4. Diffusion of phosphorus in InGaAs/InGaP/P heterostructures

In [16] co-diffusion of Ga and P was investigated, and it was shown that co-doping strongly affects the diffusion of phosphorus. The interest to Ga and P co-diffusion appeared with the developments of multicascade solar cells.

In last two decades, germanium is considered as the most suitable material for the first cascade of multiple solar cells based on A<sup>3</sup> B<sup>5</sup> compounds that is intended for transformation of the infrared solar spectrum [23]. Germanium cascade of the multiple solar cells is formed by phosphorus diffusion into heavily gallium-doped germanium substrates. It was found that p-n junction depth weakly depends on the diffusion time. In [24, 25], P and Ga profiles in the heterostructure In0.01Ga0.99As/In0.56Ga0.44P/Ge were investigated. p-n junction of this element was formed at 635�C by phosphorus diffusion from In0.56Ga0.44P buffer layer having thickness of about 24 nm to heavily doped of Ga germanium substrate (CGa = 2\*1018 cm�<sup>3</sup> ). The diffusion time was 2.6 min. SIMS has been applied to obtain profiles of P and Ga in heterostructure.

Figure 4 shows P, Ga, and free carrier concentration distribution in the Ge part of heterostructure. To calculate free electron concentration electroneutrality, equation was solved in the form of

$$
\mathcal{C}\_P^+(\mathbf{x}) + p(\mathbf{x}) - n(\mathbf{x}) - \mathcal{C}\_{\text{Ga}}^-(\mathbf{x}) = \mathbf{0} \tag{21}
$$

where Fermi integral of order ½:

valence band, respectively.

tions of P and Ga.

described [25].

parameter. Then

F<sup>1</sup>=<sup>2</sup>ð Þ¼ η

shaped curve with <sup>D</sup>Ga = 1.4 � <sup>10</sup>�<sup>15</sup> cm<sup>2</sup>

Drift term includes continuity equation:

component in the charged particle diffusion.

2 ffiffiffi <sup>π</sup> <sup>p</sup> � ð ∞

0

ε<sup>1</sup>=<sup>2</sup>dε εε�<sup>η</sup> þ 1 ; <sup>ε</sup> <sup>¼</sup> <sup>E</sup> � EC

where F is the Fermi level and Ec and Ev are bottom of the conduction band and top of the

Numerical calculations of Fermi level were made by Newton method for defined concentra-

It was found that Ga diffuses insensitive to Ge substrates together with P. The higher solubility of Ga than P was found on the InGaAs/Ge interface as it was also noted earlier [27] that leads to formation of two p-n junctions. The shallow p-n junction was formed at a depth of 30 nm and the second one at a depth of 130 nm. Diffusion part of Ga profile demonstrated Fickian-

As it was expected, phosphorus profile has two parts: Fickian type near the surface in p-region (CGa > CP) and box-shaped between p-n junctions where n > ni. Unfortunately using diffusion coefficient with quadratic and cubic dependencies, the P profile could not be accurately

Two methods of diffusivity calculations were used [28]. The first one was Sauer-Freise (SF) method based on the Boltzmann-Matano calculation of diffusivity [1]. The second one was

In the CDD method, two parameters are introduced that describe a probability of hopping process ϕ(x) and probability that the nearest vacant place for diffusion is empty γ(x). Then diffusivity D(x) and drift velocity V(x) are expressed through these parameters and average distance between neighboring places λ. We have taken λ = a = 0.566 nm as a germanium lattice

∂ϕð Þx

∂CX

Figure 5 shows calculated dependencies of P diffusivity on x for both methods. Positions of pn junctions are presented. As we can see, diffusivity calculated using SF method is comparatively higher than using CDD method. That may be a consequence of existing a strong electric field in the sample in the p-n junction regions that leads to appearance of a strong drift

DX �

<sup>∂</sup><sup>x</sup> � <sup>ϕ</sup>ð Þ<sup>x</sup>

<sup>∂</sup><sup>x</sup> � V xð ÞCx � �

method of the analysis of coordinate-dependent diffusion (CDD) [29].

