high field transport in gaas, inp and inas

Solid-Stare Elecrronics Vol. 27, No. 4. pp. 347-357, 1984 0038-I lOI/ $3.00 + .oo Printed in Great Britain. Pergamon Press Ltd.

HIGH FIELD TRANSPORT IN GaAs, InP AND InAs

KEVIN BRENNAN and KARL Hess Department of Electrical Engineering and Coordination Science Laboratory, University of Illinois at

Urban-Champaign, Urbana, IL 61801 U.S.A.

(Received 13 June 1983; in revised form 13 August 1983)

Abstract-Calculations of the steady state and transient electron drift velocities and impact ionization rate are presented for GaAs, InP and InAs based on a Monte Carlo simulation using a realistic band structure derived from an empirical pseudopotential. The impact ionization results are obtained using collision broadening of the initial state and are found to fit the experimental data well through a wide range of applied fields. In InP the impact ionization rate is much lower than in GaAs and no appreciable anisotropy has been observed. This is due in part to the larger density of states in InP and the corresponding higher electron-phonon scattering rate. The transient drift velocities are calculated under the condition of high energy injection. The results for InP show that higher velocities can he obtained over 1000-l 500 A device lengths for a much larger range of launching energies and applied electric fields than in GaAs. For the case of InAs, due to the large impact ionization rate, high drift velocities can be obtained since the ionization acts to limit the transfer of electrons to the satellite minima. In the absence of impact ionization, the electrons show the usual runaway effect and transfer readily occurs, thus lowering the drift velocity substantially.

In the last few years, attempts have. been made to construct new types of semiconductor devices which are capable of ever higher speed. Most of the pre- dicted advantages are based on the high field-transient transport properties of III-V compounds[l-51 and the associated higher drift velocities. These may give rise to the possibility of high current-low power applications. It is the purpose of this paper to present a detailed study and comparison of the (transient) transport properties of InAs, GaAs and InP. The study includes collision-free transport, high energy injection and velocity enhancements and impact ionization phenomena in the context of their importance for device performance[6-91.

The III-V semiconducting compounds encompass a wide range of materials with very different physical properties. Many of these properties play a significant role in the high field transport behavior of the material. The threshold energy for impact ionization strongly depends upon the energy band gap of the material, which differs widely amongst these compounds. Narrow band gap semiconductors, such as InAs, have a low impact ionization threshold leading to a high ionization rate at low electron energies. In larger band gap materials, InP and GaAs, impact ionization only occurs at high electron energies. Each type of material provides different advantages in various applications[9].

Representative of the large disparity in transport properties of the III-V compounds are GaAs, InP and InAs. As mentioned above, the impact ionization rate in these materials is vastly different over a wide range of applied fields. The electron drift velocities have a wide range of values as well. It has been known for some time that very high steady state drift velocities can be attained in GaAs[2,3]. This has sparked much interest in practical device applications of GaAs[lO,ll]. By use of velocity overshoot[l2] it is possible to achieve extraordinarily high drift veloci-

ties and currents through a very small distance and at low power levels. Velocities more than four times the steady state value can be achieved under certain conditions in device size, applied electric field, and initial electron energy [ 131. Recently InP has been considered as a candidate for high speed device applications based on velocity overshoot. Velocity overshoot can be attained over a much wider range of applied fields and launching energies in InP than in GaAs [ 141 making it a more promising material for device considerations. We will see that very high drift velocities can be attained over a very short distance in InAs as well. However the effect of impact ionization in InAs severely limits the practical use of this material in real device structures.

As we shall see, velocity overshoot can lead to a substantial gain in the transient drift velocity from the steady state value. However the overshoot does not persist over a very long distance and only occurs under very limited conditions[l3,14]. In order for the device to benefit from velocity overshoot these limited conditions must be met throughout the entire device structure.

In our simulation, we use a pseudopotential band structure for each material [15]. We have extended the impact ionization theory of Shichijo and Hess[lS] to all three materials and have found excellent agreement between the theory and experiments [ 16,171. The previous work by Tang and Hess [ 131 and Brennan et al. [ 141 has been elaborated on and been extended to InAs. We will show how the impact ionization rate can limit the electron drift velocities and we will provide a complete theory of transient transport in III-V compounds.

