PHYSICAL REVIEW B VOLUME 52, NUMBER 7

Self-oscillations at photoinduced impurity breakdown in GaAs

15 AUGUST 1995-I

M. Kozhevnikov, B.M. Ashkinadze, E. Cohen, and Arza RonSolid State Institute, Technion Isr—ael Institute of Technology, Haifa 32000, Israel

{Received 17 February 1995)

The effects of microwave {MW) irradiation on photoexcited free and donor-bound electrons are stud-ied in semi-insulating, bulk GaAs in the temperature range of 2—20 K. The main observations are as fol-lows: an abrupt increase in the photoinduced microwave absorption (PMA), a concurrent decrease ofthe donor-acceptor pair photoluminescence and self-oscillations in the PMA when the incident MWpower exceeds a threshold value. The PMA self-oscillation frequency varies with photoexcitation inten-sity and incident MW power in the range of 0.5—3 kHz. These nonlinear phenomena are observed underspatially uniform MW irradiation and photoexcitation, without any electrical contacts. A model isdeveloped for the dependence of the free and donor-bound electron densities on the microwave powerand photoexcitation intensity under the conditions of donor impact ionization (breakdown) by the MWheated free electrons. A linear dependence of the electron temperature on the free-electron density is as-sumed. The self-oscillation frequency is determined by the remote donor-acceptor recombination rate.The model-calculated steady state and self-oscillations of the free-electron density agree well with the ob-served PMA behavior. It is thus shown that spatially uniform self-oscillations are an intrinsic propertyof the photoinduced impurity breakdown.

I. INTRODUCTION

In the early 1980s a renewed interest arose in hot-electron phenomena in semiconductors. In particular,electric current instabilities, self-oscillations, and chaoswere considered as results of the complex behavior of anonlinear dynamic system that is easily affected by an ap-plied electric field and by photoexcitation. ' At lowtemperatures these instabilities usually occur when asharp transition from a nonconducting to a conductingstate is induced by the applied electric field. The basicphysical mechanism governing these effects is the impactionization of electrons (holes) bound to shallow impuri-ties by free carriers that are heated by the electricfield. Similar breakdown phenomena were also ob-served in pure or compensated semiconductors underphotoexcitation. In this case the donors (or acceptors)are ionized in the dark, and the photoexcitation leads totheir neutralization and subsequently to impurity or exci-tation impact ionization. '

Regular and chaotic electric current self-oscillations,that are often observed at impurity breakdown, are asso-ciated with an S-shaped current-voltage characteristic[namely, with a negative differential conductivity(NDC)]. In the NDC region, current or electric-field dis-tributions are unstable with respect to a fluctuationgrowth of current filaments or fields domains. ' Amotion of the current filaments (or field domains) or anexternal circuit will cause tem. poral instabilities of theelectric current when a breakdown is induced by an ap-plied, direct electric field.

In several studies, microwave (MW) irradiation wasused for charge-carrier heating instead of a dc electricfield. ' ' ' This contactless method allows the study ofimpurity breakdown under spatially homogeneous condi-tions that exclude both electric-field and electric current

redistributions, and eliminates effects due to the externalcircuit. Growth of any carrier density fluctuations is alsoexpected to be attenuated when the electric field varies atmicrowave frequency. Nevertheless, self-oscillations inthe electron density were observed in photoexcited Ge,Si, and GaAs crystals, at T & 10 K, under the spatiallyhomogeneous conditions of MW irradiation. ' '

