A quantum description of drift velocity overshoot at high electric fields in semiconductors

Solid-State Electronics Vol. 32, No. 12, pp. 1411-1415, 1989

0038-1101/89 $3.00+0.00 Copyright © 1989 Pergamon Press pie

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A QUANTUM DESCRIPTION OF DRIFT VELOCITY OVERSHOOT AT HIGH ELECTRIC FIELDS IN SEMICONDUCTORS

F. Rossi and C. Jacoboni Dipartimento dl Fisica dell' UniversitY, Via Campi 213/A, 1-41100 Modena, Italy

ABSTRACT A quantum description of transient charge transport in semiconductors is presented. In particular, we discuss the drift velocity overshoot in GaAs. The paper is intended to show how typical quantum features, such as intracollisional field effect and multiple collisions tend to modify the transient behavior of the system as predicted by semiclassical transport. This analysis has been performed by means of a quantum Monte Carlo procedure which takes into account the GaAs band structure through a many-valley model. The results of the quantum simulation, as regards drift velocity and upper valley population, have been compared with those of the classical theory and this comparison shows that in the case of GaAs quantum features are not relevant. For a better understanding, a semiconductor model, characterized by a very strong electron-phonon coupling constant, has been considered where quantum effects are appreciable and from this analysis it is possible to identify physical systems for which a full quantum treatment is required.

KEYWORDS Hot electrons; quantum transport; density matrix; Gallium Arsenide; velocity overshoot; Monte Carlo method.

INTRODUCTION When a physical system is characterized by typical lengths of the order of the carrier coherence length, and typical times comparable with carrier relaxation times, the semiclassical transport theory based on Boltzmann equation is no longer adequate for its description. Such physical conditions are being rapidly reached by modern semiconductor device technology, and a quantum description of transient transport can improve device modelling. In the present paper we investigate the overshoot of the drift velocity of electrons in semiconductors which may influence the high frequency performance of submicron devices. In this case quantum effects are expected to modify the value of the drift velocity as obtained from a traditional ensemble Monte Carlo procedure (EMC). In particular, energy non conserving transitions, allowed at short times in a quantum scheme, are expected to anticipate the population of upper valleys and therefore to reduce the velocity overshoot; on the other hand the intracolllsional field effect,by reducing the efficiency of all scattering mechanisms, is expected to act in the opposit direction. The results, obtained for a simplified model of GaAs, will show that in ordinary working conditions quantum effects are not relevant. Physical conditions for which a quantum description is required will be discussed.

THEORETICAL APPROACH AND NUMERICAL PROCEDURE In order to study the properties of the drift velocity in a quantum scheme, we shall apply a Quantum Monte Carlo (QMC) procedure recently developed (Brunetti and others, 1989) by the authors, briefly summarized in the following. ~Let us consider a non interacting electron gas in a semiconductor crystal, coupled to the phonon gas. The system is assumed to be homogeneous, and its Hamiltouian is given by H = He + H a + H p + H e p ,

(1)

where He is the term corresponding to an electron in a perfect crystal, HB ----eE • r is the term due to the electric field and Hp is the Hamiltoulan of the free-phonon system. The electron-phonon interaction Hxmiltouian Hep has the general form

1411

1412

F. ROSSl and C. JACOBONI

He, = E ihF(q){aqeiq'"

-

atqe-'q"} = H.b + H~,,~,

(2)

q

where Hat, and Hem refer to phonon absorption and emission, respectively, and F ( q ) is a function of the phonon m o m e n t u m q whose explicit form depends on the particular scattering mechanism considered. Let us consider the set of time-dependent basis vectors ] ko, {nq},t} represented by

v~ve~[k<')"e-' fo d,-,.,Ik(<

I {'D.q}, t>,

(3)

where k(t) = ko - -K-t ,E and w(k(t)) represents the electronic band structure. They are direct products of electronic accelerated plane waves, normalized to 1 over the crystal volume V, and the phonon states I {nq},t}. If we now consider the density matrix p of the system in the representation of the set in eqn (3), the Liouville-von Neumann equation that describes its time evolution contains only the perturbation Hamiltonian: 0

ih--~p(X,

X , t) --= [H,p(t),

p(t)]x,x,,

(4)