Since our calculations are performed using a pseudopotential generated band structure it is necessary that the parameters used in the calculation be consistent with the pseudopotential results. The possibilities of simultaneously adjusting the valley separations,

SSE Vol. 27, No. LD

348 K. BRENNAN

band gap, and effective masses to measured values are limited and we did have to compromise. The phonon energies and coupling constraints are also chosen consistent with the pseudopotential band structure. From our experience we believe that the deviations of these quantities from measured values are immaterial for the results presented, i.e. the results will change by no more than a few percent if the parameters are varied within the available range of data.

2. STEADY STATE DRI@T VELOCITY THEORY

Many parameters are needed to effectively model the physical behavior of a semiconductor using the Monte Carlo technique. Parameters such as the intervalley scattering phonon energies and coupling constants are exceedingly difficult to measure directly. Their values are often ascertained by fitting the results of Monte Carlo calculations to more easily measured quantities[3,4], such as the steady state electron drift velocity. The Monte Carlo results are known to be exact at low energy[3] where they can be compared to experimental measurements of the Gunn effect[l8]. Littlejohn et a/.[41 have performed Monte Carlo calculations in GaAs that give good

and K. HESS

agreement with the Gunn effect data. From this they have determined a set of coupling constants and phonon energies for intervalley scattering in GaAs. We have used their results for the coupling constants but have found that for our pseudopotential model, we obtain a better fit to the experimental drift velocities[l8,19] (Fig. 1) if we use slightly lower values for the intervalley phonon energies. The pa-

1 2 5 10 20 50 loo zoo 5 Field (kV/cmi

Fig. 1. Steady state drift velocity in GdAs as a function of electric field. The field is applied in the (100) direction at

T = 300 K.

Table 1. Parameters for GaAs program

Parameter Value r valley L valley x val.ley

Density 5.36 g/cm3

E m 10.92

B 0 12.90

Number of valleys

Effective mass m*/mo

Non-parabolicity (eV-'1

Valley separation (eV)

Temperature 300'K

I

.063

.690

Intervalley Coupling Contacts and Phonon Energies

T-L 1.0 x 109 eV/cm .026 ev

i--X 1.0 x 109 eV/cm .026 eV

L-L 1.0 x lo9 eV/cm .026 eV

L-X .9 x 109 eV/cm .026 eV

x-x .9 x 109 ev/cm .026 eV

Polar Optical Phonon Energies

r .0x eV

L .0343 ev

X .0343 ev

4 3

.23 .43

.65 .36

.33 .52

Acoustic Scattering Parameters

Deformation Potential 8.0 eV

Sound Velocity 5.24 x lo5 cmfsec

Parameter Value r valley L valley x valley

Density 4.79 g/cm3

Ea 9.52

E 12.35

Number of valleys 1 4 3

Effective mass m*/mo 0.078 0.26 0.325

Non-parabolicity (eV_') 0.83 0.23 0.380

Valley separation (EV) 0.54 0.775

Temperature 300'K

Intervalley Coupling Constants and Phonon Energies

High field transport in GaAs, InP and InAs

Table 2. Parameters for InP program

T-L 1.0 x LO9 eV/cm

F-X 1.0 x log ev/cm

L-L 1.0 x log eV/cm

L-X 0.9 x lo9 eV/cm

x-x 0.9 x log eV/cm

Polar Optical Phonon Energies

0.0278 eV

0.0299 ev

0.0290 ev

0.0293 ev

0.0299 ev

r 0.043 ev

L 0.0423 eV

X 0.0416 ev


Deformation Potential a ev

Sound Velocity 5.13 x lo5 cm/set

349

rameters used in our computations for GaAs are The material parameters for InP used in the Monte collected in Table 1. They are similar to those used Carlo program are presented in Table 2. These have by Schichijo and Hess[l5] and Tang and Hess[l3]. been reported previously in Brennan et d. [ 141 and are The masses and nonparabolicities obtained yield an repeated here for completeness. As can be seen from analytic band structure that agrees with the pseudo- Fig. 2 the Monte Carlo calculation of the steady state potential in all directions to better than 5% at low drift velocity agrees well with the experimental energies. results[20].

