Electron-hole drag in semiconductors

Solid-State Electronics Vol. 31, No. 3/4, pp. 643~48, 1988 Printed in Great Britain

0038-1101/88 $3.00 + 0.00 Pergamon Journals Ltd

ELECTRON-HOLE DRAG IN SEMICONDUCTORS

R . A . HiSpfe 1 ( a ) Institut ftir Experimentalphystk U n i v e r s t t~it I n n s b r u c k A-6020 Innsbruek, AUSTRIA and J.

Shah

AT&T B e l l L a b o r a t o r i e s H o l m d e l , NJ 0 7 7 3 3 , USA

ABSTRACT Transport of minority carriers in semiconductor plasmas can be strongly affected by electron-hole scattering. In high-mobility carrier systems not only the presence of one carrier type, but also its drift velocity determines the transport of the other carrier type via electron-hole scattering. This effect is known as "carrier drag". In modulation-doped quantum well structures the carrier drag is strong enough to cause "negative absolute mobility" of both minority electrons and holes. In t h i s p a p e r we d e s c r i b e all-optical transport measurements, f r o m w h i c h momentum r e l a x a t i o n times by electron-hole scattering are quantitatively determined. Extremely short scattering times result for minority electrons in a hole plasma (40 to 100 £s) in contrast to the reverse case of holes in an electron plasma (2-5 ps). The physical reasons (mass ratio, degeneracy, two-dimensionality) are discussed, as well a s new p h e n o m e n a a s n e g a t i v e p h o t o c o n d u c t i v i t y and plasma instabilities in the presence of strong electron-hole drag.

KEYWOPJ~

Electron-hole scattering, negative mobility, quantum wells, electron-hole drag, carrier drag, photoconductivity.

GaAs, carrier

transport,

I. INTRODUCTION The role of scattering between electrons and holes for minority carrier transport in semiconductors was first recognized by Paige /1/ after measuring the minority carrier drift i n Ge a t l o w t e m p e r a t u r e s . I n a s u b s e q u e n t p a p e r / 2 / , McLean a n d P a i g e p r e s e n t e d a detailed theory of the effects of electron-hole scattering on the mobility, based on a numerical evaluation o£ the coupled Boltzmann equations. The mutual transfer o£ momentum t h a t r e s u l t s from the drift of the majority carriers is since then called "carrier drag" or "electron-hole dra~". For strong electron-hole scattering and weak carrier-lattice scattering, even negative mobility of minority carriers due to the carrier drag was theoretically predicted (for holes in n-InSb)/2/. Since then, several experimental /3-10/ and theoretical /11-13/ works have been reported on the influence of electron-hole scattering and carrier drag on minority carrier mobility. Recently, we w e r e a b l e t o o b s e r v e f o r t h e f i r s t time the effect of negative (absolute /14/) mobility due to carrier drag, for both minority electrons and minority holes in "modulation-doped" Ca/ks q u a n t u m w e l l s /15,16/. These observations w e r e m a d e w i t h a new all-optical transport technique, which allows the determination of electron-hole momentum scattering times. The obtained scattering t i m e s a r e e x t r e m e l y s h o r t (< 1 0 0 f s ) f o r m i n o r i t y electrons in a hole plasma, and much longer for minority holes in the electron plasma (2 to 5 ps). In a review paper /17/ more detailed descriptions of the experiments and the analysis as well as theoretical calculations are contained. We r e f e r t o t h i s p a p e r a s w e l l as refs. 15 and 16 f o r details and more explanations. I n t h e p r e s e n t p a p e r we w i l l e m p h a s i z e m o r e t h e experimental results o n momentum s c a t t e r i n g and their physical reasons. We w i l l discuss the implications for devices in respect of ballistic minority carrier transport, the photoconductivity a n d a new k i n d o f l o w - f i e l d plasma instability.

