Structured ultrafast mobility in highly photoexcited semiconductors

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PrintedS°lid StateinGreatC°mmunicati°nS'Britain.Vol.66,No.7,

pp.683-687,

STRUCTURED ULTRAFAST MOBILITY

1988.

IN HIGHLY PHOTOEXCITED

0038-1098/88 $3.00 + .00 Pergamon Press plc

SEMICONDUCTORS

V.N. Freire, A.R. Vasconcellos and R. Luzzi Departamento de F{sica do Estado Solido e Ciencias dos Materiais, Instituto de F{sica "Gleb Wataghin", Universidade Estadual de Campinas (UNICAMP), 13081 - Campinas, S.P., Brasil

(Received JAN/20/88 by R C C Leite ) We study the dependence of the mobility of the hot carriers of highly photoexcited plasmas in semiconductors on the irreversible evolution of their macroscopic state. We show that there are three different regimes of behavior of the ultrafast mobility transient: (i) structure (i.e. existence of maxima and minima) without overshoot at low fields, (ii) structure with overshoot at intermediate fields, (iii) normal evolution at intermediate to high fields. We name regimes (i) and (ii) hot carriers structured ultrafast mobility transients, and a criterion for their occurrence is given. We present results of numerical calculations appropriate for electron transport in the central valley of GaAs.

Ultrafast electron transport in highly excited plasma in semiconductors (HEPS) is a phenomenon of relevance in the physics of semiconductors, as well as a particularly interesting subject in the area of nonlinear nonequilibrium statistical mechanics. Vast improvements in ultrafast-laser-spectroscopic techniques (1) have recently a11owed its experimental study in the pico- and subpicosecond time scales. (2'3) We consider here the theoretical aspects of the question, performing an analytic study of the nonlinear ultrafast transport of carriers in HEPS arbitrarily away from equilibrium and in the presence of a constant electric field. For this purpose we have employed a nonlinear quantum transport theory derived from the nonequilibrium statistical operator method (NSOM), a powerful formalism among the existing approaches to the Statistical Mechanics of irreversible processes. (4'5) When specific calculations are performed we have used Zubarev's approach (6) to the NSOM in the form already applied by us to the study of ultrafast relaxation phenomena in HEPS. (7) We show that as a result of the irreversible processes that develop in the HEPS, and depending on the electric field intensity, the mobility of the carriers in a single valley develops maxima and minima during its evolution, an unreported phenomenon which we term hot carriers structured ultrafast

transient mobility. We consider a direct-gap polar semiconductor where a concentration n of electron-hole pairs are created by an intense pulse of laser light. These carriers are strongly departed from

683

equilibrium but are in a state of internal thermalization that is brought about in a fraction of picosecond by the Coulomb interaction (7,8) (Rot carriers). Next a constant electric field of intensity E is applied to the sample accelerating the carriers, which at the same time relax energy and momentum to the phonon field. The sample is in contact with a thermal reservoir at temperature T o . The optical phonons are heated up in scattering events involving FrShlich and potential deformation interactions with the carriers, but the acoustical phonons are only slightly warmed up. <7) To apply the NSOM to the study of this problem the first step is the choice of the basic set of nonequilibrium thermodynamic variables, that are appropriate for the description of the macroscopic state of the system. There is no universal criterion for such selection which depends on the physics of the specific problem under consideration. (4'5) As we have done in previous studies of ultrafast relaxation processes in HEPS, (7) we take in the present problem the set composed of the reciprocal of the quasi-temperatures of carriers, 8c(t) , of optical phonons, 8LO(t) and 8TO(t), and of acoustic phonons, 8Al(the latter is taken as a constant equal to T~ ), and the two other variables -Sc(t)~e(t) and -Sc(t)Bh(t) , where Be and ~h are the quasichemical potentials of electrons and holes respectively, adding now -Sc(t)Ve(t) and -Sc(t)Vh(t), where v e and v h are the components along the direction of the electric field of the corresponding drift velocities. The concentration of photoinjected carriers is taken as a constant since their rate of recombination