5 10 20 Eleclrlc FIespd &%)

200 500

Fig. 2. Steady state drift velocity in InP as a function of electric field. The field is applied in the (100) direction at

T=3OOK.

The material parameters used for InAs are listed in Table 3. Due to the lack of extensive drift velocity data in InAs we have used the same intervalley phonon energies for InP. Because of the similar deformation potentials of all three compounds [21,22], the intervalley coupling constants are taken to be the same. Since the satellite valleys are sepa- rated from the central valley by extremely large energies, the effect of intervalley scattering in InAs on the drift velocity is only important at very high applied fields. Figure 3 shows the steady state drift velocity in InAs as a function of field in the presence of impact ionization. Notice that the peak drift velocity is very high, 8.0 x lO’cm/sec, at a field of 75 kV/cm. If impact ionization does not occur (which can be achieved experimentally by using very short

350 K. BRENNAN and K. HFS

Table 3. Parameters for InAs program

PaITalUetL?f Value r valley L valley x valley

Density 5.67 g/cm3

Em 11.8

E 14.55 0

Number of valleys

Effective ma.ss m*/mo

Optical phonon energies (eV)

Non-parabolicity (&J-l)

Valley separation (eV)

Tempefature 300°K

Energy gap (eV) .36

1 4 3

.032 .206 .640

.0302 .024 ,020

1.390 ,536 ,900

1.082 1.620

Intervalley Coupling Contacts and Phonon Energies

r-L 1.0 x 109 eV/cm .0278 eV

T-X 1.0 x log eV/cm .0299 ev

L-L 1.0 x lo9 ev/cm .0290 ev

L-X .9 x 109 eV/cm .0293 ev

x-x .v x 10 9 rV/cm .0299 ev


Deformation Potential

Sound Velocity

8.0 ev

4.35 x 105 cmlsec

current pulses) then the peak drift velocity in InAs is much lower and occurs at a much lower field as can be seen from Fig. 4. In the absence of impact ionization the electron energy shows the usual “runaway” effect; more energy is gained from the field than lost to the phonons and the threshold for intervalley transfer is easily reached. The impact ionization acts to limit the electrons to low energy so they stay within the central valley. The valley separation energies are so large that few electrons survive

l”;lo”

01 I I J 0 1 10 100

Electric Field (kV/cm) 1 10

Applied Electric Field (kV/cm)

Fig. 3. Steady state drift velocity in InAs as a function of Fig. 4. Steady state drift velocity in InAs as a function of electric field when impact ionization is present. The field is electric field when impact ionization does not occur. The

applied in the (100) direction at T = 300 K. field is applied along the (100) direction at T = 300 K.

impact ionization while within the central valley. After impact ionizing, the electron loses most of its energy and starts again near the gamma point. Again the electron drifts in the field and reaches the threshold for impact ionization well before it reaches the threshold for intervalley transfer. Unless the field is

4.0x107 ,

3.5- InAs Steady State kfl Velocity w/o Impact Ion.

High field transport in GaAs, InP and InAs 351

extremely high such that the electron can reach the where G is the electron Green function, D is the intervalley threshold without collisions or by phonon Green function, and V(q) is the electron- suffering only polar optical scatterings, it will impact phonon coupling. The electron Green function can be ionize and will restart near gamma. In this way, the expressed as [27] electrons are restricted to the central valley where their drift velocities can become very large. G(& E) =

1 (2)

The peak steady state drift velocity is greatest in E - E(k) - Z(k, E) + is

InP and occurs at a much higher field than in GaAs while the phonon Green function has the form[28], or in InAs when impact ionization is not present. The polar optical scattering is much weaker in InAs than 1 - in either InP or GaAs because of the much smaller density of states in InAs. This leads to runaway of the electrons as mentioned previously resulting in a smaller drift velocity. The satellite valleys are at a higher energy in InP than in GaAs and since the polar

where 6 is a positive infinitesimal quantity. After

optical scattering rates are similar higher drift veloc- substituting the expressions for G(k, E) and D(q, hw)

ities are possible in InP. As we will see this favors InP into (1) and evaluating the integral over w by use of

for high speed device applications. contour integration, the expression for C(k, E) becomes [29]

3. IMPACT IONIZATION

Due to the relatively large band gaps in GaAs and InP impact ionization only occurs at high electron energies (-2 eV) in these materials. At such high energies, the band structure deviates drastically from parabolic and even k. p models. It is essential, in order to effectively model high energy transport, that an accurate band structure based upon a pseudopotential calculation be used [ 151.