Current and electric-field instabilities are commonlyanalyzed within the framework of the phenomenologicalimpurity-breakdown model that assumes impact ioniza-tion of both the ls and 2s neutral donor (or acceptor) lev-els by the heated free electrons. This model was intro-duced by Kastalskii' in order to explain the S-shapedcurrent-voltage characteristic of compensated Ge at lowtemperatures. Scholl ' used this model in order to de-scribe the impurity-breakdown phenomenon as a none-quilibrium phase transition. He analyzed both stationaryand dynamic current-voltage dependencies and showedthat current instabilities become possible only if both spa-tial and temporal variations of the electrical field (dielec-tric relaxation) or the slow energy relaxation of hot car-riers are taken into consideration. Recently, Quadeet al. ' carried out a Monte Carlo simulation of the mi-croscopic breakdown model for the case of p-type Ge,and Kehrer, Quade, and Scholl did the same for n-typeGaAs. Their results demonstrated that the S-shaped, dccurrent-voltage behavior and the associated instabilitiesare predicted only if a two-level model is used (in contrastto a one-level donor model). However, these authors didnot take into account some specific features of hot-electron processes at low temperatures. For example, therunaway of hot electrons (namely, an electron overheat-ing' ) and the increased screening of the electron scatter-ing by ionized impurities as the breakdown occurs werenot considered. These phenomena were believed to beessential in the earlier phenomenological models of im-

0163-1829/95/52(7)/4855(9)/$06. 00 4855 1995 The American Physical Society

4856 KOZHEVNIKOV, ASHKINADZE, COHEN, AND RON

purity breakdown. ' ' ' In these studies, impact ion-ization of only the 1s donor state was considered, and theS-shaped current-voltage curve was associated with vari-ous mechanisms leading to an increased mobility as theelectron density increased under breakdown conditions.Therefore, though the two-level model is often used forthe analysis of the nonlinear behavior and various insta-bilities of both the dark current and the photocurrent, aswell as of the photoluminescence (PL) intensity depen-dence on the electric Geld, ' '" a comprehensive micro-scopic theory is still unavailable. Moreover, there is nophenomenological model taking into account the recom-bination of nonequilibrium charge carriers and theiroverheating in analyzing instabilities under photoinducedbreakdown conditions.

In this study we present an experimental investigationof the photoinduced breakdown of shallow donors innominally undoped semi-insulating GaAs at low temper-atures under spatially homogeneous MW irradiation con-ditions. The stationary and temporal dependence of thephotoinduced MW absorption (PMA) and of the excitonand donor-acceptor pair photoluminescence (PL) on in-cident MW power and light excitation intensity as well ason temperature were studied. Abrupt variations of bothPMA and PL intensities at a threshold MW power andself-oscillations in the PMA in a certain range of the MWpower and photoexcitation intensity were observed. Ex-perimental results are analyzed in terms of a phenomeno-logical model for the spatially homogeneous photoin-duced breakdown of shallow donors. Impact ionizationof only the 1s donor level is considered, and the depen-dence of the electron temperature on the free-electrondensity is explicitly introduced. The steady-state and dy-namic solutions of the rate equations for the free anddonor-bound electron densities are obtained. We showthat the self-oscillations in both free and donor-boundelectron densities are an intrinsic feature of the photoin-duced breakdown phenomenon. Good agreement is ob-tained by comparing the model calculations of the self-oscillations in the electron densities with those observedexperimentally. Our model also provides an estimate ofthe semiconductor parameter ranges, namely the donorconcentration, electron-hole recombination rates, and lat-tice temperature in which self-oscillations are predicted.

The paper is laid out as follows: In Sec. II, the experi-mental setup, the samples used, and the experimental re-sults are presented. A model for the photoinduced donorbreakdown and an analysis of the resulting steady state aswell as dynamics of the free and donor-bound electrondensities are detailed in Sec. III. A discussion and com-parison of the model with the experimental results aregiven in Sec. IV. An analysis that determines the condi-tions on the model parameters for which the self-oscillations are expected is presented in the Appendix.

II. EXPERIMENTAL PROCEDUREAND RESULTS

We investigated several samples of nominally undoped,semi-insulating bulk GaAs, grown by the liquid-encapsulated Czochralski (LEC) technique. The samples,

with dimensions of 4 X 6 X0.5 mm, were cut fromdiFerent wafers (that are used as substrates for quantumstructure growth). Electron heating is achieved by mi-crowave irradiation (instead of the commonly used directelectric field). Using microwaves allows us to apply anelectric field in a contactless manner, whereby we obtaina macroscopically uniform electric-field distributionwithin the sample. The samples are placed in the an-tinode of the microwave electric field in an 8-mmwaveguide which is short circuited at one end. Thewaveguide was immersed in liquid He or in cold He gasso that the temperature was varied in the range ofT=2—30 K. Laser light (He-Ne and tunable Ti-A1203lasers as well as a quartz-halogen Glament lamp wereused) illuminated the sample through a pinhole in thewaveguide. This excitation was modulated at a frequencyvarying in the range of 10—400 Hz.

