1. Introduction

Impurity diffusion in semiconductors is one of the main processes for electronic device manufacturing, but on the other side, it could badly influence a semiconductor structure in multistage high-temperature electronic device manufacturing processes. Dopants, as phosphorus, at diffusion temperatures are ionized; therefore they actively interact with ionized lattice defects creating charged complexes. These complexes are formed and destroyed in the diffusion process that

© 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited. © 2018 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.

leads to the appearance of generation and recombination components in a continuity equation that describes a diffusion process [1, 2].

concentration was not determined in the [5]. Another problem revealed in [5] was deviation of experimental values of p-n-junction depth in Sb diffusion (as the most studied dopant) from

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

for estimation of the diffusion coefficient, a low diffusion time was used. Decrease of a penetration depth against expected one was attributed to diffusant evaporation in the diffusion process. These problems connected with the integral nature of a method of D coefficient

In [7], the phosphorus profiles were determined using layered etching and sheet resistance measurements. Profiles of P in Ge that were made by vapor phase diffusion process were obtained for two surface phosphorus concentrations: less than and more than intrinsic carrier density ni and at four diffusion temperatures—600, 650, 700, and 750�C. This allows to characterize temperature dependence of D. At low surface concentrations, the profile is described

At high surface concentration profiles which were extended, later [8] a name "box shaped" appears. For diffusivity calculations, authors applied Boltzmann-Matano method [1]. A dependence of the diffusion coefficient on the local phosphorus concentration was discussed. For the

Experimental data did not fit well into Arrhenius curves, especially for data at high phosphorus concentrations. With the temperature increase, the diffusion activation energy also increased.

Similar results were obtained in [8]. SIMS method was used for concentration profile measurements. Phosphorus diffusion was carried out at temperature range 600–910�С. Surface

"box-shaped" profiles. Boltzmann-Matano method also was used for evaluating the concentration dependence of P diffusivity. The observed concentration dependence was approximately in agreement with results of [7]. The strong concentration dependence in D was attributed to dependence of D on Fermi level or due to strain effects caused by the difference

kT � �cm2 � <sup>s</sup>

<sup>D</sup><sup>1</sup> <sup>¼</sup> <sup>330</sup> � exp � <sup>3</sup>:<sup>1</sup>

Dh <sup>¼</sup> <sup>0</sup>:<sup>01</sup> � exp � <sup>2</sup>:<sup>1</sup>

concentration-independent part, there was an expression obtained:

concentration of phosphorus was higher than 10<sup>19</sup> cm�<sup>3</sup>

In [8], temperature dependence of phosphorus diffusivity was found as

However the data of the paper allowed to derive another D(T):

<sup>D</sup><sup>1</sup> <sup>¼</sup> <sup>ð</sup>0:<sup>009</sup> � <sup>0</sup>:025Þ � exp � <sup>2</sup>:<sup>1</sup> � <sup>0</sup>:<sup>2</sup>

<sup>D</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>:<sup>21</sup> � exp � <sup>2</sup>:<sup>53</sup> � <sup>0</sup>:<sup>2</sup>

in ionic radius of P in Ge.

<sup>t</sup> <sup>p</sup> ) at large time values. Therefore

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

33

kT � �cm<sup>2</sup> � <sup>s</sup>�<sup>1</sup> (3)

�<sup>1</sup> (4)

; therefore all samples were showing

kT � �cm2 � <sup>s</sup>�<sup>1</sup> (5)

kT � �cm2 � <sup>s</sup>�<sup>1</sup> (6)

calculated dependence of p-n-junction depth on time (<sup>d</sup> � ffiffi

determination.

by Fick law, and diffusion coefficient is

Germanium is an important element to development of semiconductor theories and practice, and also it is a subject of many diffusion process researches. In this chapter, we focus on a narrow question: phosphorus diffusion in germanium, one of the main dopant of this material. Descriptions of diffusion processes were developing simultaneously with research of the crystalline and defect structure of this material and with improving of dislocation-free crystal growth technology together with development of measurement techniques and mathematical description of diffusion processes. That is why results that are 40 or 50 years old could be significantly different from contemporary ones. All these questions are under study and development. Progress in the first principal calculations together with the development of experimental techniques such as atomic force and scanning tunneling microscopy that allows to distinguish individual atoms and their lattice position will lead to the refinement of mechanisms and characteristics of diffusion processes. Our goal is to present the available data and knowledge about diffusion of phosphorus in germanium, possibly noting the problems and limitations of the representations used.

## 2. Phosphorus diffusion: first steps

Phosphorus, as a p-element of the group V of the periodic table, is a shallow donor impurity in germanium. The first works on phosphorus diffusion are about 1952–1954 years [3–5], and their review is in [2, 6].