At high electron energy an additional complication arises from the large increase in the electron-phonon scattering rate. In the semi-classical approximation the scattering processes are treated as transitions between sharp unperturbed momentum states. This approximation breaks down in the presence of a very high electron-phonon scattering rate[23]. The electron can no longer be considered as a classical particle. Instead a quasiparticle picture (electron dressed by phonon cloud[23]) is more appropriate. A “dressed particle” or quasiparticle can be viewed as an electron which continuously absorbs and reemits virtual phonons. This interaction introduces an additional energy to the particle, the self-energy,

c(kN241. Figure 5 shows diagrammatically the self-energy of

the quasiparticle. The self-energy, C(k), can be calculated neglecting the vertex correction as follows[25,26]

C(k, E) = i_ c d3q dw (27L)4 V*(G)D(@, hw)G(F - 4, E - hw)

Z(&,E)= 2 s (2x)’

X G2(E + cj)

0 E-hwa-E($)-Z

> . + ia

(4)

Using the approximation of a constant deformation potential, g*(k + q) becomes gz and (4) can be re- expressed as an integral over energy as

C(E) =g* dE’p(E’)

E-ho-E’-Z(E-hw)+id (5)

where we have taken w to be independent of q and p(E) is the electron density of states. If we consider the weak coupling limit (g e 1) then the term C(E - hw) can be neglected in the denominator of eqn (5). Applying the Principal Value Theorem to the above integral yields

C(E) = P s

dE’g*p(E’)

E-hw -E’ - ing*p(E - hw). (6)

From (6) the imaginary part of C(E) is clearly

f(E) = ng*p(E - hw) (7)

while the real part is

A(E)=P s

dE’g*p(E’)

E-hw -E” (8) 1

(1) Physically, the real part, A (E), corresponds to a level

D(;,hw) shift of the energy eigenstates while the imaginary part, T(E), gives rise to a finite lifetime of the

P-t

state[29]. Since the state has only a finite lifetime, via the Uncertainty Principle, the energy of the level is broadened.

It is easy to show[29] that the lifetime of the state can be expressed as

Fig. 5. Diagrammatic representation of the electron-phonon h

interaction. rk = 2r(k)’ (9)

352 K. BRENNAN and K. HESS

To first order l/rk corresponds to the total scattering rate of the particle in the state k[29]. Neglecting the vertex correction[25], Z(E(k)) is simply (k(Z’lk) where T is the transition matrix[30]. For elastic scattering the scattering probability is conserved and the optical theorem gives [30]

B TOT = - & Zm (LITIL) = - $ZmZ(E(K)). (10) k k

The total scattering rate is then

2r (E(Q) l/r(E) = LrTOTVk = 7’ (11)

Substituting for Zm(Z(E)) one obtains in the limit of weak coupling

l:r(E)=;g’y(E--hw) (12)

where p(E - hw) is the density of states and g is the coupling constant. For stronger coupling, the integral eqn (5) must be solved. This has been accomplished by Tang[29] and Chang[31]. In the Monte Carlo calculations the total scattering rate at high energies is replaced by the above relation. The self-energy effectively reduces the overall scattering rate.

The impact ionization data for both GaAs and InP have been calculated using this modified scattering rate. The results are presented in Figs. 6 and 7 respectively. For GaAs, the calculated values match the experimental results of Bulman et aZ.[16] over a wide range of applied fields. Calculations are made for two applied field directions (100) and (111). At high electric fields no appreciable anisotropy exists in agreement with our previous findings. At low electric fields, the calculation is extremely time consuming because only few ionization events occur. Therefore there exists a large statistical uncertainty in the value of the calculated impact ionization rate. Nevertheless

Em (V/cm) 60 5.0 40

lo: IO l/Em (cm/V) x10-6

Fig. 6. Impact ionization rate in GaAs as a function of inverse field. The error bars are based on convergence error

estimates from the calculation.

