3. Continuum theoretical calculations of dopant diffusion in semiconductors

The most detailed theory that describes dependence of dopant diffusivities on vacancy concentration in different charge states can be found in [10]. Indirect diffusion mechanisms, which involve vacancies Vk , are described by the following reaction:

$$(PV)^{j} + (j - m - k)e^{-} = P^{m} + V^{k} \tag{7}$$

The local equilibrium is characterized by

Phosphorus and Gallium Diffusion in Ge Sublayer of In0.01Ga0.99As/In0.56Ga0.44P/Ge Heterostructures http://dx.doi.org/10.5772/intechopen.78347 35

$$\frac{\mathbb{C}\_{P^n} \cdot \mathbb{C}\_{V^k}}{\mathbb{C}\_{(PV)^i} \cdot n^{j-n-k}} = \text{const} \tag{8}$$

where Pm-phosphorus in substitution position, V<sup>k</sup> -vacancies in k-ionization state, (PV)<sup>j</sup> vacancy-phosphorus complex in j-ionization state.

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 reaction is reached.

For the conditions near equilibrium:

$$D\_P = \frac{\mathbb{C}\_{(PV)^j} D\_{(PV)^j}}{\mathbb{C}\_{P^m} + \mathbb{C}\_{(PV)^j}} \tag{9}$$

If n ≈CP > ni,

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

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

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,

The most detailed theory that describes dependence of dopant diffusivities on vacancy concentration in different charge states can be found in [10]. Indirect diffusion mechanisms, which

� <sup>¼</sup> <sup>P</sup><sup>m</sup> <sup>þ</sup> <sup>V</sup><sup>k</sup> (7)

and it was expected that it is in direct proportion to charged complex concentration.

3. Continuum theoretical calculations of dopant diffusion in

, are described by the following reaction:

þ ð Þ j � m � k e

ð Þ PV <sup>j</sup>

[8] were calculated by Eq. (6).

semiconductors

involve vacancies Vk

The local equilibrium is characterized by

D in [7, 8] for high phosphorus concentrations.

Figure 1. Diffusivity dependence on phosphorus concentration.

34 Advanced Material and Device Applications with Germanium

$$D\_{P\_s^m}^{\rm eff} = (m+1)D\_{(PV)^j} \left(\mathbb{C}\_{P\_s^m}\right)^{m-j} \tag{10}$$

Thus, for <sup>m</sup> ¼ þ1, j ¼ �1, Deff <sup>P</sup> � DPVj � <sup>n</sup>2, if <sup>j</sup> ¼ �2, Deff <sup>P</sup> � DPVj � <sup>n</sup><sup>3</sup>

In one dimension, the diffusion equation takes the form:

$$\frac{\partial \mathbf{C}\_{\mathbf{x}}}{\partial t} + \frac{\partial f\_{\mathbf{x}}}{\partial \mathbf{x}} = \mathbf{G}\_{\mathbf{x}\prime} \tag{11}$$

where C<sup>x</sup> and Jx, respectively, are the concentration and flux of point defect X (P<sup>m</sup> s, V<sup>k</sup> , P-V<sup>j</sup> ) as a function of time t and position x. Possible reactions between X and other defects are taken into account by Gx. If flux is determined by the diffusion of X, that is,

$$J\_x = -D\_x \frac{\partial \mathcal{L}\_x}{\partial x} \tag{12}$$

The diffusion equation is given by

$$\frac{\partial \mathbf{C}\_x}{\partial t} - \frac{\partial}{\partial x} \left( D\_x \frac{\partial \mathbf{C}\_x}{\partial x} \right) = \mathbf{G}\_x \tag{13}$$

