Andrzej Wolkenberg, "A Model of Numerical Calculation of Conductivity for III-V MBE Epilayers Using a Hall Device", Journal of Materials, vol. 2013, Article ID 260982, 16 pages, 2013. https://doi.org/10.1155/2013/260982
A Model of Numerical Calculation of Conductivity for III-V MBE Epilayers Using a Hall Device
Andrzej Wolkenberg^{1,2}
^{1}Institute of Electron Technology, Aleja Lotnikow 32/46, 02-668 Warsaw, Poland
^{2}Institute of Materials Engineering, Czestochowa University of Technology, Aleja Armii Krajowej 19, 42-200 Czestochowa, Poland
Academic Editor: Ram Katiyar
Received13 Nov 2012
Accepted10 Jan 2013
Published25 Mar 2013
Abstract
An electrical conduction versus temperature model using a Hall device was developed. In the case of InAs, InGaAs, and GaAs MBE epilayers, the prediction agrees well with the experimental results. Herein, we explain here how these calculated fractions of total conductivity describe the measured values. The method allows for the calculation of the carrier concentration and mobility of each component of a multicarrier system. The extracted concentrations are used to characterise the different components of charge transport in the active layer. The conductance values G [S] of these components of charge transport were obtained. Also the scattering events for the investigated samples are presented. The analysis of the experimental results for three semiconductor compositions and different concentrations demonstrates the utility of our method in comparing the conductance of each component of the multilayered system as a function of temperature.
1. Models
1.1. Two Layer Model
We take as an example a physical two-layer model according of thin InAs MBE layers. After etching step by step Grange et al. [1] obtained a 1 μm InAs MBE layer with the concentration profile shown in Figure 2 (▼). Near the interface with the GaAs substrate, the researchers measured the concentration to be as high as /cm^{3}.
This result is in agreement with the TEM image presented in Figure 1 obtained by Fawcett et al. [2]. We proposed that the MBE layer of InAs consists of two parts:(i)a strongly defected layer measuring in thickness, (ii)a correctly formed layer measuring in thickness, where is the metallurgical thickness.
We present in Figure 2 data regarding the concentration values obtained by many authors. The following equation is used to relate the experimental concentration data with the InAs layer thickness:
where , , and .
Therefore one can convince one’s self that the two-layer model may be practically used in our calculation.
1.2. Calculation Background
Electron and hole concentrations are derived by the numerical solution of the neutrality equation:
where is the total conductivity electron concentration,
is the net concentration, is the concentration of the donor-like states, is the concentration of the acceptor like state, is the effective density of states in the valence band, is the ionisation energy of the acceptor-like states, is the Boltzmann constant, is the hole concentration, is the concentration of shallow donors with activation energy , is the concentration of deep donors with activation energy , and is the concentration of acceptors with activation energy .
To obtain the mobility, we numerically integrate the equation
where the number of used values depends on values presented in Tables 4, 8, and 12 and the scatterings times are presented in Table 1.
Semiconductor
Scattering on ionized donors and acceptors
Scattering on acoustic phonons (deformation potential)
Scattering on acoustic phonons (piezoelectric potential)
Scattering on optics polar phonons
Scattering on space charge
Scattering on dislocations
InAs
;
;
In_{0.53}Ga_{0.47}As
;
;
GaAs
;
;
2. Introduction
Structural inaccuracy, especially in layered structures, is one factor that affects how the magnitude of a magnetic field influences measured charge transport properties [3]. While studying the transport properties of HgCdTe layers, Gold and Nelson [4] reported on the influence of different charge carriers using Hall characterisation. An exhaustive review concerning these phenomena was undertaken by Meyer et al. [5]. Such disorder was first described and interpreted by the so-called mobility spectrum by Beck and Anderson [6] and many papers have focused on this problem [7–20]. For the proper characterisation of a multicarrier conduction system, the number of independent measurements must match the number of desired physical quantities. In actuality, the complete characterisation of multicarrier conduction is extremely complex. Even for two-carrier conduction, one must make four independent measurements to calculate the individual carrier concentration and mobility of each component [21]. To this end, other methods, such as the maximum entropy approach [22] or the application of Bryan’s algorithm [23], must be considered. The concept most often used in the characterisation of mobility spectra has been described in many papers by Antoszewski and collaborators [12, 24, 25]. The main features of the above-mentioned methods are as follows.(1)The Hall voltage of the investigated semiconductor is not proportionally dependent on the magnetic field.(2)Such behaviour in the conductivity process results in charge carriers (electrons and holes) with different mobilities. (3)The mentioned authors [6, 12, 24, 25] used different mathematical approaches to obtain mobility spectra from mobility and conductivity data.In cases in which this concept is applicable, the calculations are supported by the conductivity tensor dependence on the magnetic field [6]:
where ; or and and are a pair of conductivity density functions.