E, (V/cm)

Fig. 7. Impact ionization rate in InP as a function of inverse field. The error bars are based on convergence error esti-

mates from the calculation.

the results for the (100) crystallographic direction are in good agreement with the experiments. In the limit of very low field, an anisotropy seems to develop and the results for u in the (111) direction are somewhat lower than the (100) result. However, only six ionization events have been simulated at an expense of a day CPU time on a VAX 750 super- mini-computer. There has been some controversy concerning the anisotropy of a in the past[32,33]. Notice that the calculated anisotropy is at much lower fields than observed by Pearsall et al.[34] and also much smaller in magnitude. We therefore have to conclude that the measured anisotropy is caused by effects not included in our simulation (e.g. transient phenomena or impurity correlations etc.). The small anisotropy seen from the Monte Carlo calculation seems connected with the fact that at low enough fields the electron distribution is centered closer to k = 0 than at high fields. Since the distribution is cooler, those electrons which reach the ionization threshold do so only after gaining much energy from the field. This requires that the electrons not be scattered much from the field direction, similar to Schockley’s lucky electron theory[35]. A small anisotropy at low fields would be expected then because the ionization threshold may be different in different directions[36]. Also, due to the anisotropy of the band structure, an electron will gain different amounts of energy along different field directions per drift. The lucky electrons play an insignificant role at high fields[l5] because the distribution is now much hotter which results in the electrons being distributed throughout the Brillouin zone. Since the vast major- ity of ionizing electrons start from anywhere in the Brillouin zone, the directional dependence of the rate vanishes. It is possible that the proper inclusion of the final state broadening may smear out the observed anisotropy at low fields.


Energy CeV)

Fig. 8. Density of states of the first conduction band in GaAs and InP as a function of energy.

The impact ionization rate is much lower in InP than in GaAs as can be seen from a comparison of Figs. 6 and 7. In our calculation of the impact ionization rate in InP we use the same intervalley coupling constants as were used in GaAs. This is appropriate since the band structures of the two materials are very similar[37] and it is believed that the deformation potentials are also [21,22]. The threshold for impact ionization is roughly 20% higher in InP than in GaAs. The impact ionization threshold energy is determined for GaAs to be 1.70 eV while for InP it is 2.10 eV. The impact ionization rate includes a multiplicative factor as well[lS] which is the same for both materials, 0.5. As can be seen from Fig. 8, the density of states rises more abruptly and reaches a higher peak in InP than in GaAs. The scattering rate is roughly proportional to the density of states re-

Ix: 20 Electyo Field (kV/cm)

5 I” ’ I

Impact Ian : InAs . T= 300°K

. F in (100)

. 4_

g 5;

.

.

103 I I I 0 5 10 15 20

l/E X 10e5 (cm/V)

25

Fig. 9. Impact ionization rate in InAs as a function of inverse field. Since many events occur, the convergence of the program is excellent and the resulting error is small.

Electric Field (kV/cm) D403020 10 5

I I I

3 In As

0 F In < lOO> 0 T:77”K

0

3 1 I I I

0 5 10 15 20

l/E X10e5 (cm/V)

Fig. 10. Impact ionization rate in InAs as a function of inverse field at T = 77 K.

sulting in a higher scattering rate within InP than in GaAs. Also since the density of states is much greater in InP below the impact ionization threshold energy it is more difficult for an electron to drift to states at and above threshold. Hence fewer electrons will reach high enough energies for impact ionization to occur. The large difference in the impact ionization rate between these two materials therefore is mainly due to the density of states at high energy and to the higher threshold in InP.

It should be noted that the anisotropy in the impact ionization rate does not appear in InP. This is because the scattering rate is higher so the electrons are distributed throughout the Brillouin zone re- moving any directional dependence to the ionization rate. As in the case of GaAs, an anisotropy may appear for lower fields but due to the lower ionization rates the Monte Carlo simulation is too time consuming.

Due to its narrow band gap, impact ionization in InAs occurs at much lower fields than in either GaAs or InP. The threshold, E,,,, for impact ionization in InAs can be readily calculated using the Anderson and Crowell criteria[38] since the parabolic approximation to the energy band is acceptable at low energy. E,,, is found to be 0.383 eV at 300 K. Since the satellite valley separation energies are greater than 1 eV most of the ionizations involve electrons within the central valley. In pure InAs the only important scattering mechanism in the central valley is pro- duced by polar optical phonons[39]. Polar optical scattering is much weaker than intervalley scattering[39]. Therefore in the range in which impact ionization occurs in InAs the self-energy effect is negligible. The impact ionization results for InAs are presented in Figs. 9 and 10. The impact ionization rate is determined at both 300 and 77 K. The rate

354 K. BRENNAN and K. Hms

deviates strongly from the exponential l/E law as the temperature varies from 77 to 300 K. Notice that the impact ionization rate at both temperatures is extremely high at rather low electric fields.