II. The minority

EXPERIMENTS

carrier

drift

velocity

is measured

by a

643

new

all-optical

technique,

which

is

644

based on the following principle: The luminescence from injected minority electrons is timeand spatially resolved. Because of the high majority hole concentration, luminescence only occurs where minority electrons are, so t h a t t h e s t u d y o£ t h e l u m i n e s c e n c e g i v e s d i r e c t information on the minority carrier dynamics. The technique is described in r ef. 15: Electrons are excited in the semiconductor hole plasma by a focused laser pulse. The spatial dependence of the luminescence (the luminescence "profile") is measured with an "imaging" technique. The laser beam is focused inside the cryostat. The magnified luminescence image (x13) is scanned with a mechanical slit ( w i d t h 2 5 ~m), a n d s p e c t r a l l y analyzed to detect only the intrinsic band-to-band recombination. The semiconductor structures in our modulation-doped m u l t i p l e q u a n t u m well p a r a m e t e r s g i v e n in Figs. 1 t o 3.

III. Figure

experiments were high-mobility nand p-type structures, g r o w n by m o l e c u l a r b e a m epitaxy, w i t h the

RESULTS

1 (main)

shows typical

luminescence

images for

several

lattice

temperatures

T L. C u r v e s

b and c show distorted images into the direction of the negative electrode. This means that, at low t e m p e r a t u r e s a n d low e l e c t r i c fields, photoexcited electrons have negative absolute mobility, i.e., electrons drift towards the neKatively char~ed electrode. The luminescence profiles with applied fields (curves b - e in Fig. 1) are then fitted by a straightforward procedure (described in re£. 15) with the drift velocity vd as the only shape parameter. The time

dependence

n£ t h e

a characteristic

are

luminescence

lifetime

of

(Fig. l,

1.0 ns in

inset

top right)

the whole range

shows an exponential

Data from several measurements at different temperatures s h o w n i n F i g . 2 . A t low t e m p e r a t u r e s (T L = 15 K) a n d

and different electric fields low e l e c t r i c fields, the electron

drift mobility is negative, down t o - I 1 5 0 0 c m 2 / V s . The n e g a t i v e electric field ( a t 15 K l a t t i c e temperature) and with increasing becomes even positive. We

performed

the

same e x p e r i m e n t

decay with

o£ t e m p e r a t u r e s .

with n-modulation-doped

mobility lattice

quantum

decreases both with temperature, and

well

structures,

5000 - G a A s ' A I 0 " 4 8 G a 0 ' 5 2 A s

with

MQW

P0=1.6 x 1011cm"2 n < 3 x 1 0 1 0 c l n "2

~

0

I

~

/

~ool-

T-,5~

o -10,000

I,~t

::~

-,o,ooo I~ .,5,ooo _

-15,000 0

I 100

,

,

,

200 400 600 ELECTRIC FIELD (Vlcm) I 200

I 300

TEMPERATURETL(K )

0I

I

I

I

I

I

I

I

I

I

-40

-30

-20

-10

0

10

20

30

40

! I

F i g . 2: M i n o r i t y e l e c t r o n mobility as a function of the lattice temperature for low e l e c t r i c fields. Circles and solid line: experiment. Dashed line: gajority hole mobility multiplied by -1. Inset: Electron mobility as a function of electric field at lattice t e m p e r a t u r e T L = 15 K.

DISTANCE x (~m) Fig. i: T i m e - i n t e g r a t e d l u m i n e s c e n c e images (one-dimensional scans) for d i f f e r e n t lattice t e m p e r a t u r e s T L = 15 K, e l e c t r i c f i e l d E = 0 (curve a), E = 20 V/cm (curve b), T L = 50 K, E = 120 V/cm (curve c), T L = 90 K, E = 280 V / c m (curve d), T L = 150 K, E = 600 V / c m (curve e). The solid b l a c k lines are the c a l c u l a t e d image shapes a c c o r d i n g to ref.15. The insets show the t i m e - i n t e g r a t i o n of the d r i f t i n g m i n o r i t y c a r r i e r s (top left) and the time dependence of the l u m i n e s c e n c e i n t e n s i t y (top right) as m e a s u r e d w i t h a Si a v a l a n c h e p h o t o d i o d e w i t h a time r e s o l u t i o n of 0.44 ns.