Equation (5) allows for the treatment of the nonlinear magnetic field dependence on the Hall effect [10]. Many experimental results have been analysed using this method [8–12, 24, 25]. Antoszewski et al. concluded that a major advantage of mobility spectrum analysis (MSA) and quantitative mobility spectrum analysis (QMSA) over other methods is that these procedures are inherent not arbitrary that is, no prior assumptions are required [24, 25]. These investigators suggested that their improved (QMSA) method has sufficient reliability, versatility, and sensitivity for their proposed analysis which is fully computer-automated, using magnetic field dependence data as input.
Our approach accounts for several sources of charge carriers as components contribution to the total conductivity of a semiconductor layer [3]. These sources form the parts used to solve the neutrality equation, and the solution is obtained by numerical computer calculation. The values of such charge carrier sources are chosen such that the calculated concentration curve approximates the experimental data [3, 26–32], as demonstrated for our samples (Table 2) by the example in Figure 3. We believe that the measured conductivity is affected by the presence of different donors in the semiconductor layers. Our model contains four charge carrier sources: two donors—one shallow (), located under the conductivity band (), and one deeper (); one unknown , which in our approach, can be donor-like or acceptor-like, depending on the temperature; and one acceptor () as the source of holes. The elements , , and are located in the bulk of the epitaxial layer while is located at the interface between the substrate and the semiconductor within the first atomic layers (for an explanation of this concept, see Section 1—Models). We numerically solve the neutrality equation, thereby combining the values of , , , and together with their energies of activation, until we obtain the curves or lying as closely as possible to the experimental data at temperatures of 4–300 K. With values for and and using well-known physical relations (see Section 1—Models), we can calculate all transport properties needed to determine the conductance values. The negative values of the calculated transport parameters show that at the appropriate temperatures, charge carriers show hole-like properties. We suppose that our approach does not lose the physical interpretation of each step. The results were carefully analysed to ensure that the numerical calculations provided an accurate physical model of the studied effect.
Investigated semiconductor
Energy gap [eV]
Intrinsic concentration [cm^{−3}]
Average distance between impurities [Å]
Donor state Bohr radius [Å]
Donor state ionisation energy [meV]
InAs
0.354
~1 · 10^{15}
384.5
367
1.05
1.36
In_{0.53}Ga_{0.47}As
0.743
8.5 · 10^{11}
1224.6
180
6.8
2.8
GaAs
1.43
2.2 · 10^{6}
809.5
104
7.8
5.25
3. Experimental
The procedure for epitaxial layers of InAs is described as an example. The layers were obtained by molecular beam epitaxy (MBE) in a Riber 32P reactor on GaAs (001) substrates. The preparation conditions are presented in Table 3.
No. of procedures
Thickness [μm]
Relation between V/III
Substrate temperature [°C]
92
4
5
440
298
5.6
4.5
450
301
4.7
3.7
500
330
9.05
5.8
506
No.
Parameter
InAs 330 9.0 μm
InAs 301 4.7 μm
InAs 298 5.6 μm
InAs 292 4 μm
1
Mobility [cm^{2}/V · s]
, , , ,
, ,, , ,
,
,
The layers exhibited very smooth surfaces and good crystallographic quality (RHEED results) and electrical properties (Hall measurements). The electrical properties are measured “ex situ” by the Hall effect (measurements currents of A– A in a 0.6 T magnetic field). The Hall device had a square-shaped configuration with dimensions of 5 5 mm. The ohmic contacts of the device were made of In 99% + Sn 1% alloy in a Bio-Rad Micro-science-Division (RC2400 Alloying Furnace). The linearity of the contacts characteristics were verified before each Hall effect measurement. All layers were intentionally undoped. The measured n-concentration of the wafers resulted from technical factors of preparation. As layers were deposited by MBE on SI-InP and GaAs as well as SI-GaAs. We carried out the calculations as described previously [3] for a few samples of InAs, As and GaAs with different charge carrier concentrations [3, 26–32]. The procedure consisted of the following steps:(1)numerical solution of the neutrality equation,(2)numerical calculation of the mobility of each component of the neutrality equation versus temperature; (3)after determining the values for concentration and mobility versus temperature, we could obtain the conductivity values.
To validate the accuracy of the calculation, we compared the resistivity versus temperature, that is, calculated the curve using experimental data as in Figure 3 (resistivity was measured without a magnetic field; see Figure 4).