4. TRANSIENT ELECTRONIC TRANSPORT

In high speed-low power device applications it is essential that the carrier transit time through a structure be small. Collision-free transport and near collision-free transport have been proposed as a means of reducing carrier transit times[4044]. Re- cent work by Tang and Hess [13] and Brennan et al. [ 141 has explored the possibility of near collision- free transport in GaAs and InP respectively. Their investigations have shown that a collision-free window, CFW, exists for different applied electric fields and injection energies[l3,14]. They have found that higher velocities can be attained in InP over a longer distance than in GaAs for higher fields and launching energies[l4]. This gives a distinct advantage to InP because higher applied voltages can be used in device applications.

High applied fields can produce velocity overshoot over small distances by driving the electrons to velocities above the corresponding steady state velocity. Figure 11 shows the transient electron velocity in GaAs as a function of distance for various applied fields at zero launching energy. At low applied fields, 1 kV/cm, the velocity does not overshoot the steady state value by a large amount. Not much is gained in the average speed of the carriers over that for the steady state at low fields and low injection energy. As the field is increased, the velocity overshoots the steady state significantly. This can be seen for GaAs at fields of 10 and 30 kV/cm. The electron transit time at these fields will be substantially reduced by the overshoot from that for electrons at the steady state velocity. However as the applied field is increased more, the overshoot dramatically decreases due to the transferred electron effect. Owing to the large density of states within the satellite valleys, upon transfer- ring, the electron drift velocity decreases sharply. Clearly there is only a limited range of applied fields

14;10’ , / /

0 0 ZOO 400 600 800 1000 1200 1400 1600

Distance Along the Dwce Cd,

Fig. 11. Average drift velocity versus the device length for various fields in GaAs at zero launching energy and 300 K.

149’ , I , I , , I , / I , InP

12 E_o=OeV k,=iO.O,O.O,O.O)

-:10- F ,n <lOO> T= 300°K

F:lOOkV/cm

ova I 1 L I 1 1 / I I 1

0 200 400 600 800 1000 1200 1400 1600

Dastonce Along the Device (8,

Fig. 12. Average drift velocity versus the device length for various fields in InP at zero launching energy and 300 K.

that will lead to substantial velocity overshoot through a range of 100~15008, in GaAs.

Since the valley separation energy is greater in InP, it is expected that velocity overshoot can be attained at higher applied fields than in GaAs. Figure 12 illustrates the electron drift velocity as a function of distance in InP for various fields at zero launching energy. As is readily seen from the figure, high drift velocities are maintained over long distances at higher applied fields in InP. Devices made of InP can be operated then at greater applied voltages and still show velocity overshoot.

Velocity overshoot can be accomplished in a different way by launching the electrons at energies above the gamma point. High energy injection at various energies is possible using heterobarriers with different band edge discontinuities. In this way, the electrons start with velocities much larger than the

steady state drift velocity. Figure 13 shows the electron velocity as a function of distance for various launching energies in GaAs at an applied field of 10 kV/cm. At zero and low launching energies the overshoot is very small and little is gained from the

300°K 1

__

1

m EoW)

0 00: 0 -- b 004 t 003 007 d 004 011 e 0.05 0 16 f 0.06 021 9 0.07 027 h 006 034 -

OU I 1 1 I 0 500 1000 1500

Distance (%)

Fig. 13. Average drift velocity versus the device length with the launching energy as a parameter in GaAs for an applied

field of 10 kV/cm and 300 K.