645

similar results /17/:

The hole

value

cm2/Vs

of

~h = -40 000

mobility is

is negative

measured

at

the

at

low fields

lowest

and

possible

low temperatures,

temperature

a

and electric

field. Summarizing the experimental results, we c a n s a y for both minority electrons in a high mobility hole plasma and minority holes in a high mobility electron p l a s m a i n CaAs q u a n t u m wells at low temperatures and low electric fields: The injected minority carriers drift in the "wrong" electric-field direction minority electrons towards the negative, minority holes towards the positive electrode. The negative minority carrier mobilities decrease with increasing temperature (and electric field), and follow the same trend as the majority carrier mobilities (multiplied by -1, Fig. 2).

IV. ANALYSIS The a m b i p o l a r mobility is equal to the true minority minority electrons): n . ~ e << p . ~ p / 1 8 / . T h i s c o n d i t i o n

carrier mobility, if (for the case for minority carrier transport

of is

fulfilled in both experiments (n- and p-doped samples). Therefore the space charges of the injected and drifting minority carriers are screened by dielectric relaxation of the majority carrier plasma. For the analysis o f o u r d a t a we u s e a h y d r o d y n a m i c approximation /19/ with average relaxation times t h a t account for the various scattering processes. From t h e hydrodynamic equations of motion and taking into account momentum c o n s e r v a t i o n within the p l a s m a / 1 7 / we o b t a i n f o r t h e e l e c t r o n and hole mobilities



n-



- P-' m h_

- mh -



P. +

- + e . ( - me Pe =

~-~re_h>

)

n.me.

e.(

, (I)

and

Ph =

1 + + P.mh.

< r e _ L > a n d a r e

t h e momentum r e l a x a t i o n function)

distribution

of electrons relaxation

in time

and hole

masses,

the resting of

by lattice

hole

holes

plasma

in a restinH

n and p are

the

total

times

by

Coulomb

electron electron

+

-

and holes

(2)

(averaged

over

t h e momentum r e l a x a t i o n

m e a n d mh a r e sheet

me

n.me.

scattering,

plasma,

-

P.mh.

i s

and hole

-



of electrons

scattering.

-

n-m e

1 + ~

energy

-

mh

is

the

the effective

concentrations

per

the time

momentum electron

layer.

From E q s . ( 1 ) a n d ( 2 ) i t b e c o m e s c l e a r , when the minority carrier mobility becomes negative: For, e.g., n << p , P e i n E q . ( 1 ) i s n e g a t i v e , when the electron mobility relative to the hole

plasma,

e./me,

is

smaller

than

the

hole

mobility,

lattice-scattering times known (from refs 20-22, f i g u r e of the minority carrier mobilities ~e and ~h' the values

2) of

e./m h .

With

all

, and the experimental values the electron-hole scatterin~

times and can be obtained from eqs. (i) and (2).

As

shown

in Fig • 3 , the values of are below 10 -13 s (< I00 fs) for all temperatures,

with best values for of 80 ± 20 fs around I00 K. The values of are orders

of

magnitude

larger:

At

low

temperatures

almost

two

we obtain = 4 ± 1 ps, at higher

temperatures the relaxation time stays in the same range. The momentum relaxation times an_~d differ by roughly a factor of 50. The experimental accuracy sets a lower limit to this factor of at least 30 and an upper limit of about I00, only slightly on temperature.

