Non-equivalent minima scattering in n-type Ga1−xAlxAs alloys from Monte Carlo methods

Volume 89A, number 7

PHYSICS LETI'ERS

31 May 1982

NON-EQUIVALENT MINIMA SCATTERING IN n-TYPE Ga I -x Alx As ALLOYS

FROM MONTE CARLO METHODS Ashok K. SAXENA 1 Department o f Electronics and Electrical Engineering, The University o f Sheffield, Mappin St. Sl 3JD, Sheffield, UK Received 11 November 1981

Using the known conduction band structure for Gal_xAlxAs alloys, the electron drift mobilities in the P, L and X minima have been calculated as a function of pressure using Monte Carlo methods. These mobilities are found to decrease first, show a minimum at a pressure near the P-L minima crossoverand then increase again with pressure due to strong nonequivalent intervalley scattering. The calculated Hall mobility also shows similar behaviour and the results are in qualitative agreement with the experimental observations reported earlier.

For high field devices, the Ga 1 -x Alx As alloy system has the very useful property that the electron drift velocity v versus electric field e characteristics change with the A1 content x. For low values of x, the v - e characteristics show a region of negative differential mobility as for GaAs, but for x values for which the satellite minima are either close to the central P-minimum or the band structure is indirect, the v - e curves exhibit a positive differential mobility at all fields. For the alloy composition x = 0.34, the characteristic has been found to be nearly saturated at all fields (0 < e ~< 70 kV/cm) [1]. The occurrence of current saturation in 3 - 5 group ternary compounds has also been reported by Majerfeld and Pearson [2] and Immorlica and Pearson [3] for GaAs 1 -x Px and Gal -x Alx As altoys, respectively. This property of current saturation may be used in current limiting and microwave switching devices. Sitch et al. [4] have proposed a high efficiency double velocity electron transit time (DOVETT) GaAs-Ga 1 -x Alx As heterostructure device, whose operation depends on the fact that the value of the saturated electron velocity in Gal -x Alx As (x = 0.34) is lower by a factor of 3 to 5 compared to that in GaAs [1]. The Ga 1 -x Alx As t

1 Presently with the Department of Electronics and Communication Engineering, The University of Roorkee, Roorkee (U.P.), India 2471572. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland

alloy system is particularly important for heterostructure devices because of the lowest lattice mismatch (~0.16 percent) between GaAs and AIAs as compared to other ternary compounds [5]. The conduction band structure of Ga 1 -x Alx As alloys has been established by Saxena [ 6 - 8 ] . For compositions near the lowest energy direct(F)indirect(X) minima crossover, the L and X minima are close in energy to the r-minimum and, therefore, additional non-equivalent intervalley scattering could be expected. This scattering shou!d lead to a reduction of the high field saturation velocities. It has been shown that the intervalley scattering does play a significant role in limiting the electron mobility in Gal -x Alx As alloys and reflects a minimum in the measured Hall mobility at 300 K as a function o f x and near the composition for the F - L minima crossover [ 9 - 1 1 ] . A similar result has been reported for GaAs l_xPx [12] and Gal_ x InxAs [!3] alloys but the data remained unexplained. The changes in the band structure of Ga I -x Alx As alloys with x are similar to those with hydrostatic pressure for a given composition [6-8,14]. It has been found that for a given decomposition with a direct energy band-gap, the measured Hall mobility first decreases with pressure, shows a minimum at a pressure near the F - L crossover and then increases again with pressure [8]. It was also suggested that this mini351

Volume 89A, number 7

PHYSICS LETTERS

mum occurs primarily due to strong non-equivalent intervalley scattering among the various minima. For careful device design, the various parameters involved in this scattering must be known to assess their performance theoretically. Pitt et al. [15] observed a similar effect for In 1 -x Gax P alloys and suggested that the L - X minima intervalley scattering was responsible for the minimum in the Hall mobility as a function of hydrostatic pressure. In the present letter, we report the results of electron drift mobilities in the P, L and X minima and the Hall mobility of Ga I -x Alx As alloys as a function of pressure as determined by the Monte Carlo method. A striking feature of the results is that the calculated drift mobilities show a minimum at a pressure near the r - L minima crosso,cer and a minimum in the Hall mobility, in qualitative agreement with the experimental observations reported elsewhere [8]. The scattering rates of electrons in Ga 1 -x Alx As alloys are supposed to be limited by the following mechanisms: (i) deformation potential scattering Xdp, (ii) polar optical scattering Xpo, (iii) ionised impurity scattering Xii, (iv) non-equivalent kne q and (v) equivalent intervalley scattering Xeq. The relevant equations for parabolic bands are given below [16] Xdp = [(2m*)3/2E21k B T/2,r~i4ps2]vrff,