V xð Þ¼ γð Þx

∂CX <sup>∂</sup><sup>t</sup> � <sup>∂</sup> ∂x kT ; <sup>η</sup> <sup>¼</sup> <sup>F</sup> � EC

Phosphorus and Gallium Diffusion in Ge Sublayer of In0.01Ga0.99As/In0.56Ga0.44P/Ge Heterostructures

kT ; <sup>ε</sup><sup>i</sup> <sup>¼</sup> EC � EV

http://dx.doi.org/10.5772/intechopen.78347

/s that exceeds data 6 � <sup>10</sup>�<sup>17</sup> - 2.3 � <sup>10</sup>�<sup>16</sup> cm<sup>2</sup>

D Xð Þ¼ <sup>ϕ</sup>ð Þ<sup>x</sup> <sup>γ</sup>ð Þ<sup>x</sup> <sup>λ</sup><sup>2</sup> (24)

� �γ<sup>2</sup> (25)

¼ 0 (26)

∂γð Þx ∂x

kT (23)

/s [4].

39

As dopant concentrations near interface are high, Fermi-Dirac distribution was used [26]:

$$n = N\_{\mathbb{C}} \cdot F\_{1/2}(\eta), \newline p = N\_V \cdot F\_{1/2}(-\eta - \varepsilon\_i) \tag{22}$$

Figure 4. Profiles of P, Ga, n and p in Ge.

where Fermi integral of order ½:

4. Diffusion of phosphorus in InGaAs/InGaP/P heterostructures

of about 24 nm to heavily doped of Ga germanium substrate (CGa = 2\*1018 cm�<sup>3</sup>

C<sup>þ</sup>

developments of multicascade solar cells.

38 Advanced Material and Device Applications with Germanium

of multiple solar cells based on A<sup>3</sup>

Figure 4. Profiles of P, Ga, n and p in Ge.

form of

In [16] co-diffusion of Ga and P was investigated, and it was shown that co-doping strongly affects the diffusion of phosphorus. The interest to Ga and P co-diffusion appeared with the

In last two decades, germanium is considered as the most suitable material for the first cascade

infrared solar spectrum [23]. Germanium cascade of the multiple solar cells is formed by phosphorus diffusion into heavily gallium-doped germanium substrates. It was found that p-n junction depth weakly depends on the diffusion time. In [24, 25], P and Ga profiles in the heterostructure In0.01Ga0.99As/In0.56Ga0.44P/Ge were investigated. p-n junction of this element was formed at 635�C by phosphorus diffusion from In0.56Ga0.44P buffer layer having thickness

time was 2.6 min. SIMS has been applied to obtain profiles of P and Ga in heterostructure.

<sup>P</sup> ð Þþ x p xð Þ� n xð Þ� C�

As dopant concentrations near interface are high, Fermi-Dirac distribution was used [26]:

Figure 4 shows P, Ga, and free carrier concentration distribution in the Ge part of heterostructure. To calculate free electron concentration electroneutrality, equation was solved in the

B<sup>5</sup> compounds that is intended for transformation of the

n ¼ NC � F1=<sup>2</sup>ð Þ η , p ¼ NV � F1=<sup>2</sup>ð Þ �η � ε<sup>i</sup> (22)

). The diffusion

Gað Þ¼ x 0 (21)

$$F\_{1/2}(\eta) = \frac{2}{\sqrt{\pi}} \cdot \left[ \frac{\varepsilon^{1/2} d\varepsilon}{\varepsilon^{\varepsilon-\eta}+1}; \varepsilon = \frac{E - E\_{\mathbb{C}}}{kT}; \eta = \frac{F - E\_{\mathbb{C}}}{kT}; \varepsilon\_{i} = \frac{E\_{\mathbb{C}} - E\_{V}}{kT} \right. \tag{23}$$

where F is the Fermi level and Ec and Ev are bottom of the conduction band and top of the valence band, respectively.

Numerical calculations of Fermi level were made by Newton method for defined concentrations of P and Ga.