A stabilized 36-GHz Gunn diode was used as the MWsource. An electrically controlled attenuator allowed usto vary the incident MW power continuously so thai thepower incident on the sample was in the range of 0.1 —20mW. The reAected MW radiation was directed onto adiode detector by a circulator.

The rejected microwave power was modulated by thephotoexcitation of the sample. This modulation signal isproportional to the microwave power absorbed by thephotoexcited free electrons. The photoinduced mi-crowave absorption (PMA) is -o e -crI', where cr is theconductivity, and c and P are the microwave fieldstrength and microwave power, respectively. The con-ductivity at the MW frequency m is given byo =o o/(1+co r ), where o o=enp is the dc conductivity,r=pm*/e is the momentum relaxation time of the freeelectrons, and m* is the electron effective mass. At lowtemperatures (T & 30 K) the dc mobility of LEC-grownGaAs is of the order of p —10 —10 cm V ' sHence cur«1 and PMA croP =-enpP. Therefore, thePMA signal is proportional to the free-electron concen-tration.

We studied the steady-state and temporal dependencesof PMA on the MW power and on the photoexcitationintensity (Ir ). Similarly, the PL and PMA spectraldependences on these parameters were also studied. ThePL and the signal from the MW diode detector were ana-lyzed by a lock-in amplifier or by an oscilloscope that hasa fast Fourier transform function.

Figure 1(a) shows the PMA dependence on P for Iz = 1

mW/cm . At a certain threshold MW power, an abruptincrease of the PMA is observed. Then a hysteresisbehavior occurs as the MW power sweep direction is re-versed. The threshold MW power depends weakly on thelight-excitation intensity (in the range of Ir =0. 1 —30mW/cm ). The threshold decrease of the PL intensity atan energy of 1.49 eV for the same MW power [Fig. 1(b)]will be discussed below.

Under cw microwave and light irradiation conditions,the PMA signal starts to oscillate when P exceeds thethreshold value (Figs. 2 and 3). The frequency of theself-oscillations depends on both P and I~, as can be seenfrom the Fourier transforms shown in Fig. 4.

The free-electron lifetime was extracted from the tem-

SELF-OSCILLATIONS AT PHOTOIND UCED IMPUR ITY s ~ ~

cf)

C3

ge~j-insulating GaAsT=2K

EL——1.96 eV

=1m% / cm 2L

Semi insulating QaAsT=2K, EL——1.49 eV, I„=lmW/cm

(j~(c) &

t/)

1—CD

1.48 1.49 1.50 1.51 1.52photon energy (eV)

0.5P (mW)

1.0(e)

t

rIiIIIItIIIIII'

g )IIt

The hotoinduced microwave pe absor tion (PMA)FIG. 1. (a) p

icrowave irradiatio~luminescence spectrara observed wit out microres old.and atP ig erh h than the breakdown thres o

Semi-insulating GaAs

T=2K, E,= 1.49 ev, P =1.5 mW

10Time ( ms )

FIG. 3. Similar to ig.F' 2 for the incident MW power valuesP (a) 1 (b) 1.5, (c) 3.5, (d) 8, and (e) 18 mW.

nse of the PMA signal to a square-s pe-sha ed

Photoluminescence spown in the inset of ig.MW irradiation are shown

'

Semi-insulating GaAs2T=2K, EL——1.49 eV, &L = ImW/ cm

(c)

(~)I)I ) IIII)|I f/(i)(II tI)( /II, II(e)

0I I

10Time (ms)

I

20I

0I I

2Frequency ( kHz )

I

4

the hotoinduced microwave ab-FIEx. 2. Self-oscillations of e p osorption (PMA). The light intensity values Il are (a) 0. ,(c) 6, and (d) 20 mW/cm

rms (c),ms of the three PMA wave formFICz. 4. Fourier transforms o t e(d), and (e) shown in Fig. 3.

