Electron mobility in InO.53GaO.47As as a function of concentration and temperature

Microelectronics Journal, 26 (1995) 653-657

0026-2692(95)00010-0

Electron mobility in Ino.53Gao.47As as a function of concentration and temperature V.W.L. Chin, T. Osotchan and T.L. Tansley Semiconductor Science and Technology Laboratories, School of MPCE, Macquarie University, N S W 2109, Australia

The electron mobility in Ino.s3Gao.47Asas a function of temperature has been calculated by the variational principle over a carrier concentration range of 1 × 10TM to 1 × 10TM cm-3 with compensation ratio as a parameter. We show that these results, which fit experimental data, can be well represented by a simple analytical function. We also give the fit coeflficientst])r this expression as a function of temperature for various compensation ratios which may facilitate device modelling.

I. Introduction n0.53Ga0.47As is ~, significant material due to its superior transport and optical properties and, at this composition, it is lattice-matched to InP. Interest has been shown in the development o f high speed devices such as high electron mobility transistors (HEMTs) [1], Schottky barrier photodiodes [2] and junction field-effect transistors [3]. These (and other) devices are often modelled by solving the Maxwell's, continuity, and charge conservation equations by iterative methods so that the device characteristics can be predicted.

Electron mobility is an important parameter in device modelling [4] and design, and can be calculated for a given material by solving Boltzmann's transport equation (BTE). There are several ways o f solving B T E including methods based u p o n the relaxation time approximation, iterative techniques, Monte-Carlo simulation, and the variational principle. Jacoboni and I~eggiani [5] have summarized some o f the advantages and disadvantages o f each m e t h o d o f calculating electron mobility.

I

0026-2692/95/$9.50 0 1995 Elsevier Science Ltd

In general, using the variational principle or iterative technique to solve B T E is the most correct m e t h o d for calculating the electron mobility. Comparison o f electron mobilities calculated using Matthiessen's rule formalism, the relaxation time approach, iterative and the variational principle [6] have shown that each can be fitted well to experimental data. T h e values o f alloy scattering potential and c o m p e n sation ratio (0) used in these methods to obtain best fit are also close enough together to suggest

653

V.W.L. Chin et al./Electron mobility in Ino.53Gao.47As

that all three satisfactorily describe the experimental electron mobility. Nevertheless, use of the variational m e t h o d gives the best correlation and sits on a sounder physical basis.

i

. . . . . . . .

i

. . . . . . . .

I ~"

>

\

O. 15

0.45

""

2. Method of calculation

NA/No=O,O

..'" " ",

0.60

0

103

........

t

1 014

........

~

10 is

........

D

i

10 le

"

...... 10 is

1017

C a r r i e r Conch.

(1)

•

Ino.s3GOo.47As T=3OOK

_o

-1

D31e,31e

. . . . . . . .

I 04

I.)

T h e electron mobility calculated using the variational principle is given (cm2V -1 s-1) by [7]

i

. . . . . . . .

0.30

(crn -:3)

Fig. 1. The electron mobility oflno.s3Gao.47As as a function of electron density with compensation ratio 0 as a parameter: dashed curves, variational calculations; dotted

w h e r e Z 1 is the reduced optical p h o n o n energy

curves, eq. (2) at 300K. C), [12]; O, [13]; A, [14]; i , [15]; ~, [16]; V, [17]; [[], [18]; II, [19].

(licol/kT), F1/2 is the Fermi-Dirac integral, ~7 is the reduced fermi energy, and D3/2, 3/2 D are the determinants that contain the variational integrals. The calculation details using variational principle can be found in [8].

0.60 and 0.75 compensation ratio (NA/ND) as an additional parameter, since in real samples it is most likely that the material grown is lightly compensated, usually NA/ND <~0.3 [6].