14x10’ I

IllP km to -_

12- F=30kV/cm n <lOO> 0 00 0.0

Tz300”K b 002 0.033 - c 004 0092 d 008 0283 e 010 0416 f 0.12 0558

F = 100 kV/cm

=: ;4

D

2 i

InAs GzleV

ro=Coo3,00.0.0, F I” <lOO> T= 300°K

00 0 200 4cc 600 800 loo0 1zgo 1400 1600

Dlstonce Along the Device (A)

Fig. 14. Average drift velocity versus the device length with the launching energy as a parameter in InP for an applied

field of 10 kV/cm and 300 K.

Fig. 16. Average drift velocity versus the device length for various applied fields in InAs at launching energy of 0.1 eV

and 300 K. Impact ionization events occur.

steady state. If the electrons are launched at energies above or near the intervalley threshold, the electrons can be easily accelerated to energies where they will transfer to the satellite valleys. This results in a sharp drop in the velocity and there is no gain from the overshoot. In GaAs the window of launching energies that gives rise to high drift velocities over distances of 100&15OOOA is from 0.1-0.3 eV.

In InP the range of launching energies which results in a high drift velocity throughout device lengths of 1000-1500 A is greater than in GaAs as seen in Fig. 14. Again, at low energy injection the overshoot of the drift velocity is minimal. The overshoot is appreciable at launching energies from 0.1-0.5 eV in InP. At launching energies above 0.5 eV the electrons are easily transferred to the satellite valleys and the drift velocity greatly diminishes. From the previous results we conclude that InP is better suited than GaAs for devices based on velocity overshoot.

The drift velocity in InAs is strongly affected by the presence of impact ionization as mentioned previously. When impact ionization occurs, very high steady state drift velocities are possible as we have seen. Figure 15 shows the transient electron drift

velocity in InAs as a function of distance for various electric fields at zero launching energy and in the presence of impact ionization. Extremely high drift velocities are attained, greater than 1 .O x lo* cm/set, for most applied fields. Figures 16 and 17 show the effect of launching the electron at high energies. Notice that high velocities are attained throughout the entire structure for all the applied fields up until transfer becomes significant. Only at very high fields does transfer occur and the drift velocity is not lowered as drastically as it is in InP and GaAs. When impact ionization does not occur the behavior of the drift velocity is very different. Figure 18 shows the effect of neglecting impact ionization on the drift velocity following high energy injection. Transfer occurs readily for high applied fields and there is little significant overshoot in the drift velocity. InAs does not appear to be a promising material for high speed devices based upon velocity overshoot, since impact ionization is necessary in order to attain high drift velocities. This should be avoided in real device applications.

The previous results can be summarized by consid- ering the carrier transit time through the entire structure for each of the three materials. The transit time as a function of applied field for electrons injected at zero energy is plotted in Fig. 19. The transit time based on the steady state drift velocity of

16.X 10’ , , / / , , F= lOkV/cm

200 400 600 800 1ccc 1200 ,400 16aJ Distance Along the Dewce (8,

ZOO 400 600 800 loo0 1280 1400 1600 Distance Along the Device (Al

Fig. 15. Average drift velocity versus the device length for various applied fields in InAs at zero launching energy and

Fig. 17. Average drift velocity versus the device length for

300 K. Impact ionization events occur. various applied fields in InAs at a launching energy of 0.5

and 300 K. Impact ionization events occur.

356 K. BRENNAN and K. Hms

F:50 kV/cm _

F=lOOkV/cm -

InAs Wlthout Impact Ionlzatlon E,=O 5eV

F r <lOO> Dwce Length:15008 T=300”K Hilgh Launchng Energy

InAs With Impact lon~rotaon En=0 5eV

0 1 I i 1 1 I 1 1 I 0 200 400 ax 800 iooo 1200 140 B 1600 1800 2000

Distance Along the Dwce ( I

Fig. 18. Average drift velocity versus the device length for various applied fields in InAs at a launching energy of 0.5 eV

and 300 K. Impact ionization does not occur.

the electrons in InP is also plotted for comparison. As can be readily seen from Fig. 19, there exists a range of applied fields in each of the three materials in which high speed transport is possible. The range of field values is very small in GaAs but is larger in InP. The electron transit time is extremely small in InAs over a very large range of applied fields. However this is true only when impact ionization occurs. Notice that there is very little improvement in the transit time over the steady state result for very low applied fields at zero launching energy. Even at higher fields the overshoot does not drastically improve the transit time of the electrons when they are launched at zero energy. Figure 20 shows the transit time as a function of applied field at high launching energy. Again the transit time for the steady state velocity is plotted for InP. The overshoot is more substantial at high launching energy and there is a fair improvement over the steady state transit time at low applied fields. However the gain is much less than an order of magnitude. The transit time in InAs is calculated with and without impact ionization. When impact ionization does not occur, the transit time increases substantially at high fields and the overshoot is minimal.