It was found that Ga diffuses insensitive to Ge substrates together with P. The higher solubility of Ga than P was found on the InGaAs/Ge interface as it was also noted earlier [27] that leads to formation of two p-n junctions. The shallow p-n junction was formed at a depth of 30 nm and the second one at a depth of 130 nm. Diffusion part of Ga profile demonstrated Fickianshaped curve with <sup>D</sup>Ga = 1.4 � <sup>10</sup>�<sup>15</sup> cm<sup>2</sup> /s that exceeds data 6 � <sup>10</sup>�<sup>17</sup> - 2.3 � <sup>10</sup>�<sup>16</sup> cm<sup>2</sup> /s [4]. As it was expected, phosphorus profile has two parts: Fickian type near the surface in p-region (CGa > CP) and box-shaped between p-n junctions where n > ni. Unfortunately using diffusion coefficient with quadratic and cubic dependencies, the P profile could not be accurately described [25].

Two methods of diffusivity calculations were used [28]. The first one was Sauer-Freise (SF) method based on the Boltzmann-Matano calculation of diffusivity [1]. The second one was method of the analysis of coordinate-dependent diffusion (CDD) [29].

In the CDD method, two parameters are introduced that describe a probability of hopping process ϕ(x) and probability that the nearest vacant place for diffusion is empty γ(x). Then diffusivity D(x) and drift velocity V(x) are expressed through these parameters and average distance between neighboring places λ. We have taken λ = a = 0.566 nm as a germanium lattice parameter. Then

$$D(\mathbf{X}) = \phi(\mathbf{x})\gamma(\mathbf{x})\lambda^2 \tag{24}$$

$$V(\mathbf{x}) = \left(\gamma(\mathbf{x})\frac{\partial\phi(\mathbf{x})}{\partial\mathbf{x}} - \phi(\mathbf{x})\frac{\partial\gamma(\mathbf{x})}{\partial\mathbf{x}}\right)\gamma^2 \tag{25}$$

Drift term includes continuity equation:

$$\frac{\partial \mathbb{C}\_{\mathcal{X}}}{\partial t} - \frac{\partial}{\partial \mathbf{x}} \left( D\_{\mathcal{X}} \cdot \frac{\partial \mathbb{C}\_{\mathcal{X}}}{\partial \mathbf{x}} - V(\mathbf{x}) \mathbb{C}\_{\mathbf{x}} \right) = \mathbf{0} \tag{26}$$

Figure 5 shows calculated dependencies of P diffusivity on x for both methods. Positions of pn junctions are presented. As we can see, diffusivity calculated using SF method is comparatively higher than using CDD method. That may be a consequence of existing a strong electric field in the sample in the p-n junction regions that leads to appearance of a strong drift component in the charged particle diffusion.

Figure 5. Diffusivity dependence on depth for T = 635�C.

Both methods of diffusivity calculations show two parts of D on x dependence: when x = 0– 100 nm, diffusivity increases, and at higher values of x, diffusivity decreases. Width of the p side depletion region of the shallow left p-n junction on the Figure 5 is of the order of 5–8 nm ðjCGa �CP<sup>j</sup> <sup>&</sup>lt; 1019cm�3); both sides of right p-n junction in Figure 5 are of the order of 50–80 nm (CGa � Cp <sup>&</sup>lt; 1017cm�3); therefore an intrinsic electric field exists in the area between p-n junction. Approximately in the middle of junctions, the electric field changes its direction. Near the surface the intrinsic electric field accelerates negatively charged particles; when x > 100 nm, it inhibits diffusion. Outside of depletion regions (x > 160 nm), drift component of diffusion is negligible and diffusivity calculated by both methods which are equal.