Figures 1 and 2 show the calculated mobilities at 77 K and 300 K, respectively (dashed curves), for electron concentrations between 1014cm and 1018cm -3 and for a set of compensation ratios NA/ND between 0 and 0.75. In this figure, we have also presented some experimental data from various sources for comparison. It can be seen that the correlation is quite good with most experimental data having a compensation ratio o f less than 0.3. W e have used an alloy scattering potential o f 0.41 eV in these calculations, which is also in excellent agreement with the value o f 0.42 eV determined experimentally [9]. In device modelling, however, it is m u c h more convenient to obtain electron mobility from an algebraic generating function embedded in the model. We present here an algorithm from which electron mobility in Ino.53Ga0.47As can be generated between 7 7 K and 3 0 0 K for an electron density from 1 0 1 4 c m to 1 0 1 8 c m -3. These mobilities are also fitted with 0.0, 0.15, 0.30, 0.45,

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T o each set o f dashed curves in Fig. 1 and 2, we have fitted (dotted lines) an empirical expression for the mobility # as a function o f n and T o f the form [10]:

. . . . . . . .

i

. . . . . . . .

|

. . . . . . . .

i

. . . . . . . .

........

i

........

NA/ND'O-O NA/ND'O-O I(n ~-

lOS

0.15

"E 10 4

T = 7 7 K

10

~

10 t4

........

i 10 Is

0.75 ........

i I0 le

1017

10 III

Carrier Concn. (cm-z)

Fig. 2. The electron mobility ofln0.s3Gao.47Asas a function of electron density with compensation ratio 0 as a parameter: dashed curves, variational calculations; dotted curves, eq. (2) at 77 K. The symbols are the same as Fig. 1.

Microelectronics Journal, VoL 26, No. 7

TABLE 1

Fitted parameters used in eq. (7) to calculate the electron mobility o f l n G a A s ; x = 0.47

I.lm:m(T)0 = al + a2 T 0 (N^/ND)

al

0.00 0.15 0.30 0.45 0.60 0.75

10 100 7673 5686 4031 2654 1504

//o(T)0

=

a2 + 82.99 4. 44.70 4. 20.39 + 13.51 4. 9.50 4- 7.79

-9.80 -5.33 -2.34 -0.407 -0.622 -0.906

4. 0.409 4. 0.220 4- 0.101 4. 0.07 4. 0.05 4- 0.04

10 (=3+a4T+as T2)

0 (NA/ND)

a3

0.0 0.15 0.30 0.45 0.60

5.719 5.682 5.632 5.567 5.484 5.354

0.75

a4 ± 4.789 + 6.549 4. 7.752 4. 1.007 4. 1.412 4. 1.722

-9.878 --9.552 --9.123 --8.612 --8.018 -7.182

x 10 -3

x 10 -3 x 10 -3

x 10 -2 x 10 -2

x 10 -2

as x 1 0 - 3 4- 5.54 X 10 -s x 1 0 - 3 4- 7.58 X 10 - s

X 10 -3 4- 8.97 X 10 -3 + 1.17 X 10 -3 4- 1.63 x 1 0 - 3 4- 1.99

× 10 - s × 10 -4 × 10 -4 × 10 -4

1.155 x 10 -5 4- 1.447 x 1168 x 10 - s 4- 1.976 x 1.149 x 10 -5 4- 2.339 x 1.105 x 10 - s 4- 3.039 x 1.048 x 10 -5 4- 4.262 x 9.542 x 1 0 - 6 4- 5.195 x

10 -7 10 -7 10 -7 10 -7 10 -7 10 -7

n ~ (T)o = 10 ("6+~r)

0 (NA/ND)

a6

0.00 0.15 0.30 0.45 0.60 0.75

14.93 14.77 14.63 14.51 14.39 14.26

a7 4- 5.35 x 10 -2 4- 4.48 x 1 0 - 2 4- 3.96 x 10 -2 4- 3.15 x 10 -2 4- 2.64 x 10 -2 4- 2.53 x 10 -2

5.33 6.12 6.32 6.38 6.26 6.04

x 1 0 - 3 4- 2.64 x 1 0 - 4 x 1 0 - 3 4- 2.21 x 1 0 - 4 x 1 0 - 3 4- 1.95 × 1 0 - 4

x 10 -3 4- 1.55 x 10 -4 x 10 -3 4- 1.30 x 10 -4 x 10 -3 4- 1.25 x 10 -4

tX(T)o = a8 + a9T + a10 T 2 + a 1 1 T 3 0 (NA/ND)

88

a9

a10

a.