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

$$PV^{-1} = P^{+1} + V^{2-} \tag{14}$$

If <sup>m</sup> ¼ þ1, j ¼ �1, then <sup>D</sup>eff <sup>P</sup> � n2

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

between 600 and 920�C for various times in vacuum. The multiple use of the dopant source leads to depletion of the source. So the maximum doping level could be changed from the values that exceed the intrinsic carrier concentration ni to values close or beneath ni at the diffusion temperature. Doping profiles with penetration depths in the range of 30–150 μm were measured by spreading resistance method. Secondary ion mass spectrometry was used to record diffusion profiles with depths of a few microns. It was confirmed that in the range of low dopant concentration, the intrinsic diffusion with the constant Din has been occurred. The extra diffusion with "box-shaped" diffusion profiles was observed when dopant concentration exceeded ni. In this case:

$$D\_{(PV)^{-}}^{\mathcal{eff}} = D\_{(PV)^{-}}(n\_i) \left(\frac{n}{n\_i}\right)^2 \tag{15}$$

In Figure 2, a temperature dependence of the intrinsic diffusivity for cubic and quadratic models, experimental results in intrinsic diffusion regime [5] are presented. Figure 3 demonstrates concentration dependence D for two models together with experimental dependence [5] if proposed n = CP. As we can see, calculated by Boltzmann-Matano values of D differ from estimations of Din from Fickian's part of diffusion curve, as it was done both in [17] for

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

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

37

Figure 3. Dependencies of diffusivity for cubic and quadratic models. Dashed lines are from experimental results [7].

quadratic and [22] for cubic diffusion mechanisms.

Figure 2. Intrinsic diffusivity for different models.

$$D\_p(n\_i) = 9.1\_{-3.4}^{+5.3} \exp\left(-\frac{(2.85 \pm 0.04)eV}{k\_B T}\right) c m^2 s^{-1} \tag{16}$$

Eq. (16) was calculated from Fickian-like profiles at low P concentrations. Then (15) were used for continuity equation, and a good agreement between experiment and calculations was achieved.

A "box-shaped" P profile was also detected under ion implantation procedure [18–21]. The "quadratic model" was used to describe diffusion process.

In [21], the phosphorus distribution in germanium after ion implantation and annealing at temperatures 523 and 700�C was measured by SIMS method. It was shown that neither quadratic nor constant diffusion coefficient models cannot be used for profiles at 700�C annealing and longtime annealing for both temperatures.

Later a cubic dependence of the P diffusivity on the electron concentration was proposed [22]. The equations and dependencies used were.

$$\frac{\partial \mathbb{C}\_P}{\partial t} = -\frac{\partial J\_P}{\partial \mathbf{x}} \; J\_P = -D^{\sharp \mathcal{f}} \cdot \frac{\partial \mathbb{C}\_P}{\partial \mathbf{x}} - D^{\sharp \mathcal{f}} \cdot \frac{\mathbb{C}\_P}{n} \cdot \frac{\partial n}{\partial \mathbf{x}} \tag{17}$$

$$D^{\prime \#} = D^{2-} \left( \frac{n}{n\_i} \right)^2 + D^{3-} \left( \frac{n}{n\_i} \right)^3 \tag{18}$$

$$D\_i = D^{2-} + D^{3-} \tag{19}$$

$$\begin{aligned} D\_i &= 44.3 \cdot \exp\left(-\frac{3.01 \pm 0.04}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \\ D^{2-} &= 11.1 \cdot \exp\left(-\frac{2.93 \pm 0.01}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \\ D^{3-} &= 5.7 \cdot \exp\left(-\frac{2.92 \pm 0.02}{kT}\right) \text{cm}^2 \cdot \text{s}^{-1} \end{aligned} \tag{20}$$

There was a satisfactory conformity between experimental data and calculations for results of these authors and also with experimental data from [17] with this cubic model.