684

STRUCTURED ULTRAFAST MOBILITY IN HIGHLY PHOTOEXCITED

is of the order of ns -1 and we are considering ultrafast transport phenomena in the ps scale. The nonlinear coupled set of integrodifferential generalized transport equations that govern the time-evolution of these eight nonequilibrium thermodynamic variables follow from the NSOM in the form of far reaching generalizations (4) of either the Hilbert-ChapmanEnskog method (e.g. those in Zubarev's approach (4)) or the Mori-Langevin equation method (e.g. those in Robertson's approach(9)). Using the linear approximation in the NSOM-theory of relaxation (IO'6) we obtain the equations of evolution for the quasi-temperatures and quasi-chemical potentials which are identical to those of reference 8, except for the presence of the electric field that produces a shift in the distribution functions of the carriers, and adds a Joule heating effect in the equation for the rate of variation of the carriers energy. Further, two new equations appear in the present problem:they are those for the drift velocities, namely d d-~ IVa(t)l

= ( e / ma) E - y

a

(t)

Iva (t) I

(I)

SEMICONDUCTORS

Vol. 66, No. 7

where ft (t) = I d t ' a JO

(5)

y (t')

a

Applying elementary calculus to Eqs. 3, 4 and 5, we demonstrate that maxima and minima of the drift velocity in its transient regime should be observable depending on the evolution of the nonequilibrium macroscopic state of the HEPS. In fact if at time tex , v(t) has an extremal value, then the acceleration is null, and from Eq. (3) it follows that d

d-~ lVa(t)lt ex

= (e/m)E[l-z a

a

(t

ex

)y (t )]=0 a ex

(6) or T (t ) = Y-l(t ), and the extremum is a a ex @ . ex maximum or a mlnlmum if the second derivative 2 d 'Va(t)' t =-(e/ma) E ~ a ( t e x ) d 2 dt ex

,a(t)I

t

ex (7)

where a = e or h for electron or hole respectively. In this equation y (t) is the reciprocal of the a instantaneous relaxation time of the carriers linear momentum~ P , resulting from collisions with phonons a via the energy interaction operator H', given by 0 Ya(t)Va(t)= 2 [ dt' e Et' Tr{[H'(t'),[H',P ]p (t,O)} j -® a cg where p (t,O) is the NSOM-coarse-grained c distribution (5'6,7). For the case of a semiconductor with inverted parabolic bands, assuming that FrShlich interaction is the relevant one, using a dispersionless Einstein model for the LO phonons, and in the limit 8 m v << I, we obtain c a a

is negative or positive respectively. We shall then state the following criterion: for HEPS

arbitrarily away from equilibrium and in the presence of an electric field, a maximum (minimum) in the carriers mobility occurs whenever there exist a cross-over of the evolution curves for transport time, T (t), and • momentum relaxatzon time, y-1 (t), and athe latter is decreasing (increasing) ~t the cross over. In typical pump-probe approximately write

~ya l(t )

d__ Xa(t)l dt

~a(t)/~

=z(t)3/2e-Z(t)/2{ [ l+Vo(t) (eZ(t)+l) ]KI (z(t)/2)+ ao o + [I - ~(--~]Ko(Z(t)/2)}

(2)

where z(t) = 8 (t)he , ~(t) = [exp z(t)-l] -I, X = (2/3~) eCE ( 2 ~ /~m )-I/2, with E being a£ oa o ~I Frohlich field intensity,( ) Vo(t) is t~ a instantaneous population of LO phonons, mo the LO phonons frequency, and K are Bessel functions of the second kind. The differential equation (1) can be transformed into the equivalent integral equation (t)E

,

(3)

which has a non-linear Drude-type form with an instantaneous relaxation time for transport _~a(t) a(t) = e

t

ex

=_y2(t ) a

* ~T (t) c

dT * (t) c dt

t

' ext (8)

v (t)

Iv (t)l = ( e / m ) T a am

experiments we can

it Ca(t') Iodt ' e

,

(4)

since at the extrema ~ = O, and the rate of a change of T* is much larger than that of the c (/) phonons quasi-temperatures. Hence, Eq. (8) i m p l i e s t h a t a maximum (minimum) of V a ( t ) f o l l o w s a t t h e c r o s s - o v e r when t h e c a r r i e r s s y s t e m s i s c o 6 1 i n g down i f y71 i n c r e a s e s ( d e c r e a s e s ) w i t h T~, i.e. dy-I/dT~"> 0 or dy-i/dT~ < 0 res~yctively The dependence of,Ta(t ) of Eq. (2) with Tc, for fixed values of TLO , is shown in Fig. 1 (quasitemperatures are normalized in terms of Einstein temperature 8 o = ~ o / k , and the reciprocal of the momentum relaxation time is expressed in units of Yao ) Fig. 1 clearly shows that ~T~I/~Tc~ goes from positive to negative values on cooling from high temperatures. To test the results just discussed we consider the specific case of GaAs with a photoinjected carriers concentration of 2xlO 18

Vol. 66, No. 7

STRUCTURED ULTRAFAST MOBILITY

IN HIGHLY PHOTOEX~ITED

SEMICONDUCTORS

685

m~ I--

,~

Z

N

I.d " "

~ W ~

o Z

~

"- ~

15

" 0.95

- / - ~ (OF-

,.o

I

0 .
/

I

/

/

/

/

0.55

//

I -

_

/,

I

/

,L.< . . . .