0.00

1.024 + 3.17 x 10 -2

- 6 . 1 9 × 10 -3 + 5.93 x 10 -4

3.33 × 10 - s -4- 3.37 x 10 -6

- 5 . 3 1 × 10 -8 + 5.91 x 10 -9

0.15

1.044 & 1.85 x 10 -2

- 6 . 2 6 x 10 -3 -4- 3.47 x 10 -4

3.20 x 10 -s + 1.97 x 10 -6

- 4 . 9 6 x 10 -8 + 3.47 x 10 -9

0.30

1.042 4- 1.23 x 10 -2

- 5 . 8 2 X 10 .3 + 2.30 x 10 -4

2.77 X 10 - s + 1.31 x 10 -6

--4.04 X 10 -8 4- 2.30 X 10 -9

0.45

1.036 4- 2.61 × 10 -2

--5.35 X 10 -3 4- 4.88 x 10 -4

2.39 X 10 -s 4- 2.77 × 10 -6

--3.31 x 10 -8 4- 4.88 x 10 -9

0.60

0.985 4- 4.12 X 10 -2 0.812 4. 4.74 X 10 -2

--3.98 x 10 -3 + 7.70 X 10 -4 - 1 . 2 2 X 1 0 - 4 4. 8.87 X 10 -4

1.52 x 10 -s 4. 4.38 X 10 -6 8.10 X 10 -6 4- 5.04 X 10 -6

--1.76 x 10 -8 4. 7.69 X 10 -9 --2.50 X 10 -8 4- 8.85 X 10 -9

0.75

655

V.W.L. Chin

et

al./Electron mobility in Ino.s3Gao.47As

/~o(T)

#(n,

T)lo =//min(T) -[" [1 + (n/nref(T)) ~]

(2)

:h

re #s ~ is the m i n i m u m mobility (in e v - 1 -mi), #o is the difference between maxim u m and m i n i m u m mobility (in c m 2 V -1 s-l), nref is the reference carrier concentration (in c m -3) and ~ is an exponent that controls the slope at n -~ nref. In order to calculate the electron mobility o f Ino.s3Gao.47As at any temperature, T, we have also fitted/~mi,, #o, nref and ~ to polynomial expressions o f temperature for various 0s, given as:

I

=0"0

o•NA/No

Co

7> O4

E (J

o

v

%

o°

>~ °m ° _

.El 0 3;

//min(T) = 0(1 + 0~2T 10 4

/io(T) = 10 (~3+~4r+~Sr2) n r e f ( T ) = 10(~6 + =7T)

(3)

n=l.5xlOlecm

,

101

.

.

.

.

.

.

.

-~

i

10 2

0~(T) = ~8 + G(9T + 0q0 T2 + C(IIT3 The parameters, ai (i = 1 to 11) are the coefficients o f best fit as T is varied and we have used a m i n i m u m order polynomial in T to obtain a satisfactory fit to the variation calculations. W e have tabulated the best fitting parameters in Table 1, obtained from the LevenbergMarquardt method. T h e '+' after each coefficient indicates the standard deviation o f that particular parameter. Figures 1 and 2 show that the two sets of curves are in good agreement, especially for low compensation ratios (0 < 0.60) where the difference is minimal. Since the compensation ratios of MBE and M O V P E grown samples are usually less than 0.30, use of eq. (2) to calculate electron mobility should be sufficiently accurate over the entire electron concentration and temperature range. W e have also calculated the electron mobility o f In0.53Ga0.47As as a function of concentration using variational principle at various intermediate temperatures from 300 K to 77 K. Since most experimental data are reported at 300 K ad 77 K, only these results are shown. Figure 3 shows the only k n o w n experimental

656

Temperature (K) Fig. 3. The electron mobility of lno.53Gao.47As as a function of temperature. Dashed curves are calculated from the variational principle while solid curves are generated from eq. (2). Experimental data are from [11]. mobility as a function of temperature obtained from [11]. T h e dashed curves are variational principle calculations using n = 1.5 x 1016 c m -3 for a compensation ratio o f 0.0 and 0.15. T h e solid curves are calculated using eq. (2) with the same electron concentration and compensation ratios. It can be seen that the two curves are in good agreement, as well as with the experimental data. Actually, a compensation ratio of 0.10 would fit the experimental data better [6]. However, for device modelling, we believe the current expression is sufficient, while interpolation between the fitted coefficients (a 1 to a11) for compensation ratios other than that presented may be possible to obtain better correlation. In conclusion, we have calculated the electron mobility o f In0.53Ga0.47As from variational principle and fitted it to a simple analytical expression. T h e correlation between the electron mobility calculated by this expression and by the

Microelectronics Journal, VoL 26, No. 7

variation principle is good and is also in remarkably good agreement: with the experimental data. We have also presented this set of fitted coefficients to calculate the electron mobility of In0.s3Gao.47As using this analytical expression, which may facilitate device modelling.