depending

V. DISCUSSION The first point of discussion is the question, why d o e s e l e c t r o n - h o l e drag cause dramatic transport effects for minority carriers in quantum wells, but not in bulk semiconductors. The answer is that the strong electron-hole-drag effects observed in our experiments are due to both the high mobility of the majority carriers and their high concentration, realised by modulation-doping. The high concentrations cause short electron-hole scattering times, the high mobility allows a large effect of a given electron-hole scattering time on the minority carrier drift mobility, according to Eqs. (1) and (2). The v a l u e s

of


t h e momentum r e l a x a t i o n

time of minority

electrons

in a hole

plasma

of

p = 1 . 5 t o 1 . 9 x lO l l cm - 2 w i t h i n l l 2 A q u a n t u m w e l l w i d t h , a r e c o m p a r a b l e t o t h e r e l a x a t i o n t i m e s o£ e l e c t r o n s in highly doped n-CaAs, where ionized-impurity scattering dominates. The values below 100 fs are furthermore consistent w i t h t h e low r o o m - t e m p e r a t u r e mobility of

646

600

I

I

10-11

~

t 00

0 0

0

0

0

0

0

0

500

400

HOLES IN

10-12 Z O

n o = 3.0 x 1 0 1 1 c m - 2

I,<

x ,<

d

t

(EXP.)

IoOoo~ o

~

.." <~%,o,~,

I[~

/

uJ

.=,

x 1010cm'2

I 200

T L = 300K

p n d

300

TEMPERATURE (K)

0

300

Fig. 3: Momentum relaxation time of minority electrons and minority holes by electron-hole scattering as a function of temperature.

minority electrons

~ ~ .C.." ~

""

1 0 0 -I 100

,'I

200

.OLE P'ASMA

.L P 0 = 1 . 6 x 1011cm "2

10-14

f f

>-

ELECTRONS IN

n<3

]~

300

I 1"

(THEOR.)

o=

p < l x 1 0 1 0 c m "2

X

0

o_.__~_

-e - ph

I-

ELECTRON PLASMA

~ 10"13

-

0 0

in p-doped quantum wells,

= 4.2x1011 cm-2 = 5xl0gcm -2 = 90]~

I I I 400 500 600 ELECTRON TEMPERATURE Te(K )

700

F i g . 4: E n e r g y r e l a x a t i o n times of minority e l e c t r o n s a s a f u n c t i o n of e l e c t r o n t e m p e r a t u r e , by electron-hole scattering (dotted line), phonon e m i s s i o n ( d a s h e d ) . and t h e t o t a l energy r e l a x a t i o n t i m e ( f u l l l i n e and c i r c l e s ) .

(~e ~ 1 500 cmm/Vs,

corresponding

to ~ 57

fs), as reported at the previous conference /23/. The reverse case, the momentum relaxation of minority holes in an electron plasma, differs in three important ways from that of minority electrons in a hole plasma. First, the effective masses of electrons and holes are much different, which directly affects the momentum transfer in the two-component plasma: For an estimate we assume the same relative concentrations for the two cases and the same screening conditions, and in first order proportional gives

to

1/p

(Conwell-Weisskopf-formula).


>

= mh/m e .

Then momentum conservation /17/ directly (3)

This means that from classical plasma kinetics we expect linear

mass

to be

by

a

factor

ratio larger than . Since the majority electron concentration

of

the

in our case

is even higher than the majority hole concentration, the ratio should be even smaller, thus / should be smaller than a value o£ 6 (m e = 0.0665 m o, mh ~ 0.4 m o, m ° ... free electron mass),

in contrast

to the experimental

result of about 50.

A second difference is the fact that the majority electron plasma is degenerate up to temperatures above I00 K, whereas the Fermi energy of the hole plasma, due to the larger effective mass, is much lower. Since at low temperatures scattering is only possible by electrons near the Fermi level, degeneracy causes a further enhancement o£ relative to for T < 100 K. Finally, in the experiments reported above, there is an asymmetry in geometry between the p-type and n-type samples that increases relative to for all temperatures. The carrier concentration in the n-type sample is higher, but the n-type quantum wells are considerably wider (d I = 258 A). The p-type quantum wells are thus close to the 2D limit, whereas the n-type sample is more nearly three-dimensional. Coulomb scattering is known to be more effective in 2D than in 3D - at low momentum transfers screening is weaker in 2]] than 3D, at high momentum transfers the 2D Coulomb interaction falls off less rapidly than the 3D interaction. These effects are calculated in 5el. 17, and yield a further contribution to the large ratio of /. Thus all three effects, the mass ratio, degeneracy of the electron system, and two-dimensionality,

contribute

to the electron-hole momentum relaxation.