0.75

/

_1

o35

_

I

I

7

14

•

i~ o ,~,~,

/

II 0.15 21

,o

.

Tc IO o

CARRIER

QUAS

I- T E M P E R A T

URE

Figure I: Dependence of the reciprocal of the momentum relaxation time (normalized) on the carriers quasi-temperature for several values of the LO-phonons quasi-temperature (Both quantities are normalized in the form indicated in the text).

-3 cm and a kinetic excess energy of 2.4 eV per pair, and in contact with a reservoir at T o = 30OK. Figures 2, 3 and 4 summarize the results obtained once the set of NSOM-nonlinear generalized transport equations is solved. Figure 2 shows the time evolution of the carriers quasi-temperature, Figure 3 displays the dependence on the electric field intensity of the steady-state electron drift velocity, and Figure 4 shows the evolution of the electron drift velocity for four values of the electric field. Figure 4 numerically confirms our predictions and tells us that there are three different types of behaviour in the carriers ultrafast transport transient: i) Structure (maximum and minimum) without overshoot (the drift velocity at the maximum is smaller than its stationary value) for E > 4.5 kV/cm, ii) Structure with overshoot (v e at the maximum is larger than its stationary value) for 4.5 < E < 9.4 kV/cm, iii) Normal evolution (v e monotonicaly increasing with time) for E > 9.4 kV/cm. Almost identical curves are obtained for Vh(t) except that the scale for v h must be reduced by

a factor of near fifty. It should be noted that the steady-state mobility is near Ohmic in regimes (i) and (iii) [Cf. Figure 3], i.e. in those regimes with absence of overshoot, but with different slopes. Further, Vst shows a steep increase for values of E around 9.4 kV/cm. The existence of a structured mobility transient is a result of the behavior of the momentum relaxation time and the rate of energy transfer during the evolution of the HEPS from its initial nonequilibrium state. Because the minimum of y-I is at, roughly, 2e o (~ 920 K in GaAs), if the phot~excitation is such that the initial value of T c is larger tha~ 28 o and the carriers cool down to values of Tc smaller than 28 o , structure shall appear. It is accompanied by overshoot if y-i at the steady-state temperature is larger than y-i at the initial conditions. On the other hand, if the field is intense enough for Joule heating to prevail over the transfer of the carriers energy to the phonon field the carriers keep heating up and, when the initial T~ is larger than 2Co, y-I continuously grows and no structured mobility transient is possible. For the specific case we considered this field is near 9.4 kV/cm. Summarizing, we have demonstrated that a structured ultrafast transient mobility of carriers

686

Vol. 66, N o .

STRUCTURED ULTRAFAST MOBILITY IN HIGBLY PHOTOEXCITED SEMICONDUCTORS

7

20 i 0 5 I_

,30 KV/cm

16 KV/cm

//

E o

12 KV/cm

>F-

W

J

W n

i

I I I I

>

rh

93 KVlcm

I REGIME

(SEE TEXT)

Io (i)

(ii)

/I

(iii)

Z O n-

0

~:

I

W .---..-- 9.5 KVlcm

w F-

W

15

103

O w _J W

9 KV/cm

n~

5

8 KV/cm

o

6 KV/cm

L_

2 KV/cm

2

I

I 10

5

I 15

I 20

4

FIELD

6

8

10

12

14

KV/cm )

INTENSITY

TIME (psec) Figure 2: Evolution of the quasi-temperature T*(t) c

of

the

highly

excited

carriers

in

Figure 3: Dependence of the hot electrons drift velocity in the stationary regime on the electric field intensity.

GaAs f o r

several electric field intensities. I0

E (.) p,.

8

I

I

I

I

J

--

/

o

/

x >-

~-

6

O 0

_J W >

9 K V/cm

b_

6 K V/cm

E: O Z

o

2 2 KVlcm

I-W _J W

I

I

I

I

5

I0

15

20

TIME

(ps)

Figure 4: Evolution of the drift velocity of the highly excited electrons in GaAs for several electric field intensities.