It was previously mentioned that III and V group elements have a smaller diffusion coefficient than other groups of elements, and changes are mostly due to the frequency factor D0. This was explained by their smaller ionic radius [5]. However, for elements of V group in germanium, this tendency was not confirmed (unlike that in silicon). Phosphorus, for example, having smaller ionic radius than any other V group element, has a smaller diffusion coefficient. For all shallow dopants (except of B), the activation energy is estimated as about 2.5 eV, and it slightly increased with decreasing diffusion coefficient in the range of As—Sb—P [5].

For a long time, constant diffusion coefficients were used for a fixed temperature [2–6]. These results were fairly expected, as in the absence of a reliable dopant profile measurement method, the diffusion coefficient was determined by p-n-junction depth; therefore it is in

$$D\_P = 1.2 \cdot \exp\left(-\frac{2.5}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \tag{1}$$

[5] and taking into account the semiempirical Langmuir-Dushman formula:

$$D\_P = 2 \cdot \exp\left(-\frac{2.48}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \tag{2}$$

At the same time, Ref. [5] already mentioned that high phosphorus concentration can lead to errors in calculations because of a tendency of this element to segregate. The surface concentration was not determined in the [5]. Another problem revealed in [5] was deviation of experimental values of p-n-junction depth in Sb diffusion (as the most studied dopant) from calculated dependence of p-n-junction depth on time (<sup>d</sup> � ffiffi <sup>t</sup> <sup>p</sup> ) at large time values. Therefore for estimation of the diffusion coefficient, a low diffusion time was used. Decrease of a penetration depth against expected one was attributed to diffusant evaporation in the diffusion process. These problems connected with the integral nature of a method of D coefficient determination.

leads to the appearance of generation and recombination components in a continuity equation

Germanium is an important element to development of semiconductor theories and practice, and also it is a subject of many diffusion process researches. In this chapter, we focus on a narrow question: phosphorus diffusion in germanium, one of the main dopant of this material. Descriptions of diffusion processes were developing simultaneously with research of the crystalline and defect structure of this material and with improving of dislocation-free crystal growth technology together with development of measurement techniques and mathematical description of diffusion processes. That is why results that are 40 or 50 years old could be significantly different from contemporary ones. All these questions are under study and development. Progress in the first principal calculations together with the development of experimental techniques such as atomic force and scanning tunneling microscopy that allows to distinguish individual atoms and their lattice position will lead to the refinement of mechanisms and characteristics of diffusion processes. Our goal is to present the available data and knowledge about diffusion of phosphorus in germanium, possibly noting the problems and

Phosphorus, as a p-element of the group V of the periodic table, is a shallow donor impurity in germanium. The first works on phosphorus diffusion are about 1952–1954 years [3–5], and

It was previously mentioned that III and V group elements have a smaller diffusion coefficient than other groups of elements, and changes are mostly due to the frequency factor D0. This was explained by their smaller ionic radius [5]. However, for elements of V group in germanium, this tendency was not confirmed (unlike that in silicon). Phosphorus, for example, having smaller ionic radius than any other V group element, has a smaller diffusion coefficient. For all shallow dopants (except of B), the activation energy is estimated as about 2.5 eV, and it

For a long time, constant diffusion coefficients were used for a fixed temperature [2–6]. These results were fairly expected, as in the absence of a reliable dopant profile measurement method, the diffusion coefficient was determined by p-n-junction depth; therefore it is in

> kT

kT 

At the same time, Ref. [5] already mentioned that high phosphorus concentration can lead to errors in calculations because of a tendency of this element to segregate. The surface

cm<sup>2</sup> � <sup>s</sup>�<sup>1</sup> (1)

cm2 � <sup>s</sup>�<sup>1</sup> (2)

slightly increased with decreasing diffusion coefficient in the range of As—Sb—P [5].

DP <sup>¼</sup> <sup>1</sup>:<sup>2</sup> � exp � <sup>2</sup>:<sup>5</sup>

DP <sup>¼</sup> <sup>2</sup> � exp � <sup>2</sup>:<sup>48</sup>

[5] and taking into account the semiempirical Langmuir-Dushman formula:

that describes a diffusion process [1, 2].

32 Advanced Material and Device Applications with Germanium

limitations of the representations used.

their review is in [2, 6].

2. Phosphorus diffusion: first steps

In [7], the phosphorus profiles were determined using layered etching and sheet resistance measurements. Profiles of P in Ge that were made by vapor phase diffusion process were obtained for two surface phosphorus concentrations: less than and more than intrinsic carrier density ni and at four diffusion temperatures—600, 650, 700, and 750�C. This allows to characterize temperature dependence of D. At low surface concentrations, the profile is described by Fick law, and diffusion coefficient is

$$D\_1 = 330 \cdot \exp\left(-\frac{3.1}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \tag{3}$$

At high surface concentration profiles which were extended, later [8] a name "box shaped" appears. For diffusivity calculations, authors applied Boltzmann-Matano method [1]. A dependence of the diffusion coefficient on the local phosphorus concentration was discussed. For the concentration-independent part, there was an expression obtained:

$$D\_h = 0.01 \cdot \exp\left(-\frac{2.1}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \tag{4}$$

Experimental data did not fit well into Arrhenius curves, especially for data at high phosphorus concentrations. With the temperature increase, the diffusion activation energy also increased.