where we have used the symbolic compact notation X = (ko, { n q } ) . We are here interested in the evaluation of expectation values of electron quantities which are diagonal in the electronic part of the states in eqn (3); thus we focus our attention on the diagonal elements p(X, t) = p(X, X, t) of p. Furthermore, we will assume a diagonal initial condition for p decoupled in electron and phonon coordinates. After a formal integration of eqn (4), a perturbative expansion for p is easily obtained by iterative substitutions:

p(X,t) = p(X,O) +

/0'dt,[7"lt,(tl),p(O)lx,x + = pIo)(x,t) + ~ p ( l / ( x , t )

dr,

dt2[Tt~,(t,),[~.p(t2),p(O)]]x,x + ...

+ Ap(2)(x,t ) + ...,

(5)

where 7-[ep = /~Hep. There is a simple and useful way of reading the terms that come from the commutators in the above equation. It is possible to show that, by application of 7~ep (or ~*p) the first (or the second) argument of p is changed from the second argument of "Hep to the first one. Since each application of ?-L,p changes the phonon state by one unity, in order to start from a diagonal element and end up to another diagonal element a mode q absorbed (or emitted) by one argument must be absorbed (or emitted) also by the other argument or reemitted (or reabsorbed) by the same argument. In the language of field theory, we refer to the first kind of processes as to "real" emissions and absorptions, while the other ones are "virtual processes". Their diagrammatic representation (Brunetti and others,1989) allows us to regard each term of the perturbative expansion as a sequence of quantum processes which correspond to single scattering events in classical transport. The electronic band-structure has been included in the effective mass approximation, with a many-valley model consisting of spherical and parabolic bands: w(k) = ~(k - ko) 2

2rn,~

+ ",:,,

(6)

where c, represents the valley index, m~, the corrisponding effective mass, k~ and hw,~ are the center and the bottom of the band, respectively. Using the above equation, it is possible to evaluate the time integral in eqn (3) analitica/ly as follows:

/otdT"w(k('))=(w° + w~)t + 2 ~ k o ' k t 2 + ~

~¢2ta,

(7)

J

where wo = w(k0) and 1~ = ~ . This result has been used within the procedure to obtain the matrix elements of the electron-phonon Hamiltonian, which are involved in the evaluation of the perturbative expansion in eqn (5). However, in describing the electron state in terms of transitions between indipendent valleys, one can realize that such a model, in presence of multiple collisions, gives rise to contradictions. As an example, let us consider the sequence of two real intervalley processes shown in Fig. 1 . At time t = 0 the electron is in a valley and the two indexes of p (represented by the two horizontal axes) are the same; at time t4 the first process (from valley a to valley a ' ) begins, changing only the first index of p from a to a'; at time ta the second process (from valley ct' to valley a) begins while the first one is not yet completed and it changes the same index of p from a to a'. The contradiction arises at time t2 when the second process, which represents an inter-valley transition from a ' to ct, seem, on the contrary, to change the second index of p from a to a'.

A quantum description of drift velocity overshoot

0~

1413

m t

t4

t2

t

t1

Fig. 1. An example of a diagram showing the inconsistency of the many-valley model in a quantum description (see text). This contradiction arises from the fictitiousness of our model and it could be avoided by means of the whole band structure e(k). However, such processes will generally describe transitions very far from energy conservation. For this reason we preferred to keep the simplicity of a many-valhy model and negiet these terms in the simulation. RESULTS AND CONCLUSIONS The QMC procedure, discussed above, has been applied to the study of the drift velocity overshoot in GaAs. We used a simplified model consisting of a F valley (mr = .063rao) and four equivalent L valleys (mL - .222m0). The bottom of the L valleys is taken as .31 eV, the value of the crystal density is 5.3g/cm s and the optical and static dielectric constants are 10.92 and 12.90, respectively. The electron-phonon interaction is given by two scattering mechanisms: an intra-valhy scattering due to polar optical phonons (Tvov = 410 K) and an inter-valhy scattering due to to non-polar optical phonons (Tov = 323 K and D = 6 x 10s eV/cm). The value for the electric field used in the quantum simulation has been taken as 40 Kg/cm, and this choice is based on the analysis of the overshoot in classical terms. In Fig. 2, the drift velocity, as given by a traditional EMC simulation, has been compared with its corrisponding perturbative sdution obtained by a Backward Monte Carlo procedure (Jacoboni and others, 1988). We can realize that the perturbative solution up to the second order in the scattering rates (i.e. the ensemble of trajectories with up to two scattering events), gives us a good approximation of the drift velocity peack. Therefore we expect to be able to describe velocity overshoot in quantum terms with the perturbative expansion described above up to the fourth order in the hamiltonian.