Vol. 66, No. 7

STRUCTURED ULTRAFAST MOBILITY IN HIGHLY PHOTOEXCITED SEMICONDUCTORS

in the conduction and valence band valleys in HEPS can be produced depending on the time evolution of its nonequilibrium macroscopic state. We have established a criterion for the existence of such structured transient, and it was shown that overshoot occurs between a lower and an upper value of the electric field intensity, accompanied by a non-Ohmic steady-state current. It should be noted that intravalley scattering, not considered in our model, is also present and in competition with the influence of the nonequilibrium evolution

687

of the HEPS. Maxima and minima also may result as a consequence of the transference of carriers to upper valleys, but Monte Carlo calculations show that this effect is smoothed out by electron-hole interactions. (12) One of us (VNF) is a Fundaqao de Amparo a Pesquisa do Estado de Sac Paulo (FAPESP) pre-doctoral fellow; the others (ARV, RL) are research associates of the Brazilian National Research Council (CNPq). We thank M.E. Foglio for a critical reading of the manuscript.

REFERENCES

i. R.R. Alfano, in: New Techniques and Ideas in Quantum Measurement Theory, edited by D.M. Greenberger pp. 118-126, (The New York Academy of Sciences, New York, 1986);

Semiconductors Probed by Ultrafast Laser Spectroscopy, vol. I and II, edited by R.R.

2.

3.

4.

5.

Alfano (Academic, New York, 1984); C.V. Shank, Science 219, 1027 (1984). C.V. Shank, R.L. Fork, B.I. Greene, F.K. Reinhardt, and R.A. Logan, AppI. Phys. Lett. 3__88, 104 (1981). R.B. Hammond, in: Proc. 4th. Int. Conf. on Hot Electrons in Semiconductors, edited by E. Gornik, G. Bauer, and E. Vass (Innsbruck, 1985), Physica 134B, 475 (1985). E . g . R . Zwanzig, in: Perspectives in Statistical Phy8~8, edited by H.J. Raveche, pp. 123-134, (North Holland, Amsterdam, 1981); Kinam (Mexico) ~, 5 (1981). A.C. Algarte, A.R. Vasconcellos, R. Luzzi, and A.J. Sampaio, Rev. Brasil. Fis. i_~5, 106 (1985), presents a unified description of the NSOM based on a variational principle; for specific cases see references 7 a n d 9 , and also S.V. Peletminskii and A.A. Yatsenko, Zh. Eksp. Fiz. 53, 1327 (1967) [Soviet Phys. JETP 2_~6, 773 (1968)]; J.A. McLennan, Phys. Fluids ~, 1319 (1961); H. Grabert, Zeit. Phys. B27, 95 (1977).

6. D.N. Zubarev, Neravnovesnaia Statisticheskaia Termodinamika, (Nauka Press, Moskwa, 1970) [English Translation: Nonequilibrium Statistical The~vnodynam~s (Consultants Bureau, New York, 1974)]. 7. (a) A.C. Algarte and R. Luzzi, Phys. Rev. B27, 7563 (1983); (b) R. Luzzi and A.R. Vasconcellos in: Semiconductors Probed by Ultrafast Laser Spectroscopy, edited by R.R. Alfano, Vol. I, pp. 135-169 (Academic, New York, 1984); (c) R. Luzzi, in: High

Excitation and Short Pulse Phenomena, edited by M.H. Pilkuhn, pp. 318-332, (North Holland, Amsterdam, 1985). 8. (a) A. Elci, M.D. Scully, A.L. Smirl, and J.C. Matter, Phys. Rev. BI__~6,191 (1977); (b) J. Collet, T. Amand, and M. Pugnet, Phys. Lett. 96___AA,368 (1983). 9. B. Robertson, Phys. Rev. 144, 151 (1966); ihid i0___88,171 (1967). iO. V.P. Kalashnikov, Teor. Mat. Fiz. 3_~4, 412 (1978) [Theor. Math. Phys. (USSR) 34, 263

(1978)]. II. E. Conwell, Solid State Phys., Suppl. 9, 1 (1967). 12. M.A. Osman, U. Ravaioli, and D.K. Ferry, in: High Speed Electronics, edited by B. K~llb~ck and H. Beneking pp. 32-34 (Springer, BerlinHeidelberg, 1986).