and of the number of e-h pairs created on average per interaction.

Concerning the first metric, the number of SEEs susceptible to be created in an electronic circuit (of the same geometry as the target for simplicity) is at most equal to the number of interaction events that can deposit enough energy, i.e., create a sufficient electrical charge able to disturb the circuit operation (in the case of an SRAM memory, this minimum amount of charge susceptible to upset a memory cell is called a critical charge). This number of SEEs is not easy to evaluate but it can be upper bounded by the number of events that can deposit energy and therefore generate an electrical charge greater than or equal to the critical charge of the circuit. To roughly estimate this metric from the histogram of the energies/charges deposited per event, a simple transformation is necessary, as illustrated in **Figure 12**. **Figure 12** (left) shows the energy histograms (frequency and cumulative frequency histograms) for all the events produced by 2.45 MeV and 14 MeV neutrons in GaP. The relationship between energy (bottom scale) and charge (top scale) considers the energy value of the creation of electron-hole pairs, as given in **Table 2**.

At fixed energy or charge, the cumulative frequency histogram directly indicates the number of events for which the total energy deposited is below this value. In practice, it is more convenient to have the complementary value, i.e., the number of events above this value is plotted in **Figure 12** (right), and directly expressed in electrical charge instead of energy. Applying this transformation to all targets gives the ensemble of cumulative histograms, as shown in **Figures 13** and **14**.

These figures can be directly used to estimate the number of events able to deposit an electrical charge superior to a given value. From this data and knowing a minimum of geometric and electrical characteristics of the circuit to be evaluated from a radiative point of view (number and geometry of the sensitive volumes, critical charge, etc.), it would be possible, in future studies, to estimate the number of events susceptible to produce SEEs at both 2.45 and 14 MeV.

**Figures 13** and **14** provide also other interesting information about the difference in the neutron susceptibility of a given material at these two energies—the greater this

#### **Figure 12.**

*Left: Histogram and cumulative histogram of the energy deposited per event for all interaction events in GaP target. Right: Reversed cumulative histogram from the left figure expressed in charge deposited per event (MeV).*

difference, the more the material, and therefore, the circuit based on this semiconductor will be likely to present a significant difference in its radiative SEE response. This difference can be appreciated following two criteria—the respective amounts of events at the lowest charge deposited per event (here at 10<sup>2</sup> fC) and the "distance" between the shoulders of the curves at the highest charge values. For example, a circuit based on diamond or GaN semiconductor may logically presents less difference between its SEE responses at 2.45 MeV and 14 MeV than a circuit based on germanium, InSb, or InAs. Of course, this difference in neutron sensitivity is also dependent on the critical charge value of the considered circuit. In the case of GaAs, for example, a circuit with a 10 fC of critical charge should be quasi-immune to

2.45 MeV neutrons but sensitive to 14 MeV. To go further in this analysis and be quantitative, it would be of course necessary to consider the cumulative histograms of the charge collected at each event. The passage from the deposited charge to the collected charge requires numerical simulations on a given architecture and modeling of transport, recombination, and collection mechanisms, a work currently in progress.

Concerning the second metric, **Figure 15** gives the number of e-h pairs created in average per interaction for the different targets at the two neutron energies. At 2.45 MeV, this value is minimum for AlAs (78,000) and maximum for diamond (29,000) and Si (27,000). The average level and the relative low dispersion of these values indicate that recoil nuclei, very widely produced in these elastic and inelastic reactions, deposit small amounts of charges over very short distances of a few nanometers to a few hundred nanometers (data not shown). At 14 MeV, the results of **Figure 15** (right) show higher values (approximately 3 to 12 for the different materials, and 20 for Si) due to the contributions of nonelastic reactions that produce the most energetic secondaries, notably protons and alpha particles. Surprisingly and contrary to the case of irradiation with atmospheric neutrons [10], Silicon sensibly differs from the other materials at 14 MeV since it has the greatest multiplying factor between

2.45 and 14 MeV. This is due to a relatively low number of events (86 kilo events) at this energy with respect to the other semiconductors (**Table 6**) combined with an important level of production of protons and alpha particles (**Table 3**) and also with an energy value of e-h pair creation inferior to the average value (around 5 eV) for all

*Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

**Figure 13.**

*Reversed cumulative histogram of the energy deposited per event for all interaction events in GaP, GaN, InSb, GaSb, InP, AlAs, GaAs, and InAs targets subjected to 2.45 MeV and 14 MeV neutrons. For a fixed value of the deposited charge, the curves give the number of interactions able to deposit at most this energy in the corresponding target.*

materials (**Table 2**). The other materials which exhibit high values are SiC, GaP, and InP. Statistically, in these four materials, a neutron-target interaction at 14 MeV will deposit more charge than in the other materials, which potentially makes these events more threatening for the creation of SEEs. The balance between all these factors is, therefore, not obvious when one remains at the only "material" level for these studies.

