Hot electron microwave conductivity of wide bandgap semiconductors

Solid-State Ekctmnics,

1976, VoL 19. pp. SSl-BSS.

Pergmon

Press.

Printed in Oreal Britain

HOT ELECTRON MICROWAVE CONDUCTIVITY OF WIDE BANDGAP SEMICONDUCTORS P. DAS Electrical and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12181,U.S.A. and

D. K. FERRY Officeof Naval Research, Arlington, VA 22217,U.S.A. (Received 26 December 1975; in revisedform 8 March 1976) Ahstrac-Hot electron microwave conductivity of the wide baudgap semiconductors GaN, SE and Diamond has been calculated using displaced Maxwellianapproximation for the electron distribution function. The effects of both the energy and momentum relaxation times due to scattering by acoustical, optical intervalley phonons and by ionized impurities are included in the derivations. Numerical results for the microwave conductivity and the change in dielectric constant as a function of frequency and bias electric field are presented. It is found that signiticant change in the conductivity and dielectric constant contribution for a tixed bias field occurs at very high frequencies on the order of lo’* Hz, which is well beyond the range of current microwave device interest.

INTRODUCTION

Wide bandgap semiconductors such as Gallium Nitride (GaN), Silicon Carbide (Sic) and Diamond (C) have been pointed out as having desirable properties from the point of view of designing microwave solid state devices[l]. Whatever the particular type of device in which one is interested, it will invariably involve hot electron transport. In choosing the optimum material, a tigure of merit which includes the saturation velocity and the breakdown field is usually defined. This is shown in Table 1 for the three widegap semiconductors GaN, Sic and Diamond, and for Silicon. It is obvious that whether one uses Johnson’s[2] or Keyes’[3] criteria, the wideband gap semiconductors offer a potential of an order of magnitude improvement in the tlgure of merit that could be achieved. It is well-known that the microwave conductivity of semiconductors varies as a function of frequency. The functional dependence becomes quite complicated when hot electron transport is included. This occurs because one must consider not only the momentum relaxation time of the carriers but also the energy relaxation time, which does not come into the picture at low electric fields and is usually ignored. Even if the saturation velocity is very high for actual microwave devices, there is no guarantee

that its microwave conductivity will have a desirable value. In this paper, the real and imaginary contributions to the hot electron microwave conductivity have been calculated for B-Sic, GaN and Diamond. In the next section, a short theoretical formulation of the problem is given. This is followed by the extensive numerical results and conclusion. TEFLBRY The energy and momentum balance equations for a

semiconductor under the assumption of a displaced Maxwellian approximation to the distribution function of the carriers are given by[4,5] eF= mud l-m + a/at eodF=kTIre+8/8t(kTp),

vsa,b/sec)

1 x lo7

Ebd~WCrn) EC3

Johnson’s

rm= C rmi.

GaN

Diamond

2.5 x 107

2.8 x 107

300

750

2000

1000

12

9.7

9.5

5.93

1.5

>

1.5(?)

6.6

FM.*

(vsE&d

Kbs/EJ

a.5 x 10-f

(2)

Similarly, the total effective energy relaxation rate F, = TC’ is given by

re = z rei. I

K (W/cm-%)

Keyes’

sic

(1)

where F is the total applied electric field, m is the effective mass of the carriers, ud is the displaced Maxwellian drift velocity, k is Boltzmann’s constant, e is the electronic charge, and I, = rm-’ is the total effective momentum relaxation rate, given by a sum over the various interaction as

Table 1. Figures of merit for devices si

(mud),

4.8 x loll

I2 3 x 10

a x loI2

4.5

x 1013

2.4

1.4

4.2

2.3

x 108

F.1:. 112

x 107

x 106

x 107

The individual scattering mechanisms considered in this work are due to interactions of the electrons with acoustical, optical, and intervalley phonons and with ionized impurities. The scattering rates for these various interactions are tabulated in the Appendix of Ref [l]. 851

P. DASand D. K. FERRY

a52

Equations (1) and (2) are, in general, valid whenever a moment can be applied to the Boltzmann equation. These equations thus rely upon the general validity of the Boltzmann equation itself. It should be added that the moment equations given here are linearized equations. For application to semiconductors, however, this restriction does not imply any loss of generality. In the present work, the moments will be taken using a drifted Maxwellian distribution function. In calculating the microwave conductivity, one considers that a small a.c. field is added to a large d.c. field as F = Fo + FI e’“‘.

tity and its frequency dependence is not a simple function of frequency, since both terms in 07~ and wrm are important. The microwave conductivity is given as h(o)

= noeRe(p),

(17)

where no is the background carrier density. The imaginary part of CL,contributes to the real part of the dielectric constant, as K = KL +yIm(p).