4858 KOZHEVNIKOV, ASHKINADZE, COHEN, AND RON 52

low-energy PL band (around 1.49 eV) is due to donor-acceptor-pair (DAP) recombination, and the PL bandnear 1.51 eV is due to bound exciton recombination.The low-temperature intensity ratio between these twobands provides a rough estimate of the impurity concen-tration. Similar PL spectra were reported for undopedGaAs samples with shallow donor concentrations es-timated to be about (2—5) X 10' cm

The PL intensity decreases when the MW power Pexceeds the same threshold as that required to cause theabrupt increase in the PMA [Fig. 1(b)]. This decrease isobserved for all PL bands. A hysteresis loop is observed,similar to that of the PMA.

The PMA nonlinear behavior and its self-oscillationsare also observed at photoexcitation energies lower thanthe band-gap energy. Figure 5 shows the PMA excitationspectrum observed at low microwave power (P=0.1

mW). The appearance of the PMA for an excitation en-ergy as low as 0.8 eV is attributed to an electron excita-tion from the EL2 state [which is associated with thedouble As (Ga) antisite defect ] to the conduction band.The band at —1.50 eV is associated with an electrontransition from acceptor levels to the conduction band.The sharp decrease of the PMA at photoexcitation ener-gies near and above Eg p is due to the strong surfacerecombination of the electron-hole pairs. It should benoted that there are some differences between the resultsof this work and our preliminary report. ' These are dueto a difference in the sample structure (there we studiedsamples with thin epilayers of pure GaAs).

We studied the effect of increasing the ambient temper-ature on the observed PMA nonhnearities. The PMAself-oscillations and the threshold dependence in the

Semi-insulating GaAs

PMA and the PL disappeared at temperatures above 10K.

Self-oscillations of the PMA were not observed in otherGaAs samples with higher impurity concentrations, forwhich exciton emission was not observed.

III. MODEL AND ANALYSIS

dm =yn (N~ —m) —Pnm-dt

—y mls„. (lb)

Here yn (ND —m) is the rate of free-electron captureby the ionized donors, XD is the donor density, and y isthe capture rate coefFicient. The recombination rates offree and donor-bound electrons with holes (mainly boundto acceptors) are given by the terms

nlrb,

and

mlle&,

where ~, and ~2 are the respective recombination times.The term ymS„describes the thermal excitation of elec-trons from the donor levels to the conduction band.N„=N, exp( E; Ik T), where —N, is the density of

We consider a highly compensated semiconductor withone type of donor and acceptor (Fig. 6). In the dark, thedonors are completely ionized and the acceptors are oc-cupied by electrons. Photoexcitation generates electronsin the conduction band at a rate of 6 (with partial neu-tralization of the donors). Only free electrons will beconsidered, since the holes are rapidly trapped bycharged acceptors and, therefore, the hole concentrationis small in comparison with that of the free electrons.The applied electric field c. heats up the free electrons,which in turn impact ionize the bound electrons in the 1sdonor level. In this analysis, the electrons are assumed tobe heated by either a microwave or a dc electric field.Then the rate equations for the free-electron density n (t)and for the donor-bound electron density m (t) are

dn =G+Pnm yn(ND——m) — +ymN„,n

+1

e

L

CL

G

pnm ~(N, -~), ~N„D

CB

---- ------- EL2

I

0.8 1.0 1.2 1.4Photoexcitation energy (eV)

apA

VB

FIG. 5. The photoinduced microwave absorption (PMA) ex-citation spectrum observed at low microwave power.

FIG. 6. Energy-level scheme and dynamic processes of themodel used to analyze the nonlinear behavior of the free anddonor-bound electron density ( n and m, respectively).