Phosphorus and Gallium Diffusion in Ge Sublayer of In0.01Ga0.99As/In0.56Ga0.44P/Ge Heterostructures http://dx.doi.org/10.5772/intechopen.78347 37

Figure 2. Intrinsic diffusivity for different models.

between 600 and 920�C for various times in vacuum. The multiple use of the dopant source leads to depletion of the source. So the maximum doping level could be changed from the values that exceed the intrinsic carrier concentration ni to values close or beneath ni at the diffusion temperature. Doping profiles with penetration depths in the range of 30–150 μm were measured by spreading resistance method. Secondary ion mass spectrometry was used to record diffusion profiles with depths of a few microns. It was confirmed that in the range of low dopant concentration, the intrinsic diffusion with the constant Din has been occurred. The extra diffusion with "box-shaped" diffusion profiles was observed when dopant concentration

ð Þ PV � <sup>¼</sup> <sup>D</sup>ð Þ PV � ð Þ ni

Eq. (16) was calculated from Fickian-like profiles at low P concentrations. Then (15) were used for continuity equation, and a good agreement between experiment and calculations was achieved. A "box-shaped" P profile was also detected under ion implantation procedure [18–21]. The

In [21], the phosphorus distribution in germanium after ion implantation and annealing at temperatures 523 and 700�C was measured by SIMS method. It was shown that neither quadratic nor constant diffusion coefficient models cannot be used for profiles at 700�C

Later a cubic dependence of the P diffusivity on the electron concentration was proposed [22].

∂CP <sup>∂</sup><sup>x</sup> � <sup>D</sup>eff �

<sup>þ</sup> <sup>D</sup><sup>3</sup>� <sup>n</sup> ni <sup>3</sup>

kT 

kT 

kT 

There was a satisfactory conformity between experimental data and calculations for results of

JP ¼ �Deff �

ni <sup>2</sup>

<sup>D</sup>eff <sup>¼</sup> <sup>D</sup><sup>2</sup>� <sup>n</sup>

Di <sup>¼</sup> <sup>44</sup>:<sup>3</sup> � exp � <sup>3</sup>:<sup>01</sup> � <sup>0</sup>:<sup>04</sup>

<sup>D</sup><sup>2</sup>� <sup>¼</sup> <sup>11</sup>:<sup>1</sup> � exp � <sup>2</sup>:<sup>93</sup> � <sup>0</sup>:<sup>01</sup>

<sup>D</sup><sup>3</sup>� <sup>¼</sup> <sup>5</sup>:<sup>7</sup> � exp � <sup>2</sup>:<sup>92</sup> � <sup>0</sup>:<sup>02</sup>

these authors and also with experimental data from [17] with this cubic model.

Dpð Þ¼ ni <sup>9</sup>:1þ5:<sup>3</sup> �3:<sup>4</sup>exp � ð Þ <sup>2</sup>:<sup>85</sup> � <sup>0</sup>:<sup>04</sup> eV

n ni <sup>2</sup>

> cm<sup>2</sup> s

CP n � ∂n

Di <sup>¼</sup> <sup>D</sup><sup>2</sup>� <sup>þ</sup> <sup>D</sup><sup>3</sup>� (19)

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

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

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

kBT  (15)

(18)

(20)

�<sup>1</sup> (16)

<sup>∂</sup><sup>x</sup> (17)

Deff

"quadratic model" was used to describe diffusion process.

annealing and longtime annealing for both temperatures.

∂CP <sup>∂</sup><sup>t</sup> ¼ � <sup>∂</sup>JP ∂x

The equations and dependencies used were.

exceeded ni. In this case:

36 Advanced Material and Device Applications with Germanium

In Figure 2, a temperature dependence of the intrinsic diffusivity for cubic and quadratic models, experimental results in intrinsic diffusion regime [5] are presented. Figure 3 demonstrates concentration dependence D for two models together with experimental dependence [5] if proposed n = CP. As we can see, calculated by Boltzmann-Matano values of D differ from estimations of Din from Fickian's part of diffusion curve, as it was done both in [17] for quadratic and [22] for cubic diffusion mechanisms.

Figure 3. Dependencies of diffusivity for cubic and quadratic models. Dashed lines are from experimental results [7].