(18)

(5)

As a consequence, the various responses in the electron properties can be expanded as (6)

It should be remarked that in general, Im (~1) < 0. It has been popular to assume that the a.c. mobility is just the differential mobility of a static velocity field curve. This, in general, occurs only if 07~ and 07~ < 1. In fact, for o +O, this differential mobility can be found from (17) and (15) to be

(7) (19)

In addition, the scattering rates also vary due to their dependence on T,. These can be expressed as where

710

is q (O = 0) and pdc is e/m rmo, the d.c. mobility. NUMJMICAL CALCULATIONS

where Imo and IO0are the respective values at T, = T,o. The formulas for these derivatives with respect to the electron temperature are given in the Appendix. We can now use (5)-(9) in (1) and (2) to obtain the zero order terms eFo = mv.iormo,

(10)

eudoFo= klrcor.0.

(11)

Similarly, the first order terms yield

In Table 2 are shown the values of the various physical constants utilized for B-Sic, GaN and Diamond in these calculations. The drift velocity in high electric fields was determined in a previous paper[ 11.In Fig. 1, we repeat the curves for the case of cubic Sic. The coupling constants for the intervalley and optical phonons are adjusted to provide a best fit to the variation of low field mobility as a function of temperature. The curves in Fig. la show curves for three sets of coupling constants and a curve representing the data of Nelson et al. [6]. The high electric field velocity is then found using the values found by the above procedure. Such curves are shown in Fig. lb for the three sets of coupling constants used in the low field fit. Values found for the best fit to the low field mobility versus temperature are used throughout the remainder of Table2. Materialparametersutilized

eFI = mVdlrmO(l

t iO/Imo) t

mVdoTel

(12)

eVdlFo t eVdoF1= kT.ryT,o,

Diamond

5.5

(13)

0.25

where

25.0

y=l+(iw/reo)t~

( > g

o.

0

(14)

us.5 23

Equations (lo)-(14) can now be used to solve for just the a.c. mobility. This is found to be Vdl p'=FY=mrmO

e

l-17(0) ltiO/rmOt+)

1 (19

2P.O

3.5 12.1

3.5 3.5 5.33

(16) As expected, the microwave mobility is a complex quan-

5.93

Microwave conductivity of wide bandgap semiconductor

I 200

loo

I

I I III1 400 600

TEMPERATURE

85-3

IO00

(OK)

ELECTRIC

FIELD (kV/cm)

Fig. 1. (a) Calculated mobility as a function of temperature for Sic for several different values of the phonon parameters. The dotted line is the data of Nelson et ol.[61.(b) The calculated higb field velocity for the same set of coupling parameters. In these figures the curves labeled a are for a coupling constant of 1 X lb eV/cm for the two intervalley phonons, 11.5eV for the acoustic phonons, and 250kV/cm for the polar optical phonon. In curve b a stronger coupling is assumed for the polar phonon, and in curve c, one of the intervalley phonons is weakened and the acoustic phonon is strengthened.

this paper. In Fig. 2, we show the variation of r., r,, (ar,/XT,), and (ar,/aZ’,) with the applied d.c. electric field. In Figs. 3-5, the values of the real and imaginary parts

of the a.c. mobility are shown for cubic Sic, Hexagonal GaN, and Diamond. The imaginary portion of the mobility is represented in each case by its contribution to the dielectric constant. As expected, the a.c. mobility is not simply the slope mobility of the velocity field curve, but deviates from this at frequencies above and near to the energy and momentum relaxation rates. In the warm electron region, near the knee of the velocity field curve, the energy and momentum relaxation rates differ considerably, and an interesting anomally occurs for frequencies

(a)

8 & d

a

3.00

2.00

2

1.00

IO8

IO'O ID’

IO9

IO"

IO"

lOI

FREQUENCYHz

100 4

SILICON .2

CARBIDE

r kV/cm

22.2

IO

(b) I -.6

-

I IO8

1

IO9

IO” IO’O FREPUENCYHz

SILICON E-FIELD

fig.

2.

IO”

IO@

lOI

lOI

CARBIDE

v/cm

Variation of r., r,,,, (ar.laT.1 and (ar,JaT.) of SiC as s1 function of applied d.c. electric field.

Fig. 3. The real part of the a.c. mobility (a) and the carrier contriiution to the dielectric constant (b) for cubic silicon carbide. The d.c. electric field in kV/cm is the parameter.