SELF-OSCILLATIONS AT PHOTOINDUCED IMPURITY. . . 4859

conduction-band states, E; is the ionization energy ofdonors, and T is the lattice temperature. The term Pnmdescribes the impact-ionization rate of neutral donors byhot electrons, where P is the impact-ionization coefficient.

In general, the coefficients y, r„~2, and P depend onthe hot-electron energy (E, ). We assume that the maineffect of the hot electrons is due to the dependence of Pon E, . Hence, following Refs. 6 and 9, we neglect thedependence of y, ~„and r2 on E, . Under electron heat-ing there are variations in the free-electron-energy-distribution function, and the mean value of E, increaseswith increasing c. We thus have to determine the depen-dence of P on s. If the hot electrons have a Maxwelliandistribution function with an effective electron tempera-ture T, =2E, /3k, then P may be approximated byP=Poexp( E;/k—T, ), where Po=ma&v„ab is the donorBohr radius, and u, is the electron velocity. Then, theelectron temperature dependence on c. can be found fromthe following balance equation: '

dT.dt

2 2T T

ep(T„n)c. —

The first term on the right-hand side describes theheating of free electrons by the applied electric field c,and the second term describes the hot-electron cooling.

is the energy-relaxation time. Since the electron-energy-relaxation rate is much larger than the rate ofelectron-density variation, we can approximate the elec-tron temperature by the value obtained from the steady-state solution d T, /dt =0:

these mechanisms is the following: with increasing free-electron density there is an increased screening of thecharged-impurity electric field, and this results in a de-crease in the electron-ionized impurity scattering rate.

We tried to overcome these difticulties by phenorneno-logically taking into account the runaway effect and theincreased screening of the random electric field of thecharged impurities. We assume a linear dependence ofthe electron temperature T, on the electron density, anda quadratic one on the electric field s (T, ~ na ) for thefollowing reasons. First, a numerical estimate of the mo-bility increase with increasing n shows that a linear ap-proximation of the Brooks-Herring expression forelectron-ionized impurity scattering is reasonable in thefree-electron density range of 10"—10' cm . Second,the runaway effect can also be qualitatively described bythis dependence. For an electric field lower than thebreakdown value, the free-electron density is low andthere are few high-energy runaway electrons that canproduce an impact ionization. With increasing electrondensity due to an impact ionization, the increasedelectron-electron-scattering rate [which becomessignificant for n ) 10" cm (Ref. 28)] redistributes theenergy of the overheated electrons among all the freeelectrons, and this results in an increase of the mean elec-tron energy. Hence we assume a linear increase of theeffective electron temperature T, as the electron densityincreases. Using Eq. (3), we introduce both these process-es in our model by setting the impact-ionizationcoefficient as

T(ns)= T+ r( Tn)p( Tn) s (3) P =Poexpe

1

(1+An)P,(4)

In order to solve Eq. (3) for T„ it is necessary to deter-mine the dependencies p, (T„n) and r, (T„n). In LEC-grown GaAs crystals with impurity concentrations ofabout 10' —10' cm, the electron mobility at tempera-tures T & 30 K is determined by ionized-impurity scatter-ing, and, therefore, p ~ T, . At T, & 50 K theelectron-energy losses are due to their interaction withacoustic phonons and to impact ionization of impuri-ties. In GaAs, the electron-phonon interaction is pri-marily via the piezoelectric potential with an energy-relaxation time ~, ~ T,' . Substituting these temperaturedependencies of p and r, into Eq. (3) results in the so-called runaway effect: an electron heating rate [first termin Eq. (2)] increases with s faster than the energy-loss rateto the phonons. ' ' Impact-ionization losses can limitthis effect and the dependence T(s) will then show an S-type behavior. Therefore the electron temperature in-creases sharply at a certain c., and can induce the irnpuri-ty breakdown. ' The solution of the nonlinear Eq. (3) de-pends very strongly on the values of the parameters p and'T and, implicitly, on XD . Therefore, the dependenceT, (n, e)cannot be de, termined explicitly. Moreover, it isnecessary to take into account the dependence of the elec-tron mobility on the free-electron density. Various mech-anisms leading to a p(n) dependence were discussed inRefs. 8 and 9 in connection with S-shaped current-voltage curves in the case of impurity breakdown. One of

with

P& =—'poe~, c /E, .