(1)

Xpo = [e2(m* )l/2 ~lo/,V~h ](e--1 - e-61)E-1/2

31 May 1982

where Ni/ ='[exp(~16oiJk B T) - 1 ] -1 ;

keq = [(Z i - 1)(m~)3/2D2/vr2-ffp~2~3 ] X Nij [E(k) + P/~oi/]1/2

(absorption),

heq = [(Z/. - 1)(m~)3/2D2/x/~O ¢o2h3 ]

X (Nij + 1)[E(k) - broil] 1/2

_~+

~

(absorption),

?'po = [e2(m*)l/2a'h_o/~/2?i](e---1 - CO1) E-l~2 X (N O + 1) In ~

_~~

(emission), (2)

where N O = [exp(h6Olo/k B T) - 1] -1 ;

Xii = [2~r~Nie4(m*)l/2/e2~2~2 ]E -1/2 ,

(3)

where 3 = 2eOrNi/ekB T) 1/2 ; X eq =

X Nil [E(k) -- A/- + A/+/i~ii ] 1/2

(absorption),

Xneq tzi(mD3/2 z~,~/'c%-p ~o~ 3] =

X (Nil + 1)[e(k) - A/+ &i - 1f6oi1]1/2

(emission),

(4) 352

(5)

where m* = electron effective mass, E 1 = acoustic deformation potential, p = material density, s = sound velocity, E = electron energy, e** and e 0 = high and low frequency dielectric constants respectively, 6Olo = angular frequency of longitudinal optical phonons, N i = impurity concentration, Z/= number of equivalent minima, Dil and co//= deformation potential field and angular frequency of phonons respectively for scattering from state k in valley i to a state in valley], A~/= energy of the minima from the appropriate valley. We have neglected the piezoelectric and alloy scatterings since the scattering rates due to these mechanisms have been found to make a negligible contribution in comparison to the mechanisms considered above [11]. It has also been assumed that the F-valley is spherical at all energies of interest, but we have taken account of the non.parabolicity by relating the energy E to the wave-vector k through the relation [17],

ti2k2/2rn * = r(E) = E(1 + oLE), X N O In ~

(emission),

(6)

where a = E~I(1 - m~/mo)2, with E r the energy of the r minimum. The equations ( 1 ) - ( 5 ) were modified to include the non-parabolicity as in ref. [18]. Also the overlap integrals between the periodic parts of Bloch functions were properly accounted for. For the calculation of various parameters involved in the computations, models have been developed and discussed elsewhere [9,11]. For composition x = 0.34, these are tabulated in table 1. The energies of the various minima have been taken from refs. [6-8] and are found to vary with pressure with best coefficients [6] of: dEr/dP = 12.6, dEL/dP = 5.5 and dEx/dP = -1.5 meV/kbar. It has been shown that in Ga 1 -x Alx As, the potential E~ of 8.6 eV remains unchanged with x and that f o r x = 0.34, E X ~ 8.6 eV [11,19]. The value o f E L = 12 eV has been taken from Ge as reported by Fawcett and Herbert [20]. The optical polar phonon temperature f o r x = 0.34 has been determined to be

Volume 89A, number 7

PHYSICS LETTERS

Table 1 Various material parameters for the alloy compositionx = 0.34.