There are two regions of weak dependence of <sup>D</sup> on <sup>n</sup>. The first one at <sup>n</sup> < 2 � <sup>10</sup><sup>18</sup> cm�<sup>3</sup> corresponds to intrinsic diffusivity and is quite expected. The second is observed at high n in the region where the electric field exists. To understand the weak dependence of DP on n (3–4 on Figure 6), we shall consider the equations for P-V complexes forming. In Table 1, the equations and parameters k and j that lead to different dependencies of <sup>D</sup>ð Þ PV <sup>j</sup> on <sup>n</sup> (see (7)) are presented

Figure 6. Diffusivity dependence on electron concentration for T = 635�C. 1: 0 < x < 25 nm, 2: 25 < x < 33 nm, 3: 33 < x < 60

Phosphorus and Gallium Diffusion in Ge Sublayer of In0.01Ga0.99As/In0.56Ga0.44P/Ge Heterostructures

http://dx.doi.org/10.5772/intechopen.78347

41

The first is the same as in [17] when n = CP+; the second is for the case of CP<sup>+</sup> = const as it is in

Assumption that n = CP is valid in a material with one type of impurity. In a strongly compensated material, the concentration of free charge carriers is significantly lower than the concentration of the impurity. Between p-n junctions in the measured heterostructure, phosphorus concentration changes slowly and we may suggest CP<sup>+</sup> = const. Phosphorus atoms in substitution positions are fully ionized, so m = +1; vacancies may be single, double, and triple ionized, that is, k = 0, �1, �2, and � 3, VP pairs—single and double ionized (j = 0, �1, �2). Which type of reaction will be realized depends on the position of the Fermi level of the material, which controls the ratio of the centers in different charge state. The greater the electron concentration, the greater the charge state of acceptors, that is, for the condition n = CP+, the most probable dependence of diffusivity of the complex is proportional n or n<sup>2</sup>

our case P-V complex should be charged, that is, j = �1 and �2. For CP<sup>+</sup> = const a weak

ð Þ PV �<sup>1</sup> <sup>¼</sup> <sup>P</sup><sup>þ</sup> <sup>þ</sup> <sup>V</sup>�<sup>2</sup> (27)

dependence of the diffusion coefficient on n is possible most likely for the reaction:

. In

our samples between p-n junctions (see Figure 6). We propose that <sup>D</sup>ð Þ PV <sup>j</sup> � <sup>C</sup>ð Þ PV <sup>j</sup> [17].

for two cases.

nm, 4: 60 < x < 100 nm, 5: x > 100 nm.

Figure 6 shows dependencies of P diffusivity on n for Sauer-Freise, coordinate-dependent diffusion calculations, and different diffusion data from the literature.

An expected increase of the diffusivity with the free electron concentration was observed in both methods. Diffusivity produced by CDD calculation has two regions. The first one belongs to intrinsic diffusion (<sup>n</sup> <sup>&</sup>lt; ni = 3.2 � 1018 cm�<sup>3</sup> [24]). As we can see, the lowest values of this part are equal to intrinsic diffusivity [5, 17, 30]. The second one corresponds to the diffusivity in the n-side of the p-n junctions and is higher than predicted both cubic and quadratic diffusion mechanisms. But the highest values of <sup>D</sup> = 2 � <sup>10</sup>�<sup>12</sup> cm2 /s at the <sup>n</sup> = 7 � 1018 cm�<sup>3</sup> well correspond to maximum values [7], calculated by Boltzmann-Matano method. These values are observed in the electric field region of p-n-p structure that is formed in the germanium near the interface. Diffusivity dramatically drops at the ends of this structure in the p-region that may be connected with the shape of intrinsic electric field that in the case of linear p-n junction depend on x quadratically and drops sharply in the end of the depletion region. We can assume that the electric field causes not only the appearance of a drift component in diffusion but also increases the diffusivity of P-V pairs <sup>D</sup>ð Þ PV <sup>j</sup> .

Phosphorus and Gallium Diffusion in Ge Sublayer of In0.01Ga0.99As/In0.56Ga0.44P/Ge Heterostructures http://dx.doi.org/10.5772/intechopen.78347 41

Figure 6. Diffusivity dependence on electron concentration for T = 635�C. 1: 0 < x < 25 nm, 2: 25 < x < 33 nm, 3: 33 < x < 60 nm, 4: 60 < x < 100 nm, 5: x > 100 nm.