647

VI. PHOTOCONDUCTIVITY A further point of discussion is photoconductivity in the range of "negative absolute mobility": As i t w i l l b e s h o w n i n t h e f o l l o w i n g f o r a n n - m o d u l a t i o n - d o p e d quantum well, the photoconductivity signal should be negative in this range, the extreme electron-hole drag should reduce the conductivity of the sample. The change of the sheet conductivity ha by injecting minority carriers (concentration p) into the majority carrier plasma of concentration n o ( P o = O) i s g i v e n a s ha = n.e.p e + p.e.pp with

~e-L

scattering).

the

- no.e.Pe_ L

majority

Substituting

electron equations

(zt) mobility (1) and

/17/. The photoconductivity becomes negative condition is valid independent of the lattice

without (2)

for

the

presence

Pe and pp gives

of holes the exact

exactly when the mobility scattering parameters.

(only result

lattice for

becomes negative.

ho This

A negative photoconductivity s i g n a l w i t h t h e same d e c a y t i m e a s t h e p h o t o l u m i n e s c e n c e could no__! b e o b s e r v e d . However, both in the n-type and in the p-type sample, negative photoeonductivity is experimentally verified. The decay times, the absolute values, the spectral dependences, and the temperature dependences show that the observed negative pbotoconductivity is of different physical origin. As s h o w n i n r e f s . 2 4 a n d 2 5 , t h e n e g a t i v e photoconductivity originates from excitation of minority electrons into ionized impurities in A1GaAs a n d s u b s e q u e n t r e c o m b i n a t i o n of the minority holes (n-doped sample, ref. 24), and from trapping of the additional majority holes in the potential m i n i m a o f A1GaAs ( p - t y p e sample, ref. 25), which in the latter case leads to huge values of the negative photoconductivlty in the spectral range of absorption i n A1GaAs.

VII.

ENERGY TRANSFER

In a different s e t o f e x p e r i m e n t s we m e a s u r e d t h e m i n o r i t y e l e c t r o n energy distribution at high electric fields in the presence of electron-hole scattering. The results of these experiments are relevant for this paper, since from these experiments the energy transfer rate between electrons and holes can be determined, and subsequently the energy relaxation times of hot electrons in a cold (room temperature) hole plasma, by electron-hole scattering. The experiments are described extensively in refs. 17 a n d 26, the essential results are contained in Fig. 4: T h e e n e r g y relaxation times are shown as a function of electron temperature. The total experimentally measured energy relaxation-time is around 200 fs, increasing with temperature. The total energy relaxation of hot minority electrons consists of the contributions of both electron-hole scattering and eleetron-phonon-emlssion. With theoretical calculation of the pure electron-phonon energy relaxation / 1 7 / we o b t a i n e n e r g y relaxation times by electron-hole scattering between 200 and 500 fs.

VIII.