c~ v')-

p SSSS"

"G "~S'~"

/ ./"

o

c~ c~ 0.0

50.0

i00.0

15'o.o 20o.o 25'o.o 3o'o•o t Cfs)

Fig. 2. Comparison between the E.M.C. drift velocity (-) and its perturbative expansion: the zero order (- -), the first one (...) and the second one (-.-)(see text)• Results are shown in Figs• 3-5. In Fig. 3 the quantum drift velocity has been compared with the classical one: it can be seen that under these conditions, quantum features are not so relevant. Fig. 4 shows the corrisponding corrections to the ballistic velocity, and in Fig. 5 the quantum and classical upper-valley populations are presented. From Figs. 4 and 5 quantum effects can be seen; in particular, we may note the effect of energy-non conserving transitions, which causes an earlier upper-valley population.

1414

F. Ro~[ and C. JACOBONI

c~

.-f e D o

o

~o > o

o o

o.o

~6.o

86.0 t

12'o.o

~io.o

20'o.0

[fs]

Fig. 3. Comparison between the quantum (-) and the classical (- -) drift velocity overshoot (see text). "ooo x o-

E 0

> +

>e~"

o

J

60.0

9£0

12'~.0

,s'0.0

~&.0

22'0.0

t. [ f s ]

Fig. 4. Quantum (-) and classical (- -) corrections to the ballistic velocity (see text). The fact that quantum effects are not so important in this case can be explained in terms of the uncertainty relation between time and energy: the energy defect in energy non conserving transitions would be of the order of the inter-valley threshold (.31 eV) and the time interval is of the order of 100fs. Their product is much larger than h so that quantum features for our GaAs model are negligible. In order to reduce this time interval before the drift velocity overshoot, we have repeated our simulated experiments using higher values of the electric field. However, the result was not so relevant because the dynamics of electron-phonon interaction itself requires some time to produce appreciable effects. This last amount of time could be shorter if the coupling constant were stronger. d

c/"

C-c~-

8 ° 6o.0

,J--r---~-( 92.0

,2~.o ~io.o t[fs}

i~8.o

2~'o.o

Fig. 5. Comparison between quantum (-) and classical (- -) upper-valley population (see text).

A quantum description of drift velocity overshoot

1415

Therefore, in order to evidentiate quantum effects we have artificially modified the model used increasing the intervalley coupling constant (D = 2.5 x 10° eg/cra). The value of the field has been taken as 200 Kg/crn. This last result, shown in Fig. 6, shows a stronger quantum effect: the velocity overshoot is decreased by the possibility of energy non conserving intervaLley transitions, thus confirming the validity of our quantum analysis. Semiconductor materials which are characterized by strong coupling constants for intervalhy transitions may present quantum effects in the transient velocity overshoot.

q

>ca

D i

0.0

8.0

i .0

24.0

32.0

40.0

t (fs} Fig. 6. Quantum (-) and classical(- -) driftvelocity overshoot for the case of a very strong coupling constant (see text).

ACKNOWLEDGEMENT

The authors are very grateful to Wolfgang Quade for helpful discussions and a criticalreview of the paper. This research is financed by Progetto Finalizzato Materiali e Dispositivi per l'Elettronica a Stato Solido (MADESS) del Consiglio Nazionale deUe Ricerche (CNR). REFERENCES Barker, J.R. (1978). Sol. State Electron. 21,267 Brunetti, R., C. Jacohoni, and F. Rossi (1989). Phys. Rev. B39, 10781 (89) Jacohoni, C., P. Poll, and L. Rota (1988). Proc. Int. Conference on Hot Carriers in semiconductors, edited by J. Shah and G. Iafrate, Sol. State Elect. 31 (314),p.523