P, is a dimensionless control parameter that measuresthe absorbed electric power per electron in units ofE; /~, . It is defined for a constant po, the electron mobili-ty in the absence of a large free-electron density. To afirst approximation, P& is independent of the electrondensity. A, is a constant to be determined by fitting thismodel to the self-oscillation periods.

It should be noted that the impact-ionizationcoefficient dependence on the free-electron density waspreviously considered ' within the framework of thefree-electron screening of ionized donors. In Ref. 29 thedependence was assumed to be P=Poexp( E, /kT, )+zn-(where T, does not depend on n, and z is a constant), andit leads to nonlinear effects. Although this choice wasconvenient for the numerical simulations, it was not sup-ported by any physical arguments.

We rewrite now Eqs. 1(a) and 1(b) using the dimension-less units x =yam, y =y~2n, N=y~2S&, C =ywzN„,g =yr2G, r =r2/7&, r=tlr2, R =P/y, RO=PO/y, andX, =X/yr, :

ZE, COHEN AND RONKOZHEyNIKO&, A

g+yx(Z +1)—y(&+")+d~ '

=yN —y& (g +1)—x(1+C) .

(6a)

(6b)

(7a)

1 NPt = (1+A,,yz)ln r

35—EQ

c 30

Mc 25CD

o20-OCD

cD 15-MMCD 10—OMIEZ3

(a)

1+yo+ C

yo. (7b)

'bing the Photo&nduce Pim UritYThese eq&at'on 'h obtained f«e~c~tbreakdown, are Y

dence T (n) was aver similar to those o a'

sso-re the linear depen enbreakdown where6' ct of electron- o e-h l scattering on theciated with the e ec oain in a microwave egs g

e endencies o epd 1so x ) on the contro pation yc (and a so xo

P, ) are obtained from

+y. ,yo(R +1)+C+1

g) 35-DL~ 30-M

25-13

O20 -'0

CD

CD

M 15-ChCD

o 1Q-M

CD

a

0'rn 300-~CD

250—O

OCD

~ 200—

150-O

100—MMCD

50-0MC

E 00

X

I

100

100

I

200

200

Light intensity g

I

300

300

I

400

I

400

steady-state dependencies oof the freend electron densities as a uncand donor-boun

l tric power valuesnsit for various e ec r'toexcitation intensi y~ 0 F0

the same as given in ig.parameters are e

00.0

I

0.1I

0.2I

0.3I

0.4~ 300-M

CD

O

~ 200-U

CD

O

Oo~ 100DMI0M

cD 0EoX

(5)(1):g=50(2): 9=100(3):g=150(4): g=200(5):g=300

(4)

(2)

I

0.4II

0.1 0.2 0.3

of E./~ )Electric power P, (in units o

FIG. 7. (a) The calcuulated stea y-sd -state free-electron concen-r (I' ) for various pho-of the electric powert t on as a function oIa 1

=2 K. The concen tration is given inxcitation levels at T== 10' cm using the0-)(b)s l f h

tron density in units o x =&&2p 'Vz D

he dotted curve in the uppers to

and A, &=0.03. T ehe dotted- as eto g-=100 and r=4. T u h d curve correspon s

te the unsta e se bl tates for whichT 1 0= 0 K The arrows indica e

in Fi . 10.temporal simu a'

l tions are shown in 'g.

Of

V)

0)

CO

0OOCLMCO

O

Q)

E

CD

O-O

O-O

0.10 0.20 0.250.15Electric power P, (in unin units of E. ~ )

0.30

arameter {light intensity g vs electricg~ ~

contour there arelane. Inside the cdel arametersThe dimensionless mo e pth electron density. T e im

are the same as given in ig.