IMPLICATIONS

The m e a s u r e d v a l u e s o f t h e e l c t r o n - h o l e momentum r e l a x a t i o n times (Fig. 3) h a v e important implications for the question of ballistic transport of minority carriers. For collisionless (= b a l l i s t i c ) transport of minority carriers in an, e. g., bipolar transistor our results show that holes can traverse an n-type base without considerable electron-hole scattering. Hole-phonon and hole-impurity scattering will dominate at room temperature. In contrast, however, serious limits are set to ballistic minority electron transport by electron-hole scattering. The values of less than 100 fs (down to 40 fs) for the momentum r e l a x a t i o n of electrons in a hole plasma mean, that electron-hole scattering reduces the ballistic phase of the electron transport to very short distances. E.g., for average ballistic velocities of 3 x 107 c m / s a n d a n e l e c t r o n - h o l e scattering t i m e o f 5 0 f s , momentum s c a t t e r i n g is effective a l r e a d y w i t h i n 150 A n g s t r 6 m s ( 1 ) , n o t r e g a r d i n g o t h e r s c a t t e r i n g mechanisms. This has been seen very clearly in resonant tunneling spectroscopy of minority electrons /27/. A further point is the question of the validity of the Einstein relation for diffusion in the range of negative absolute mobility. It is obvious that the value of the drift mobility including the carrier drag is not the relevant mobility for the Einstein relation D = p . k T / l e I. A p a r t f r o m a m b i p o l a r e f f e c t s at high excitations /28/, the mobility relative to the drifting majority carrier plasma, is the relevant quantity for diffusion. A negative diffusion coefficient as obtained from taking the mobility including carrier drag, of course, would be unphysical in this connection. Concerning the photoconductivity, it is interesting to think of the transient photoconduetivity at temperature or electric field values where the minority carrier mobility is zero (see Fig. 2). There, according to Eq.(4), the photoconductivity i s a l s o z e r o . The transient photoeonductivity, however, on a femtosecond timescale, is not zero, since the acceleration of the injected carriers and the onset of electron-hole scattering (and the other scattering processes), in the general case, have different time-dependences. Therefore a transient photoconductive s i g n a l o f l e s s t h a n 100 f s d u r a t i o n (in p-doped samples) should be generated.

648 A further question is the stability o£ t h e n e g a t i v e m o b i l i t y a t h i g h e r e l e c t r i c fields, where the negative absolute mobility has negative differential behaviour: As o n e c a n v i s u a l i z e easily from Fig.2 (inset), the negative drift velocity (obtained from the plotted mobility) first increases with electric field (starting from zero at E = 0). The negative drift velocity then decreases f r o m E ~ 6 0 V/cm o n , a n d t u r n s t o p o s i t i v e at higher electric fields, For pure minority electron transport (n.~e<< p.pp), the transport in all field ranges is electrically stable, since all space charges are screened by the majority carriers. For higher electron concentrations, however, when s p a c e charges of fluctuating electron concentrations are not screened anymore, the situation is different. Then, in the low-field range the transport of minority electrons is unstable, for the following reason: For a locally increased electron concentration the electric field is lower towards the negative electrode, and higher towards the positive electrode. The electrons in the area of lower electric field will have lower (negative) drift velocity (in the area of higher electric field, respectively, higher negative velocity). An i n i t i a l charge density fluctuation, therefore, would build up more and more. Against this instability, screening of the majority holes and diffusion would act. Since the negative drift comes from the drag by the high-concentration and high-mobility holes, it becomes a quantitative question, in which range of the parameters the instability exists. The field of electron-hole drag opens wide areas for theoretical work. Theories of electron-hole scattering including screening and the influences of the quantum confinement (subbands and two-dimensional plasma properties) would be of high interest. A t t h i s p o i n t we want to refer to recent theoretical works on electron-hole scattering, by analytical theories /29/ as well as Ensemble-Monte-Carlo calculations /30/. Finally, it would be interesting to find other experimental systems (e.g., one-dimensional carrier systems), where carrier drag-effects can be studied further.

ACKNOWLEDGMENTS This work has been a permanent collaboration w i t h P . A . W o l f f , who a l s o p e r f o r m e d theoretical calculations on electron-hole momentum s c a t t e r i n g (Ref.17). The high-quality samples were MBE-grown b y A . C . G o s s a r d a n d W. Wiegmann a n d c h a r a c t e r i z e d b y K. B a l d w i n . One o£ u s ( R . A . H . ) likes to acknowledge support by the Fonds zur Fbrderung der Wissenschaftlichen Forschung, Osterreich (project P 6184).

REFEILENCES (a) work p e r f o r m e d w h i l e a t AT&T B e l l L a b o r a t o r i e s ,

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