Both methods of diffusivity calculations show two parts of D on x dependence: when x = 0– 100 nm, diffusivity increases, and at higher values of x, diffusivity decreases. Width of the p side depletion region of the shallow left p-n junction on the Figure 5 is of the order of 5–8 nm ðjCGa �CP<sup>j</sup> <sup>&</sup>lt; 1019cm�3); both sides of right p-n junction in Figure 5 are of the order of 50–80 nm (CGa � Cp <sup>&</sup>lt; 1017cm�3); therefore an intrinsic electric field exists in the area between p-n junction. Approximately in the middle of junctions, the electric field changes its direction. Near the surface the intrinsic electric field accelerates negatively charged particles; when x > 100 nm, it inhibits diffusion. Outside of depletion regions (x > 160 nm), drift component of diffusion is negligible

Figure 6 shows dependencies of P diffusivity on n for Sauer-Freise, coordinate-dependent

An expected increase of the diffusivity with the free electron concentration was observed in both methods. Diffusivity produced by CDD calculation has two regions. The first one belongs to intrinsic diffusion (<sup>n</sup> <sup>&</sup>lt; ni = 3.2 � 1018 cm�<sup>3</sup> [24]). As we can see, the lowest values of this part are equal to intrinsic diffusivity [5, 17, 30]. The second one corresponds to the diffusivity in the n-side of the p-n junctions and is higher than predicted both cubic and quadratic diffusion

ond to maximum values [7], calculated by Boltzmann-Matano method. These values are observed in the electric field region of p-n-p structure that is formed in the germanium near the interface. Diffusivity dramatically drops at the ends of this structure in the p-region that may be connected with the shape of intrinsic electric field that in the case of linear p-n junction depend on x quadratically and drops sharply in the end of the depletion region. We can assume that the electric field causes not only the appearance of a drift component in diffusion

/s at the <sup>n</sup> = 7 � 1018 cm�<sup>3</sup> well corresp-

and diffusivity calculated by both methods which are equal.

Figure 5. Diffusivity dependence on depth for T = 635�C.

40 Advanced Material and Device Applications with Germanium

mechanisms. But the highest values of <sup>D</sup> = 2 � <sup>10</sup>�<sup>12</sup> cm2

but also increases the diffusivity of P-V pairs <sup>D</sup>ð Þ PV <sup>j</sup> .

diffusion calculations, and different diffusion data from the literature.

There are two regions of weak dependence of <sup>D</sup> on <sup>n</sup>. The first one at <sup>n</sup> < 2 � <sup>10</sup><sup>18</sup> cm�<sup>3</sup> corresponds to intrinsic diffusivity and is quite expected. The second is observed at high n in the region where the electric field exists. To understand the weak dependence of DP on n (3–4 on Figure 6), we shall consider the equations for P-V complexes forming. In Table 1, the equations and parameters k and j that lead to different dependencies of <sup>D</sup>ð Þ PV <sup>j</sup> on <sup>n</sup> (see (7)) are presented for two cases.

The first is the same as in [17] when n = CP+; the second is for the case of CP<sup>+</sup> = const as it is in our samples between p-n junctions (see Figure 6). We propose that <sup>D</sup>ð Þ PV <sup>j</sup> � <sup>C</sup>ð Þ PV <sup>j</sup> [17].

Assumption that n = CP is valid in a material with one type of impurity. In a strongly compensated material, the concentration of free charge carriers is significantly lower than the concentration of the impurity. Between p-n junctions in the measured heterostructure, phosphorus concentration changes slowly and we may suggest CP<sup>+</sup> = const. Phosphorus atoms in substitution positions are fully ionized, so m = +1; vacancies may be single, double, and triple ionized, that is, k = 0, �1, �2, and � 3, VP pairs—single and double ionized (j = 0, �1, �2).