Similar results were obtained in [8]. SIMS method was used for concentration profile measurements. Phosphorus diffusion was carried out at temperature range 600–910�С. Surface concentration of phosphorus was higher than 10<sup>19</sup> cm�<sup>3</sup> ; therefore all samples were showing "box-shaped" profiles. Boltzmann-Matano method also was used for evaluating the concentration dependence of P diffusivity. The observed concentration dependence was approximately in agreement with results of [7]. The strong concentration dependence in D was attributed to dependence of D on Fermi level or due to strain effects caused by the difference in ionic radius of P in Ge.

In [8], temperature dependence of phosphorus diffusivity was found as

$$D\_1 = (0.009 \pm 0.025) \cdot \exp\left(-\frac{2.1 \pm 0.2}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \tag{5}$$

However the data of the paper allowed to derive another D(T):

$$D\_1 = 1.21 \cdot \exp\left(-\frac{2.53 \pm 0.2}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \tag{6}$$

CPm � CVk

Generally, reaction (7) is a fast process compared to time scale of diffusion, which typically amounts to several minutes up to several hours. For this condition local equilibrium of the

> DP <sup>¼</sup> <sup>C</sup>ð Þ PV jDð Þ PV <sup>j</sup> CPm <sup>þ</sup> <sup>C</sup>ð Þ PV <sup>j</sup>

<sup>¼</sup> ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> <sup>D</sup>ð Þ PV <sup>j</sup> CPm

∂Jx

a function of time t and position x. Possible reactions between X and other defects are taken

In [10–21], the behavior of P and Sb was consistently explained by means of the double ionized

In [17], As, Sb, and P were used for diffusion experiments. A Ge-dopant alloy source with about 1 at. % dopant content was used. Diffusion anneals were performed at temperatures

∂Cx

Jx ¼ �Dx

<sup>P</sup> � DPVj � <sup>n</sup>2, if <sup>j</sup> ¼ �2, Deff

∂Cx ∂t þ

where C<sup>x</sup> and Jx, respectively, are the concentration and flux of point defect X (P<sup>m</sup>

s

<sup>P</sup> � DPVj � <sup>n</sup><sup>3</sup>

Deff Pm s

into account by Gx. If flux is determined by the diffusion of X, that is,

∂Cx <sup>∂</sup><sup>t</sup> � <sup>∂</sup> ∂x Dx ∂Cx ∂x 

<sup>P</sup> � n2

In one dimension, the diffusion equation takes the form:

where Pm-phosphorus in substitution position, V<sup>k</sup>

vacancy-phosphorus complex in j-ionization state.

reaction is reached.

If n ≈CP > ni,

For the conditions near equilibrium:

Thus, for <sup>m</sup> ¼ þ1, j ¼ �1, Deff

The diffusion equation is given by

vacancy mechanism:

If <sup>m</sup> ¼ þ1, j ¼ �1, then <sup>D</sup>eff

<sup>C</sup>ð Þ PV <sup>j</sup> � nj�n�<sup>k</sup> <sup>¼</sup> const (8)

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


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

<sup>m</sup>�<sup>j</sup> (10)

<sup>∂</sup><sup>x</sup> <sup>¼</sup> Gx, (11)

<sup>∂</sup><sup>x</sup> (12)

¼ Gx (13)

PV�<sup>1</sup> <sup>¼</sup> <sup>P</sup>þ<sup>1</sup> <sup>þ</sup> <sup>V</sup><sup>2</sup>� (14)

s, V<sup>k</sup> , P-V<sup>j</sup> ) as

—

35

(9)

Figure 1. Diffusivity dependence on phosphorus concentration.

In the later works, a diffusion coefficient was called "intrinsic" for material, in which a dopant concentration n < n<sup>i</sup> at the growth temperature, and it was called "extrinsic" when n > ni.

In Figure 1, there is the dependence of D on phosphorus concentration from [5, 7, 8]. Data of [8] were calculated by Eq. (6).

Integral values in [5] are noticeably higher than intrinsic D in [7, 8]; however, it does not exceed D in [7, 8] for high phosphorus concentrations.

Surprisingly, the experimental papers [7, 8] did not take into account extrinsic diffusion and dopant diffusion models, suggested in 1968 [9] and developed later [10–21]. Since vacancy in germanium is mostly acceptor with charge state up to �3, then positively charged phosphorus ion makes Coulomb-coupled pair with a charged vacancy. Diffusion of such pairs goes faster, and it was expected that it is in direct proportion to charged complex concentration.