We present a more detailed example of the calculation for an InAs sample measuring 9.0 m thick (see Tables 2, 4, 5, 6, and 7, and all figures) [28]. The InAs samples have the lowest (Table 2). The results are presented in Figures 3, 4, 5, 6, 7, 8, 9, and 10 with the parameters used to solve the neutrality equation (according to [3, 32]). In addition to the results presented in [28], we provide the Hall curves calculated by using the dependence and the calculated values of , , and versus temperature in Figures 11, 12, 13, and 14.
No.
Parameter
10 K
70 K
InAs
InAs
InAs
InAs
InAs
InAs
9 μm
4.7 μm
9 μm
4.7 μm
5.6 μm
4 μm
1
[cm^{−3}]
3.8 · 10^{15}
7.9 · 10^{15}
2.7 · 10^{15}
6.62 · 10^{15}
4.52 · 10^{16}
4.085 · 10^{17}
2
[cm^{2}/V · s]
8000
3600
49000
33500
17350
*
3
[1/Ω cm]
4.9
4.56
21.17
35.53
125.63
*
4
1
1
1
1
1
*
5
[cm^{−3}]
1.7 · 10^{15}
2.01 · 10^{15}
5.03 · 10^{15}
5.04 · 10^{15}
1.495 · 10^{16}
1.43 · 10^{16}
6
[cm^{2}/V · s]
7653
3320.15
49000
33153
23972.1
*
7
[1/ cm]
2.07
1.07
21.87
26.76
57.41
*
8
0.426
0.234
1.03
0.753
0.46
*
9
[cm^{−3}]
2.17 · 10^{15}
5.94 · 10^{15}
−1.06 · 10^{14}
5.64 · 10^{15}
3.023 · 10^{16}
3.94 · 10^{17}
10
[cm^{2}/V · s]
19465
2286
−1914
7536.31
28202.4
*
11
[1/ cm]
6.76
2.175
0.033
6.81
136.58
*
12
1.4
0.48
0.0015
0.192
1.087
*
13
[cm^{−3}]
−4.63 · 10^{14}
−2.45 · 10^{14}
−2.13 · 10^{14}
−1.69 · 10^{14}
−8.66 · 10^{14}
*
14
[cm^{2}/V · s]
−100
−472.22
−2740
−1623.56
28518.2
*
15
[1/Ω cm]
0.0074
0.0185
0.093
0.044
−3.96
*
16
0.0015
0.00406
0.0044
0.0124
0.0315
*
17
[cm^{−3}]
3.407 · 10^{15}
7.705 · 10^{15}
4.7 · 10^{15}
1.05 · 10^{16}
4.43 · 10^{16}
4.083 · 10^{17}
18
[S]
4.41 · 10^{−3}
2.14 · 10^{−3}
1.9 · 10^{−2}
1.67 · 10^{−2}
7.03 · 10^{−2}
*
19
[S]
1.86 · 10^{−3}
5.03 · 10^{−4}
1.97 · 10^{−2}
1.26 · 10^{−2}
3.2 · 10^{−2}
*
20
[S]
6.0 · 10^{−3}
1.02 · 10^{−3}
2.97 · 10^{−5}
3.2 · 10^{−3}
7.65 · 10^{−2}
*
21
[S]
6.66 · 10^{−6}
8.7 · 10^{−6}
8.37 · 10^{−5}
2.068 · 10^{−5}
−2.2 · 10^{−3}
*
22
[S]
7.87 · 10^{−3}
1.53 · 10^{−3}
1.98 · 10^{−2}
1.58 · 10^{−2}
10.63 · 10^{−2}
*
23
/
1.78
0.71
1.042
0.95
1.5
*
No.
Parameter
200 K
InAs
InAs
InAs
InAs
9 μm
4.7 μm
5.6 μm
4 μm
1
[cm^{−3}]
3.28 · 10^{15}
7.86 · 10^{15}
4.68 · 10^{16}
4.055 · 10^{17}
2
[cm^{2}/V · s]
24000
21900
16600
8700
3
[1/Ω cm]
12.6
27.576
124.46
564.
4
1
1
1
1
5
[cm^{−3}]
4.0 · 10^{15}
8.12 · 10^{15}
3.41 · 10^{16}
9.88 · 10^{16}
6
[cm^{2}/V · s]
23570
29849.3
17016.3
10221
7
[1/ cm]
15.65
27.12
92.96
162.
