Corresponding author: Svetlana P. Kobeleva ( kob@misis.ru ) © 2018 Svetlana P. Kobeleva, Ilya M. Anfimov, Andrei V. Turutin, Sergey Yu. Yurchuk, Vladimir M. Fomin.
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Citation:
Kobeleva SP, Anfimov IM, Turutin AV, Yurchuk SYu, Fomin VM (2018) Coordinate dependent diffusion analysis of phosphorus diffusion profiles in gallium doped germanium. Modern Electronic Materials 4(3): 113117. https://doi.org/10.3897/j.moem.4.3.39536

We have analyzed phosphorus diffusion profiles in an In_{0.01}Ga_{0.99}As/In_{0.56}Ga_{0.44}P/Ge germanium structure during phosphorus codiffusion with gallium for synthesis of the germanium subcell in multijunction solar cells.. Phosphorus diffused from the In_{0.56}Ga_{0.44}P layer simultaneously with gallium diffusion into the heavily gallium doped germanium substrate thus determining the specific diffusion conditions. Most importantly, gallium and phosphorus codiffusion produces two p–n junctions instead of one. The phosphorus diffusion profiles do not obey Fick’s laws. The phosphorus diffusion coefficient D_{P} depth distribution in the specimen has been studied using two methods, i.e., the Sauer–Freise modification of the Boltzmann–Matano method and the coordinate dependent diffusion method. We show that allowance for the drift component in the coordinate dependent diffusion method provides a better D_{P} agreement with literary data. Both methods suggest the D_{P} tendency to grow at the heterostructure boundary and to decline closer to the main p–n junction. The D_{P} growth near the surface p–n junction the field of which is directed toward the heterostructure boundary and its decline near the main p–n junction with an oppositely directed field, as well as the observed D_{P} growth with the electron concentration, suggest that the negatively charged V_{Ge}P complexes diffuse in the heterostructure by analogy with onecomponent diffusion.
phosphorus and gallium diffusion coefficient in silicon, coordinate dependent diffusion, vacancy diffusion model
Phosphorus and gallium and the main doping impurities in germanium and therefore the interest to their diffusion emerged from the very start of the germanium p–n junction technology development. Study of diffusion in the 1950–1960’s was based on p–n junction depth and spreading resistance measurements and diffusion profile description with Fick’s laws which assume that the diffusion coefficient for a specific impurity depends only on temperature [
There were many recent works on phosphorus diffusion in germanium. It was reported [
The development of multijunction solar cell (MJSC) technology raises the interest to germanium as substrate and first stage material for А^{3}В^{5} compound base MJSC [
The specimens were grown by MOS hydride epitaxy in a Veeco E450 LDM reactor in the form of (100) gallium doped germanium substrates (N_{Ga} = 10^{18} cm^{3}) and exposed to a phosphine gas flow at 635 °C for 2.5 min. Then the In_{0.56}Ga_{0.44}P buffer layer (1 min at T = 635 °C) and the heavily doped In_{0.01}Ga_{0.99}As layer (1.6 min at the same temperature) were deposited. The gallium, phosphorus and germanium profiles were measured by SIMS on a PHI6600. As shown elsewhere [
The phosphorus diffusion coefficient D_{P} was calculated using two methods, i.e., the Sauer–Freise modification of the Boltzmann–Matano method and [
(1) experimental phosphorus profile in germanium and (2) fourth power polynomial approximation.
С_{L} = 1.41 × 10^{21} cm^{3}; С_{R} = 4.7 × 10^{17} cm^{3} using the formula
C is the concentration of phosphorus depending on the depth (x). The D_{P} (x) calculation results are presented in Fig.
Phosphorus diffusion coefficient distribution in depth: (1) Sauer–Freise method; (2) coordinate dependent diffusion method. Arrows are the direction of the electric field vector of the pn junctions.
The coordinate dependent diffusion method deals with two atom migration mechanisms, i.e., due to the concentration gradient (proper diffusion) and drift at the velocity V (x) caused by fields (electric or elastic stress). The continuity equation is written as follows:
The diffusion coefficient and the drift velocity are calculated using one parameter (the average distance between adjacent sites λ) and two variables, i.e., the probability of vacant sites for jump φ(x) and the frequency of jumps in unit time γ(x).
D = φ(x)γ(x)λ^{2};
For the calculations λ was accepted equal to the germanium lattice parameter a = 0.566 nm. The φ(x) and γ(x) functions were determined by fitting for the three profile sections: 0–28 nm, 28–165 nm and 165–200 nm from the heterostructure interface. The calculation results are also shown in Fig.
The coordinate dependent diffusion calculated D_{P} data are expectedly lower than the Sauer–Freise ones since we took into account the phosphorus atom drift component for their calculation. The only exclusion is a small portion in the hole conductivity region at the phosphorus distribution tail, but with account of the calculation inaccuracy in this region one can consider these results to be approximately equal, i.e., the drift component is negligible beyond the second p–n junction. Further analysis included free carrier concentration calculation at the diffusion temperature.
Figure
Phosphorus, gallium and free carrier concentration profiles in germanium: (1) C_{P}; (2) C_{Ga}; (3) n; (4) p.