SELF-OSCILLATIONS AT PHOTOINDUCED IMPURITY ~ ~ ~ 4861

The calculated stesteady-state dependencies of the freeand donor-bound electron densities on the in

'

and bi, respectively. An S-shaped curve

an istability are obtained for both the freec ron densities in limited regions of the pho-

culations show that the shape of thee o e curves depends on=~2 ~& . As r increases, the S-shaped de d

vanishes. Thise epen ence

first-is emonstrates a transform t' fa ion rom a

ele-order phase transition to a d- da secon -order one in the

e ectron density behavior as the ratio bee e ', as e ratio between the life-

~2 an v.) varies.

y of the steady-state solutionsAn analysis of the stabilits ows that there is a ran eintensity and electric fiels h

'ge o control parameters (light

eld) wherein they are unstable.e analysis is given in the Appendix and

'

presented in F'ix, an its results are

in ig. 9. The temporal behavior of the pho-toinduced breakdown reveals that self-os

'

an onor- ound electron densities are obtained onlor the control parameters th ta are inside the contour

shown in Fi . 9. T'g. . T"e numerical solutions of the d namequations [Eqs. (6a) and (6b) for strol

a an or some values of the con-o parameters are shown in Fi 10 Iig. . n terms of the i-

mensionless parameters the noe non inear effects occur in theo owmg ranges: yr2G =30—300, yr2ND = 100—1000,

r2/'r~ 3 —15 p/y =500—1500, and A, /yr =0.01—0.1.e p otoexcitation levels of G = 10'7—

photon/cm s used'

—10

donor concensed in the experiments and f th hor es allow

centration range of N~ = (2—5) X 10' cmbt i =(0.3—3)X10 s and, hence, y = 10 cm s

and p-10-6cm3s-i . This choice of parameters yields agood agreement of the model 'th h

~ ~

e wi t e experimental dataand is discussed in Sec. IV. Tabla e I summarizes the mod-e parameters and compares them to available v

pearing in the literature.o avai a e values ap-

Increasin the'

g ambient temperature leads to a de-creased population of donor-bound elect, dself-oscillations disappear. I d d

' sol-e ec rons, and the

n ee, our numerical sol-tions show that for T ) 10 K the self-oscillations disap-

solu-

pear using the donor ionization energy of 5 meV

0.17

CQ

~ ~~~M

Q)D

0OQ)

MQ)

O~ ~(h

Q)

0ED

C-

LL I

0.18

020

C)6 0

C) .

FIG. 10.0. umencal simulations of the self-oscillations in the electron d 't dron ensity under im-

Various'ous electric powers P& at g= 100. (b) Vari-n 1g. . a

ous light intensities g at P& =0.175.

o ~ lr 6 9 L) L 6 ~U. L»

023

lA, .CV

024

o~0 2 4

Time in units of th 2

O ~6 0 2 4

Time in units of th 2

4862 KOZHEVNIKOV, ASHKINADZE, COHEN, AND RON 52

TABLE I. Model parameters.

Parameter

y (cm s ')

Po (cm' s ')

This work

—10

—10

(0.3-3)X 10-'

From references

102X10-"2X 10-'=10

10 '-10

4114

3225

IV. DISCUSSION AND CONCLUSIONS

We have shown that nonlinear effects due to impactionization in semi-insulating GaAs at low temperaturescan be explained by a one-level donor model, assumingthat the electron temperature depends on the electrondensity. This dependence provides the simplest approxi-mation to the complex hot-electron processes that takeplace in the low-temperature impurity breakdown inGaAs, and in particular the electron runaway effect.Indeed, the breakdown in GaAs starts at very low directelectric fields (on the order of 1 V/cm); ' " theequivalent threshold microwave electric field value(E-&I' ) in our experiments is approximately the same.A simple estimate of the energy gained by a free electronin GaAs, moving over a distance equal to its mean freepath at this field value, shows that the energy gained isinsuf5cient to induce a breakdown of the donors. Howev-er, such a low threshold value can be easily understoodby taking into account the electron runaway effect. It fol-lows from Eq. (3) that an abrupt increase of T, occurs ata rather low electric field when T, -2 T, ' and this ini-tiates the impurity breakdown. This means that the ru-naway effect is essential for breakdown in GaAs. Unfor-tunately, a microscopic theory of the impurity-breakdown phenomenon driven by the runaway efFect isstill unavailable. We suggest that the T(n) dependenceassumed in our model [Eq. (4)] takes the runaway effectinto account since a good agreement between the experi-mental results and those calculated with our phenomeno-logical model is obtained.