Acknowledgement This work is supported by the Australian Research Council. References [1] S. Bollaert, P. Legnls,E. Delos, A. Cappy, P. Debray and J. Blanchet, Design, fabrication and characterisation of striped charmel HEMTs, IEEE Trans. Electron. Devfff_sED-41 (199.4) 1716. [2] J.H. Kim, S.S. Li, L. Figueroa, T.F. Carruthers and R.S. Wagner, High speed Gao.47n0.s3As/InPinfrared Schottky barrier photodiodes, Electron. Lett., 24 (1988) 1067. [3] J. Cheng, S.R. Forrest, R. Stall, G. Guth and R. Wunder, Depletion and enhancement mode In0.s3Ga0.47As/InPjunction field effect transistor with a p-InGaAs confinement layer, Appl. Phys. Lett., 46 (1985) 885. [4] C.M. Snowden, Introduction to Semiconductor Device Modelling, World Scientific, Singapore, 1986. [5] C. Jacoboni and L. Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with application to covalent materials, Rev. Mod. Phys., 55 (1983) 645. [6] V.W.L. Chin, T. Osotchan and T.L. Tansley, On the calculation of electron mobility in Ino.53Gao.47As, Solid-State Electron., :35 (1992) 1247. [7] J.R. Lowney and H.S. Bennett, Majority and minority electron and hole mobilities in heavily doped GaAs,J. Appl. Phys., 69 (1991) 7102.

[8] V.W.L. Chin, The effect of carrier densities and compensation ratios ont he electron mobilities of InAsxP.l-x, J. Phys. Chem. Solids, 53 (1992) 897. [9] J.H. Marsh, Effects of compositional clustering on electron transport in In0.53Ga0.47As , Appl. Phys. Lett., 41 (1982) 732. [10] N.D. Arora, J.R. Hauser and D.J. Roulston, Electron and hole mobilities in silicon as a function of concentration and temperature, IEEE Trans. Electron. Devices, ED-29 (1982) 292. [11] Y. Takeda, M.A. Littlejohn and J.B.. Hauser, Electron Hall mobility calculations and alloy scattering in ][no.53Gao.47As , Electron. Lett., 17 (1981) 377. [12] J.D. Oliver, L.F. Eastman, P.D. Kirchner and W.J. Scha~, Electrical characterisation and alloy scattering measurements of LPE Ga~In~_xAs/InP for high frequency device applications, J. Cryst. Growth, 54 (1981) 64. [13] C.D. Lee and S.R. Forrest, In0.s3Ga0.47As/InP heterojunction with low interface defect densities, J. Appl. Phys., 69 (1991) 342. [14] K. Ohtsuka, T. Ohishi, Y. Abe, H. Sugimoto, T. Matsui and H. Ogata, High purity In0.s3Ga0.47As layer grown by liquid phase epitaxy,J. Cryst. Growth, 89 (1988) 391. [15] T.P. Pearsall, Alloy scattering effects and calculated mobility in n-type g ~ o . 4 7 I n 0 . 5 3 A s , Electron. Lett., 17 (1981) 169. [16] R.C. Chen, G. Fomutto, C. Lamberti and S. Pellegrno, Growth of high purity InGasAs LPE layers and their characterisation, J. Cryst. Growth, 102 (1990) (477). [17] S. Kondo, T. Amano and H. Nagai, High purity LPE growth of InGaAs by adding A1 to melt, J. Cryst. Growth, 64 (1983) 433. [18] E. Kuphal and D. Fritzsche, LPE growth of high purity InP and n- and p-In0.53Gao.47As,J. Elearon. Mater., 12 (1983) 743. [19] A.A. Fadl, E.K. Stefanakos and W.J. Collis, Current controlled liquid phase epitaxial (CCLPE) growth of InGaAs on (100) InP, J. Electron. Mater., 11 (1982) 559.

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