#### **Figure 14.**

*Reversed cumulative histogram of the energy deposited per event for all interaction events in Si, Ge, SiC, and C (diamond) targets subjected to 2.45 MeV and 14 meV neutrons. For a fixed value of the deposited charge, the curves give the number of interactions able to deposit at most this energy in the corresponding target.*

#### **Figure 15.**

*The average number of e-h pairs created per interaction event in the different targets subjected to 2.45 MeV (left) and 14 MeV (right) neutrons.*

This point requires numerical simulations at the circuit level to go further in the investigation of the level of SEE in electronics based on these different materials.

### **6. Conclusion**

This work explored the radiation response of Group IV (Si, Ge, SiC, diamond) and III-V (GaAs, GaN, GaP, GaSb, InAs, InP, InSb, AlAs) semiconductors subjected to D–D (2.45 MeV) and D–T (14 MeV) neutrons. This first study was limited to the level of the neutron response of bulk materials, an essential step to be able, subsequently, to study the response of electronic components and circuits based on these materials. The response of each semiconductor has been systematically investigated through a

*Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

direct calculation using nuclear cross-section libraries, MCNP6, and Geant4 numerical simulations. The counting of reaction rates per type of reaction (elastic, inelastic, nonelastic), as well as the classification of all neutron-induced secondary products as a function of their nature and energy, gives a first very complete picture of the behavior of these materials in a fusion environment under D–D or D–T neutrons. Although all studied materials exhibit a larger total number of interaction events than silicon, both at 2.45 and 14 MeV, there is nothing to conclude at this stage that they will present a larger response in terms of SEEs at circuit level since this higher neutron susceptibility only relates to the overall number of events and not to the production of the most energetic particles, such as protons or alpha particles. Moreover, such a higher neutron response may be compensated by other mechanisms at the electronics level, for example, a higher energy value of e-h pair creation (which has the effect of reducing the charge deposited) or higher carrier mobility (which can increase the performance of transistors and strengthen the resilience of circuits to single events). Future studies at the circuit level, following a methodology similar to a first study carried out on GaN subjected to atmospheric neutrons [19], will be largely based on the interaction event databases compiled during this first work and will provide more quantitative information to precisely assess the D–D and D–T neutron radiation response of future circuits based on these alternative materials to the classical silicon of microelectronics.

## **Conflict of interest**

The authors declare no conflict of interest.

## **Appendix**



**Table 6.**

*Details of numerical values for Figures 2–4.*

## **Author details**

Jean-Luc Autran\* and Daniela Munteanu Aix-Marseille University, CNRS, University of Toulon, Faculté des Sciences, Marseille Cedex, France

\*Address all correspondence to: daniela.munteanu@univ-amu.fr

© 2022 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.

*Radiation Response of Group-IV and III-V Semiconductors Subjected to D–D… DOI: http://dx.doi.org/10.5772/intechopen.103047*

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## *Edited by Alberto Adriano Cavalheiro*

*New Advances in Semiconductors* brings together contributions from important researchers around the world on semiconductor materials and their applications. It includes seven chapters in two sections: "Calculations and Simulations in Semiconductors" and "Semiconductor Materials." The world will emerge different after the social and economic reorganizations caused by the COVID-19 pandemic and will be even more dependent on semiconductors than ever before. New Advances in Semiconductors is a book that brings together the contributions of important researchers around the world and is able to give an idea about the different characteristics of semiconductor materials and their applications. There is a section dedicated to theory, calculations and logic and another dedicated to the development and characterization of semiconductor materials of great future interest. I really hope that this book will help to spread knowledge about this research field to other researchers and students working in this area or even to those interested in starting their more advanced studies.

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New Advances in Semiconductors

New Advances in

Semiconductors

*Edited by Alberto Adriano Cavalheiro*