8
1.24
0.983
0.75
0.286
9
[cm^{−3}]
−7.5 · 10^{14}
−2.49 · 10^{14}
1.276 · 10^{16}
3.07 · 10^{17}
10
[cm^{2}/V · s]
−4765
−601.37
5692.03
31734
11
[1/ cm]
0.57
0.024
11.63
1559
12
0.045
0.00087
0.093
2.76
13
[cm^{−3}]
−9.0 · 10^{13}
5.88 · 10^{14}
−1.54 · 10^{15}
*
14
[cm^{2}/V · s]
−621
1506.67
5758.1
*
15
[1/Ω cm]
0.009
0.142
−1.42
*
16
0.0007
0.00515
−0.0011
*
17
[cm^{−3}]
3.16 · 10^{−15}
8.46 · 10^{15}
4.5 · 10^{16}
4.058 · 10^{17}
18
[S]
1.134 · 10^{−2}
1.3 · 10^{−2}
6.97 · 10^{−2}
22.56 · 10^{−2}
19
[S]
1.41 · 10^{−2}
1.27 · 10^{−2}
5.2 · 10^{−2}
6.48 · 10^{−2}
20
[S]
5.13 · 10^{−4}
1.13 · 10^{−5}
6.5 · 10^{−3}
62.36 · 10^{−2}
21
[S]
8.1 · 10^{−6}
6.67 · 10^{−5}
−7.95 · 10^{−4}
*
22
Σ [S]
1.45 · 10^{−2}
1.28 · 10^{−2}
5.77 · 10^{−2}
68.84 · 10^{−2}
23
/
1.28
0.98
0.83
3.05
No.
Parameter
300 K
InAs
InAs
InAs
InAs
9 μm
4.7 μm
5.6 μm
4 μm
1
[cm^{−3}]
4.18 · 1015
9.62 · 10^{15}
4.97 · 10^{16}
4.09 · 10^{17}
2
[cm^{2}/V · s]
16000
15800
14580
8400
3
[1/Ω cm]
10.7
24.35
116.08
550
4
1
1
1
1
5
[cm^{−3}]
5.62 · 10^{15}
8.13 · 10^{15}
4.27 · 10^{16}
1.45 · 10^{17}
6
[cm^{2}/V · s]
16240
15611
14565.7
9090
7
[1/ cm]
14.6
20.33
99.64
211
8
1.36
0.835
0.858
0.38
9
[cm^{−3}]
−1.43 · 10^{15}
−7.56 · 10^{14}
7.073 · 10^{15}
2.64 · 10^{17}
10
[cm^{2}/V · s]
−4430
−1109.67
2275.6
16535
11
[1/ cm]
1.01
0.134
2.58
698
12
0.095
0.0055
0.022
1.27
13
[cm^{−3}]
4.5 · 10^{14}
1.862 · 10^{15}
−1.51 · 10^{15}
*
14
[cm^{2}/V · s]
1965
3636.22
2302.22
*
15
[1/Ω cm]
0.14
1.085
−0.56
*
16
0.013
0.0445
−0.048
*
17
[cm^{−3}]
4.64 · 10^{15}
9.24 · 10^{15}
4.83 · 10^{16}
4.09 · 10^{17}
18
[S]
9.63 · 10^{−3}
1.145 · 10^{−2}
6.5 · 10^{−2}
22.0 · 10^{−2}
19
[S]
1.314 · 10^{−2}
9.55 · 10^{−3}
5.58 · 10^{−2}
8.44 · 10^{−2}
20
[S]
9.09 · 10^{−4}
6.3 · 10^{−5}
1.44 · 10^{−3}
27.92 · 10^{−2}
21
[S]
1.26 · 10^{−4}
5.1 · 10^{−4}
3.14 · 10^{−4}
*
22
[S]
1.417 · 10^{−2}
1.012 · 10^{−2}
5.75 · 10^{−2}
36.36 · 10^{−2}
23
/
1.47
0.88
0.88
1.65
No.
Parameter
In_{0.53}Ga_{0.47}As
In_{0.53}Ga_{0.47}As
In_{0.53}Ga_{0.47}As
In_{0.53}Ga_{0.47}As
7 μm
1.0 μm
1.1 μm
1.0 μm
1
Mobility [cm^{2}/V · s]
, , ,
, ,
, ,
, , , ,
Figure 3 presents the results of solving the neutrality equation [3, 28, 32] for the InAs sample. In this case, we used the metallurgical thickness of the epitaxial layer, which allowed us to calculate the conformity between the thickness values used to solve for the concentration and resistivity versus temperature. At low temperatures, there was an increase in concentration (Figure 3), which was probably due to the presence of impurity band conduction. The sample has an additional donor concentration at 0.13 eV below the bottom of the conduction band. This level is near for InAs (Table 2). This conductivity concentration () was used to calculate the resistivity-temperature (Figure 4) and the mobility-temperature curves (Figure 5). The calculated Hall concentration below ~100 K lies distinctly under the experimental data.