I I

15001 Device Length Zero Launching Energy

I 1 1 1 1 1 I 0 10 20 30 40 50 60 70 80 90 100 Applied Electric Field (kV/cm)

Fig. 19. Average transit time through a device of 1500 A as a function of applied electric field for GaAs, InP and InAs

at zero launching energy and 300 K.

01 1 / / 1 1 1 0 20 40 60 80 100 120 140 160 180 200

Appwd Electric Field (kV/cml

0

Fig. 20. Average transit time through a device of 1500 A as a function of applied electric field for GaAs, InP and InAs

at high launching energy and 300 K.

In device applications, in order for velocity overshoot to meaningfully influence operational speed, the entire structure will need to be on the order of 1000 A. The collecting region as well as the collecting contact can greatly influence the high speed behavior[45]. Brennan et al. [45] have shown that the contact can greatly reduce the electron drift velocity in structures of lOO(r1500 A long. It is expected that the results presented here are optimistic estimates of the effect of velocity overshoot. It is highly unlikely that velocity overshoot can be realized to any great extent in most device structures now in use. Devices such as semiconductor heterostructure versions of MOMOMs[46] or GaAs FET structures[42] may benefit from velocity overshoot but their operating conditions will be very restrictive.

5. CONCLUSIONS

We have modeled, via a Monte Carlo approach, both the impact ionization rate and the transient and steady state drift velocities for GaAs, InP and InAs. Our results for the impact ionization rate in both GaAs and InP fit the experimental data[16,17] well through a wide range of electric fields. A small anisotropy in the impact ionization rate at very low fields is found to exist in GaAs while it has not been observed in InP (it may exist at still smaller fields). The impact ionization results are obtained including collision broadening of the initial state. A more complete study of impact ionization is planned where the collision broadening of the final state will be included. Collision broadening of the final state may further reduce the extent of the anisotropy in GaAs.

We have demonstrated that velocity overshoot occurs only over very small distances for electrons injected at energies above the gamma point in all three materials. Overshoot is possible in GaAs and InP provided that the electrons are not transferred to the satellite valleys. Transfer can be avoided if the electrons are launched at energies below the L minima and are not accelerated above this energy by the applied

High field transport in GaAs, InP and InAs 357

field. Velocity overshoot does not lead to a large re-

duction in electron transit time as compared to the

transit time from the steady state velocity. InAs is not well suited for devices built on the basis of velocity overshoot because of the presence of a very high impact ionization rate at high applied fields. When impact ionization can be avoided, as in the case of very short current pulses, the overshoot is not as large and the benefits of velocity overshoot are less.

Acknowledgemenls-We would like to thank Drs. J. P. Leburton and J. Y. Tang for many helpful discussions. The use of the on board computing facilities of the U.S.S. George Wright is greatly appreciated. This work was spon- sored in part through the Office of Naval Research contract N0001476C-0806, and by the Army Research Office DAAG 29-80-K-0069.

Note added in proof-We would like to thank Dr. F. Capasso for pointing out that the electron impact ionization rate in InAs is slightly greater at 300 than at 77K. This anamolous behavior is due in part to the temperature dependence of the band gap energy. For a narrow band gap semiconductor such as InAs, as the temperature is raised from 77 to 300K the band gap decreases from 0.41 to 0.36 eV. This is a significant change which greatly influences the impact ionization threshold energy, since the threshold energy depends strongly on the energy band gap[38]. The other effect of raising the temperature from 77 to 300 K is that the phonon scattering rate increases due to the dependence of the phonon number on the temperature. How- ever, in InAs the predominant scattering mechanism is polar optical scattering since the carriers are generally confined to the gamma valley, (see text) and the increase in the phonon scattering is not large enough to counteract the effect of the reduced impact ionization threshold. According to our model the impact ionization rate is greater at 300 than at 77 K.

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high field transport in gaas, inp and inas

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