Parameter

Value

x

0.34

T P eo

300 K 4.77 g/cm s a)

e~

11.8 a) 9.82 a)

Tiro'L'x

445 K a)

Zx ZL El" Ex E~

3 b) 4 c) 8.6 eV b) 8.6 eV b) 12.0 eV d)

m~,

0.084 mo a)

m~

0.22 m0 b)

m~

0.35 m o a)

s

5.24 X l0 s cm/s e)

Ni

2 X 1016/cma c)

a) see ref. [9], b) see ref. [11], c) see ref. [7], d) see ref. [20], e) see ref. [24]. 445 K at the I" point [9] and has been considered to be the same at the X and L points in the Brillouin zone. Birman et al. [21] have given selection rules for the electron transitions between different minima in 3 - 5 group compounds. In table 2, we have listed the various coupling constants and phonon temperatures for a typical composition x = 0.34 and used in the calculations. The terms TO, TA, LO and LA stand for transverse optic, acoustic, longitudinal optic and acoustic phonons, respectively. The temperature of 113 K for TA phonons at the X points has been taken

31 May 1982

from GaAs [20] and considered to be the same for all the scatterings. The phonon temperature T = 371 K for the X - X scattering is obtained by taking a linear extrapolation between TLO = 346 K at x = 0 [20] and TLA = 380 K [22] f o r x = 0.45 because for this composition, the band structure is similar to that of GaP and is considered to be the same for L - X , P - X and P - L minima scatterings. The phonon temperature of 456 K for the L - L minima scatterings has been taken to be the same as at the L point in InP [20]. The coupling constants Di/= 5.3 X 108 eV/cm for the I ' L, I ' - X and L - X scatterings are considered to be the same and have been obtained by taking a linear extrapolation between 1.1 X 109 eV/cm f o r x = 0 and 3.4 X 108 eV/cm [20] f o r x = 0.45, respectively. For GaAs, unscreened values of the coupling constants are considered while for x = 0.45, screened values have been used. The coupling constant Di/= 6.7 X 108 eV/ cm for the L - L minima scattering has again been taken from Ge [20]. Similarly all other coupling constants have either been taken or estimated from the data given in ref. [20]. Considering the scattering mechanisms described above, we have calculated the v-e characteristics for G a l - x Alx As alloys using a computer programme on the Monte Carlo method [18]. Once these characteristics are known separately for the electrons in the P, L and X minima, the corresponding drift mobilities are then determined from the low field slopes of these graphs. The results thus obtained for a typical composition x = 0.34 cm are shown in fig. 1. The Hall mobility is calculated from the equation [23]

(nx/nr)(l~x/llr)2+ (nL/nr)OL/#r) 2 ] #H = [1 + (nx/nr)(Px/#r) + (nL/nr)(PL/l~r) ] ' # r [I +

(7) Table 2 Various coupling constants and phonon temperatures for intervalley scattering for the alloy compositionx = 0.34. Mimima involved

For LO, LA and TO phonons

For TA phonons

D//(eV/cm)

T(K)

Dq (eV/cm)

T(K) a)

F-L r-x L-L L-X

5.3 X 5.3 X 6.7 x 5.3 ×

371 371 456 a) 371

113 113 113 113

X-X

1~0 X 109

1.5 X 107 1.5 X 107 1.8 × 108 5.5 X l0 s 1.0 X 109

10 s 10 s 10 s a) l0s

371

113

a) see tel [20]. 353

Volume 89A, number 7

'I

'v0

PHYSICS LETTERS

31 May 1982

drift mobilities attain minimum values at a pressure close to the F - L minima crossover. Since none o f the scattering rates are to be expected to change appreciably with pressure and hence with A E r x and A E r L , except the non-equivalent intervalley scatterings, it becomes obvious that the minimum in the drift mobilities do occur due to this scattering mechanism. The minimum in/.t H occurs at a pressure o f ~ 2 0 kbar, in qualitative agreement with the experimental results reported earlier [8]. Thus by theoretically calculating the pressure dependence of/2 H for various alloy compositions and a comparison with the experimentally observed variation of/a H with pressure, it should be possible to extract the various scattering parameters for the three minima, which are largely unknown. The results should prove useful in assessing the material qualities and properties for various device applications.