We did not take into account the concentrations of germanium vacancies (V_{Ge}) and possible P–V_{Ge} complexes due to their negligibility compared with the doping impurity concentration [
The element concentrations at the heterostructure interface exceed the density of states in the conduction band (N_{С}) and in the valence band (N_{V}) at the diffusion temperature, i.e., germanium is degenerate in this region and hence we calculated the electron and hole concentrations using the Fermi–Dirac distribution function [
n = N_{C}F_{1/2}(η), p = N_{V}F_{1/2}(–η – ε_{i}), (4)
where F_{1/2}(η) is the Fermi integral having a value of approx. 1/2:
F is the Fermi level, E_{C} and E_{V} are the conduction band bottom and valence band top, respectively, and k is the Boltzmann constant.
Numeric calculations of Eq. (3) were carried out using the Newton method. The origin of coordinates in Fig.
The electric field of the first p–n junction is directed toward the heterostructure interface and that of the second p–n junction is opposite. One can expect that the field of the first p–n junction will accelerate the diffusion of negatively charged atoms or complexes (Ga and (V_{Ge}P) complexes) and decelerate the diffusion of positively charged (P^{+}) atoms. The field of the second deeper p–n junction will act in the opposite direction. Both methods suggest the following phosphorus diffusion coefficient behavior: D_{P} growth near the first p–n junction and decrease near the second one. This is possible if phosphorus diffuses within negatively charged complexes such as phosphorus/germanium vacancies. Germanium vacancies are acceptors with the charge state ranging from one to three resulting in the VGeP complexes being neutral, single or doublenegative charged [
To analyze the diffusion coefficient dependence on material’s parameters we plotted D_{P} vs n graphs (Fig.
D _{P} as a function of electron concentration: (1) Sauer–Freise method calculation; (2) coordinate dependent diffusion method ((I) 0 < x <25 nm; (II) 25 < x <33 nm; (III) 33 < x <60 nm; (IV) 60 < x < 100 nm; (V) x >100 nm); (3—5) calculation according to Ref. [8], [9] and [6], respectively.
or (for the quadratic mechanism):
The regions in Fig.
It can be seen from Fig.
The general D_{P} growth with the concentration n is confirmed although there are some specific features most likely associated with gallium participation in diffusion and with the effect of the p–n junction electric fields on diffusion.
In the hole conductivity region at the heterostructure interface D_{P} grows with n. The diffusion coefficient grows simultaneously with the electron concentration as one approaches the first p–n junction. Since the Fermi level tends to the middle of the band gap one can expect an increase in the share of vacancies with the highest charge state and a transition from the quadratic to the cubic diffusion mechanism. D_{P} growth deceleration may be caused by a decrease in the overall vacancy concentration since Eqs. (5) and (6) were derived in the assumption of constant overall vacancy concentration. It should be noted that the phosphorus diffusion coefficient was first studied in this work for a hole conductivity region in germanium.
In the electronic conductivity region of the structure between the two p–n junctions D_{P} depends on n but slightly. This may be for a number of reasons, primarily, V_{Ge}P complex deceleration by p–n junction fields.
The diffusion coefficient calculated as a function of a distance from the heterostructure interface using the Sauer–Freise modification of the Boltzmann–Matano method and the coordinate dependent diffusion method. It is shown that diffusion in a p–n–p structure, i.e., two p–n junctions, exhibits a D_{P} growth tendency closer to the first p–n junction whose field should accelerate the negatively charged centers, and a D_{P} decline tendency moving away from the second p–n junction whose field should decelerate the negatively charged centers. Therefore, diffusing phosphorus is bound to the negatively charged centers, i.e., V_{Ge}P complexes with a charge of 1 or 2. The Sauer–Freise method overestimates D_{P} whereas the coordination dependent diffusion method gives a better D_{P} agreement with literary data. The latter is because the coordinate dependent diffusion method takes into account both the diffusion and the drift phosphorus atom diffusion components in germanium lattice. For two p–n junctions the drift component can be associated with the charge particle movement in the p–n junction electric fields.
The D_{P} growth tendency with n also corroborates the assumption of the vacancy phosphorus diffusion mechanism in germanium. It seems that in simultaneous Ga and P diffusion, the P diffusion is also by the vacancy mechanism, similarly with phosphorus diffusion without other impurities.
The D_{P} data for the hole region also shows the D_{P} growth tendency with n, although according to the suggested phosphorus diffusion models D_{P} should be constant at phosphorus concentrations of below n_{i}. This discrepancy may originate from the former assumption made for the phosphorus diffusion formulas that n = C_{P} which is not valid for the heavily compensated p conductivity region where an increase of the Fermi level can be accompanied by a change in the charge state of the vacancies and hence a different degree of D_{P} dependence on n.The D_{P} data for the hole region also show the D_{P} growth tendency with n although according to the suggested phosphorus diffusion models D_{P} should be constant at phosphorus concentrations of below n_{i}. This discrepancy may originate from the former assumption made for the phosphorus diffusion formulas that n = C_{P} which is not valid for the heavily compensated p conductivity region where an increase in the Fermi level can be accompanied by a change in the charge state of the vacancies and hence a different degree of D_{P} dependence on n.