Which type of reaction will be realized depends on the position of the Fermi level of the material, which controls the ratio of the centers in different charge state. The greater the electron concentration, the greater the charge state of acceptors, that is, for the condition n = CP+, the most probable dependence of diffusivity of the complex is proportional n or n<sup>2</sup> . In our case P-V complex should be charged, that is, j = �1 and �2. For CP<sup>+</sup> = const a weak dependence of the diffusion coefficient on n is possible most likely for the reaction:

$$\left(PV\right)^{-1} = P^{+} + V^{-2} \tag{27}$$


Solubility of Ga in the InGaP/Ge interface is higher than of P that leads to formation of two p-n junctions. Co-doping by gallium strongly affects the diffusion of phosphorus in germanium. We propose that it occurs primarily due to the electric field of the forming p-n junctions. P-type region is formed in the thin Ge surface layer (30 nm of order) with the depletion region thickness of 8–10 nm. The electric field of this p-n junction is directed to the Ge surface and accelerates both negatively charged Ga in interstitial positions and vacancy-phosphorus pairs.

Phosphorus and Gallium Diffusion in Ge Sublayer of In0.01Ga0.99As/In0.56Ga0.44P/Ge Heterostructures

We can point out that in the case of Ga and P co-diffusion, calculations of diffusivity by Sauer-Freise and coordinate dependence diffusion methods give values an order of magnitude higher than the values, obtained for quadratic and cubic diffusion model for phosphorus diffusion. An electric field of a depletion region of p-n junctions leads to the appearance of drift components of phosphorus diffusion. At low electron concentrations in p-region near Ge surface in which there is no an electric field, phosphorus diffusivity increases with n from intrinsic diffusivity values, produced from Fickean-type profiles at low P concentration, to that one calculated by Boltzmann-Matano method for high P concentrations, while P concentration sharply decreases. We may suppose the vacancy concentration increasing as the concentration

It can be assumed that the electric field causes not only the appearance of a drift component in diffusion but also increases the diffusivity of P-V pairs. The sharp diffusivity growth and drop are consistent with the electric field direction. In the first p-n junction, it is directed to the surface and accelerates negatively charged particles including Ga and (PV). In the second

For a correct description of the Ga and P co-diffusion, it is necessary to take into account both changes in the concentration of charged centers due to a change in the Fermi level position and the formation and decay of diffusing pairs. For this, in the continuity equation, it is necessary to take into account not only the drift component but also the generation-recombination terms

[1] Mehrer H. Diffusion in Solids. Fundamentals, Methods, Materials, Diffusion-Controlled

[2] Boltaks B. Diffusion in Semiconductors. New York: Academic Press; 1963. 462 p

/s.

http://dx.doi.org/10.5772/intechopen.78347

43

That leads to comparatively high gallium diffusivity DGa = 1.4 <sup>10</sup><sup>15</sup> cm<sup>2</sup>

one, it is directed into the sample that leads to decrease of the D(PV).

corresponding to the formation and decomposition of the diffusing pairs.

National University of Science and Technology "MISiS", Moscow, RF, Russia

Kobeleva Svetlana Petrovna\*, Iliya Anfimov and Sergey Yurchuk

Processes. Berlin, Heidelberg: Springer; 2007. 535 p

\*Address all correspondence to: kob@misis.ru

of Ga and P that occupied the vacancies decreased.

Author details

References

Table 1. Equations and parameters for different dependencies of <sup>D</sup>ð Þ PV <sup>j</sup> on <sup>n</sup>.

The ionization energies of different charge states must be known to estimate a charge of a defect. It is obvious that ionization energies of vacancies and vacancy-assisted complexities depend on the temperature, but there are no reliable data of that energies [15, 31–37]. In [36] it was shown that at equilibrium conditions, half occupancy of the doubly negatively charged state of the vacancy-group-V-impurity atom pairs occurs when the Fermi level is situated at the middle of the forbidden gap. In spite of large phosphorus concentrations, n in the case of our interest is comparatively small, Fermi level is near the middle of the forbidden gap, and we may suggest that the (27) is an achievement.

As the electron density increases, the charge state of the pair can change. In the depletion region of the first p-n junction together with sharp increase of the Fermi level, the amount and charge of the pairs can be changed drastically, leading to a sharp increase in DP.