We also verified the ratio of to the density of states value in the conductivity band versus temperature curve. This ratio determines the degeneration probability in the investigated sample. Unfortunately, below ~15 K, its value is greater than 1; therefore, all calculations below that temperature that use Boltzmann statistics are problematic (Figure 6). It must be explained here that our charge transport model was very accurate only for the GaAs samples (Table 13) and the calculated values using the numerical method were sufficiently reliable. With respect to the two other semiconductors, our model was less accurate. The neutrality equation was properly solved for all investigated samples; therefore, we believe that the concentration values are consistent with the experimental results. However the InAs and InGaAs samples presented problems with the wafers exhibiting very high concentrations at temperatures below 200 K (InAs (Table 5) and InGaAs (Table 9)).
No.
Parameter
10 K
70 K
In_{0.53}Ga_{0.47}As
In_{0.53}Ga_{0.47}As
In_{0.53}Ga_{0.47}As
In_{0.53}Ga_{0.47}As
In_{0.53}Ga_{0.47}As
7 μm
7 μm
1.0 μm
1.1 μm
1.0 μm
1
[cm^{−3}]
1 · 10^{13}
1.2 · 10^{13}
2.66 · 10^{15}
1.095 · 10^{17}
1.8 · 10^{19}
2
[cm^{2}/V · s]
6000
38000
48835
4086.1
1770.3
3
[1/Ω cm]
0.01
0.073
20.8
71.6
50.98
4
1
1
1
1
1
5
[cm^{−3}]
9.5 · 10^{13}
1.5 · 10^{13}
2.4 · 10^{15}
*
*
6
[cm^{2}/V · s]
815
37300
98000
*
*
7
[1/ cm]
0.0124
0.09
37.67
*
*
8
1.24
1.22
1.8
*
*
9
[cm^{−3}]
2.0 · 10^{12}
0.0
4.92 · 10^{14}
*
*
10
[cm^{2}/V · s]
−730
−7650
−78000
*
*
11
[1/ cm]
−0.00023
0.0
−6.15
*
*
12
−0.023
*
−0.295
*
*
13
[cm^{−3}]
0.0
8 · 10^{9}
3.45 · 10^{7}
*
*
14
[cm^{2}/V · s]
−720
−6710
−53100
*
*
15
[1/Ω cm]
*
−0.000009
−2.93 · 10^{−7}
*
*
16
*
0.000123
−1.41 · 10^{−8}
*
*
17
[cm^{−3}]
9.7 · 10^{13}
1.5 · 10^{13}
2.892 · 10^{15}
*
*
18
[S]
7 · 10^{−6}
5.11 · 10^{−5}
0.00208
7.88 · 10^{−3}
5.098 · 10^{−3}
19
[S]
8.68 · 10^{−6}
6.3 · 10^{−5}
0.0038
*
*
20
[S]
−1.61 · 10^{−7}
0
−6.15 · 10^{−4}
*
*
21
[S]
*
*
*
*
*
22
[S]
8.52 · 10^{−6}
6.3 · 10^{−5}
0.003185
*
*
23
/
1.22
1.235
1.53
*
*
The resistivity curves (conductivity parameters) show very satisfactory behaviour. The calculated curve approximately agrees with the experimental data at all temperatures with some discrepancies only at temperatures below 15 K. This disagreement is most marked below 100 K, when the concentration is near 10% of the number of states in the conduction band (Figure 6). The great influence of the space-charge scattering mechanism of the product (where is the density of space charge regions and is their effective scattering cross-sectional area) must be underlined.
The Hall mobility measurements (as in Figure 5) lead to the conclusion that the theoretical disagrees with the real value at lower temperatures (approximately as low as 100 K).
We now present the results of mobility calculations of each component obtained after solution of the neutrality equation (Figures 7–10). At lower temperatures, donors named D2 marginally influenced the total mobility; their numbers were so low that they could be omitted (they have important influence at temperatures 200–300 K). It must be noted that calculated Hall effect curves (Figures 3 and 5 curves in red) lightly disagree with experimental points. The conductivity effect curves show much better compliance with the experiment, especially at low temperatures. Using the measurements procedure as described for the GaAs sample, we obtained results that allowed us to determine the components of conductivity for these samples (Table 2). The scattering mechanism used for this calculation and the pertinent values are shown: for InAs in Tables 4, 5, 6, and 7, for As in Tables 8, 9, 10, and 11, and for GaAs in Tables 12 and 13.