T=300 K X=&3Z,

4 8 12 16 20 24 28 32 3G P(K bar)

Fig. 1. The variations in the calculated electron drift mobilities # in the r , L and X minima as a function of pressure p

for a typical alloy composition x = 0.34. Also shown is the Hall mobility #H" The two bounds represent the spread in the mobilities at a given pressure.

We highly appreciate the continuous assistance of Dr. G. Hill to carry out this work and also for providing his master programme. Stimulating discussions with Dr. A. Majerfeld, Professor P.N. Robson and Dr. G.D. Pitt are acknowledged. The author is thankful to the Ministry of Education and Social Welfare, Government of India for the award o f National Scholarship to carry out this research at the University of Sheffield.

References where

(n x /n r) = (m ~ /m ~ ) 3/2 exp( - A E r x / k B T) , (nL/nr) = (m£/m~,)3/2exp(--AErL/k B T) ,

(8)

where A E r x ' I'L are the sub-band energy separations between the F - X and F - L minima respectively. The calculated result for/a H is also shown in fig. 1. It has been shown earlier that in Ga 1 - x Alx As alloys, the space charge and alloy scatterings make negligible contributions to/a H near the composition (equivalently pressure) for the d i r e c t - i n d i r e c t minima crossover [ 11 ]. Hence these scattering mechanisms have also been neglected in the present analysis. The sub-band gaps for x = 0.34 are A E r x = I00 m e V and A E r L = 90 meV. Using the best pressure coefficients for these separations, the I ' - L minima become equal in energy, at a pressure o f ~12.5 kbar. It is evident from the results shown in fig. 1 that all the 354

[1] T. Sugeta, A. Majerfeld, A.K. Saxena, P.N. Robson and G. Hill, Proc. Biennial CorneU Conf. on Active Microwave Semiconductor Devices and Circuits (IEEE, 1977) p. 45. [2] A. Majerfeld and G.L. Pearson, IEEE Trans. Electron Devices, ED-14 (1967) 632. [3] A.A. Immorlica and G.L. Pearson, Appl. Phys. Lett. 25 (1974) 570. [4] J.E. Sitch, A. Majerfeld, P.N. Robson and F. Hasegawa, Electron. Lett. 11 (1975) 457. [5] M. Neuberger, Handbook of Electronic Materials III-V Semiconducting compounds, Vol. 2 (Plenum, New York, 1971) p. 8. [6] A.K. Saxena, J. Phys. C13 (1981) 4323. [7] A.K. Saxena, Phys. Stat. Sol. (b) 105 (1981) 777. [8] A.K. Saxena, Int. J. Electron., to be published. [9] A.K. Saxena, Phys. Rev. B23 (1981). [10] A.K. Saxena, J. Appl. Phys., to be published. [11] A.K. Saxena, J. Phys. Chem. Solids, to be published. [12] M.G. Craford and W.O. Groves, Proc. IEEE 61 (1973) 862.

Volume 89A, number 7

PHYSICS LETTERS

[13] R.W. Conard, P.L. Hoyt and D.D. Martin, J. Electrochem. Soc. 114 (19~6) 164. [14] A.K. Saxena, Appl. Phys. Lett. 36 (1979) 79. [15] G.D. Pitt, M.K.R. Vyas and A.W. Mabitt, Solid State Commun. 14(1974) 621. [16] W. Fawcett, Proc. Int. Atomic Energy Agency, Vienna (1973) p. 531. [17] E.M. ConweUand M.O. Vasell, Phys. Rev. 166 (1968) 797. [18] W. Fawcett, A.D. Boardmann and S. Swain, J. Phys. C31 (1970) 1963.

31 May 1982

[19] A.K. Saxena, J. Electron. Mat., to be published. [20] W. Faweett and D.C. Herbert, J. Phys. C7 (1974) 1641. [21] J.L. Birman, M. Lax and R. Loudon, Phys. Rev. 145 (1966) 620. [22] M. Toyama, M. Naito and A. Kasami; Japan J. Appl. Phys. 8 (1969) 358. [23] D.E. Aspnes, Phys. Rev. B34 (1976) 5331. [24] H.F. Mc-Skimin, F. Jayraman and P. Andreatch Jr., J. Appl. Phys. 8 (1967) 2362.

355