The model calculations yield the recombination timerange wherein the self-oscillations are possible, namelyrz = (0.3 —3 ) X 10 s. The calculated self-oscillationperiod (Fig. 10) varies in the range of (0.3—1)Xrz andthese give the estimate of the self-oscillation frequency.The latter agrees with the observed self-oscillation fre-quency range of 0.5 —3 kHz. In addition, the self-oscillations are predicted for r~/r, =3—15. This leads toa calculated free-electron recombination time which isalso in agreement with the experimental value tz =0.3ms. Note that a recombination time ( rz ) of mil-liseconds is expected for remote donor-acceptor pairs un-der a very low excitation level.

Our experiments under microwave irradiation and themodel of the photoinduced impurity breakdown demon-strate that self-oscillations in both free and donor-boundelectron densities are an intrinsic feature of the photoin-duced breakdown phenomenon. Self-oscillations occureven when there is no S-shaped electron-density depen-dence on the electric field. The model shows that asr =~2/~& increases, the S-shape dependence disappears

ACKNOWLEDGMENTS

The research at Technion was done in the Barbara andNorman Seiden Center for Advanced Optoelectronics.B.M.A. acknowledges the support of Foundation Rich(France) and Alexander Goldberg Fund of the Technion.

APPENDIX

In order to find the conditions that the model parame-ters must satisfy so that self-oscillations of the electronicdensities will occur, we analyzed the local temporal sta-bility of a solution of Eqs. (1) in the vicinity of the steadystates. If the steady state (no, mo) is an unstable focus, aself-oscillation regime of the dynamic solutions can ariseand, similarly to Ref. 12, this leads to the following con-ditions on the parameters:

= 2(1) r= )1,(2) A„)1,(3) Qyrz(rz/r, —1)(noP/y+ no+%„)& ( 2»+ 1)/2,

where

3 )) =y&2,3PXr,nor; p+—+1

2poer, E (I+A,„) y

r 12X 6&2—no

T]—n —+1

y

D ct72

but instabilities persist. For r (1, there are no self-oscillations and the steady-state solutions are stable.

A self-oscillation behavior is possible only when thesystem exhibits both positive and negative feedbacks. Inour model, impact ionization of the donor level is a posi-tive feedback, and the free-electron —acceptor recombina-tion and donor-acceptor transitions are negative feed-backs. The impact-ionization rate coefficient P must in-crease with increasing free-electron density for self-oscillations to occur.

In summary, we observe a threshold increase in thephotoinduced microwave absorption and a concurrentdecrease of the donor-acceptor pair photoluminescence insemi-insulating bulk GaAs for temperatures below 10 K.Self-oscillations in the PMA are observed when the in-cident MW power exceeds a threshold value. The fre-quency of the self-oscillations increases with increasinglight intensity and the incident MW power, and is in therange of 0.5 —3 kHz. The self-oscillations occur in a lim-ited range of MW power and photoexcitation level. Allthese features are predicted by the one-level photoin-duced impurity-breakdown model which takes into con-sideration both free-electron and donor-acceptor pairrecombination. In both the model and the experiment,the self-oscillations occur under spatially homogeneousdistribution of the electric (microwave) field. The charac-teristic time of DAP recombination controls the self-oscillation frequency.

SELF-OSCILLATIONS AT PHOTOINDUCED IMPURITY. . . 4863

Since no and P are functions of G and P [determined byEqs. (4) and 7(a)j, it is possible to find analytically therange of control parameters (light intensity and electricfield) wherein the stationary value of electron density is

unstable (it was calculated and is presented in Fig. 9).Numerical solutions of the dynamic Eqs. (6) in this rangeof the parameters are shown in Fig. 10.

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