854

P. DASand D. K. FERRY

k

6.0

aQ A 4.0 2

20

FREQUENCYHz IO9

10’0 IO”

d2

10’3 10’4

DIAMOND

FREQUENCYHz

GALLIUM

(a)

NITRIDE 1.0

(a)

18.1 5

kV/cm

10.0 I=====% 59.4

00

\

\

\Y-

219.0

n,= 10’5/crn’

I

IO8 FREQUENCY

GALLIUM

IO9

Hz

I

I

I

d2

lOI

lOI

I

I

IO” IO’O FREQUENCYHz

DIAMOND

NITRIDE

(b)

@I

Fig. 4. The real part of the a.c. mobility (a) and the carrier contribution to the dielectric constant(b) for hexagonal GaN, with the d.c. applied electric field as a parameter.

Fig. 5. The real part of the a.c. mobility (a) and the carrier contribution to the dielectric constant (b) for diamond. The applied d.c. electric field is the parameter.

in the range

where I-00< 6J < rmo.

(20)

p= In this range, and for fields near the knee, the a.c. mobility actually increases with frequency over a narrow frequency range. This is readily understood, since in this range o /rro > 1, and y is increasing faster than o /Tmo,so that 17is decreasing. This results in an increasing real part of ~1. This continues until ~1 begins to be dominated by the term in (o/r,&. It can be readily be shown that the maximum in (Tm occurs for opcak=

P +d(P’+4NQ) 2N

J(

>

(21)

W--A3

rmormo +B) 2_%&4) mo

Q_2(B+4Z ~(2B-A+T,d+~(B+T,o)(Bz-AZ) mo

Ino

Microwave conductivity of wide bandgap semiconductor

As a check for the numerical calculation, the values of oW were numerically computed and found to agree exactly with the values shown in Fig. 34. In the dielectric constant, two effects are observed. One arises from the reduction of I~(JLI) and the other from the increase of o required for 07 > 1. For high fields the plasma edge can disappear entirely, while at intermediate fields, a double edge can occur.

855

r “““=~(clrm,~e~2k,T,)~~~ (b) Acoustic phonon scattering

(arm /XLb.. = (r,o,.JXo)

r mO.n= (ml2rY”

3m&‘keTo(keT.)‘” 2h’pv:

.

(c) Non polar optical and intervalley phonon scattering CONCLUSION

Calculations based upon a drifted Maxwellian approximation to the distribution function have been carried out for the wide bandgap semi-conductors Sic, GaN and Diamond. In microwave applications, the a.c. mobility begins to be degraded when the frequency approaches the momentum relaxation rate. For these materials, this effect occurs at frequencies well beyond the normal microwave range, in the order of 1OOGHz.

r mo.,, =& =s

ho y=k,T,;

(ey-=+ 1)x eXf2k1(x/2) (ksT,,/2~)‘n(l/(ey-‘)) ho0 x=keT.

=~(x/T.)[tanb(x-y)-$$$I.

Acknowledgement-The

authors wish to acknowledgethe numerical computationsdone by Alan Barr.

I

(d) Polar optical phonon scattering

REJERENCES

1. D. K. Ferry, Phys. Reu. B 12,236l (1975). 2. A. Johson, RCA Rev. 26, 163 (1%5). 3. R. W. Keyes, Proc. IEEE 60, 225 (1972). 4. P. Das and R. Bharat, Appl. Phys. Lett. 11, 386 (1%7). 5. P. Das, R. Rifkin and J. Poole, Electron.Left. 3, 515 (1%7). 6. W. E. Nelson, F. A. Holderand A. Rosenbloom,J. Appl. Phys. 37,333 (PM). 7. The breakdown field is calculatedfollowing tbe approachof GA. Baraff,Phys. Rev. 128,2507 (1%2); Phys. Rev. 133, A26 (1961).An analytical approximation to BaratTs numerical simulation has been developed by Y. Okuto and C. R. Crowell, Phys. Rev. B 6,3076 (1972)and it is this form that is used to calculate EM in the wideband-gap semiconductors. Recent measurements in Schottky barriers on 6H-Sic have been carried out by G. H. Glover, J. Appl. Phys. 46,4842 (1!%5),and Rive a breakdown field near to that calculated.

+ (e’-’ - l)ZL(x/2)1

3xK,(x/2)(ey-’ - 1)- K,(x/2)(eY-’ t 1)+(x - 3)ZL(x/2Mey-”- 1) (eye=t l&(x/2) t (eymX - l)ZL(x/Z) 2. Expressions for @r&T.) (a) Non-polar optical and intervalley phonon scattering Fe0= $(ey-” - 1)x e”‘K,(x/2) ii

Expressions for (aT,/aT.) and (aT./aT.) for different scattering mechanisms. For the explanation of the symbols, see Ref. [l].

Z&(x12) (b) Polar optical phonon scattering

1. Expressions for (a r,,, /aT.) (a) Ionized impurity scattering

(%i.“=-rmo I”-,+_%&~,+cE!kJ] B