Picosecond spectroscopy of semiconductors

PICOSECOND SPECTROSCOPY OF SEMICONDUCTORS D. H. AWON and S. MCAFEZE Bell Laboratories,MurrayHill, NJ 07974,U.S.A and C. V. SHANK, E. P. IPPENand 0. TEXHKE Holmdel,NJ 07733,U.S.A. Abstract-Tbe time-resolvedreflectivity of picosecond pulses from optically excited carrier distributionscan provideimportantinformationaboutthe energy relaxationrates of hot electronsand holes in semiconductors.The basic optical propertiesof non-equilibriumcarrierdistributionsare discussed, and in the specific case of GaAs, a semi-empiricalanalysisof the reflectivityspectrumis describedwhich estimatesthe contributionsfrom the principal criticalpointsof the bandstructure.Using Boltzmannfactorsto approximatethe hot carrierdistributions,it is found that the non-equilibrium reflectivityspectrum is a sensitive function of carrier temperature and that it can reverseits sian as the distributionrelaxes.These resultsare in good qualitativeagreementwith recentexperimentsemployinga

mode-lockedcw dye laser.

Picosecond optical pulses[ll offer a unique opportunity to study the dynamics of non-equilibrium carriers in semiconductors. Although indirect measurements of energy relaxation rates can be made by microwave and other techniques[2], direct time-resolved measurements are clearly desirable to elucidate the details of the relaxation mechanisms. Luminescence measurements(31 of hot carrier distributions using both pulsed and cw excitation have proved important information about hot carrier relaxation rates. In this paper, we discuss the basic features of the picosecond optical approach with particular emphasis on the interpretation of timeresolved reflectivity spectra. The experiment consists of using two optical pulses, one which is used to generate free carriers by direct band-to-band transitions, and the other to measure the induced change in reflectivity from the surface. A variable time delay between the excitation and measurement pulses enables the measuremknt of the complete dynamics of the reflectivity with a time precision comparable to the duration of the optical pulses (typically lO_” s). If the photon energy of the excitation pulse is considerably greater than the band gap, the carriers will initially be very “hot”, i.e. their mean kinetic energy will greatly exceed the thermal energy of the lattice. They will then relax to the band extrema by various mechanisms such as acoustic and optic phonon emission, until they are in thermal equilibrium with the lattice, after which they will recombine. Since the optical properties depend on the detailed nature of the carrier distribution, the time evolution on the reflectivity provides information about the relaxation rates. An example of a typical experimental result is shown in Fig. 1. This figure shows the time evolution of the incremental change in reflectivity of a GaAs crystal following excitation with a picosecond optical pulse of energy & = 4eV. The pulses were produced by a mode-locked cw dye laser[4] and its second harmonic and had durations of approximately 1.0ps. As indicated

GoAs REFLECTIVITY 4

PUMP : 3075

a

PROBE:

8

6150

LK a

5

0

V

TIME

DELAY

c

IO

(psec)

Fig. 1. Time-resolvedreflectivityfrom non-equilibriumcarriers in GaAs following excitation at 4eV. The probingwavelength was 2 eV. Opticalpulse durationswere approximately1.0ps.

in the figure, the reflectivity first decreased, and then approximately 2ps later increased and leveled off at a value of approximately AR/R = 10m3for an excitation energy density of approximately lo4 J/cm2. To interpret this result, it is necessary to fist reconsider the basit optical properties of semiconductors to allow for the influence of non-equilibrium carrier distributions. The appropriate expression for the electronic contribution to the real and imaginary parts of the optical dielectric function is[5]:

147

$2 d3W(E

n I-

f(E,)V,2E,(k) f(E..)llp

(E.-E..)~~)‘-(E,-~“,)‘]

I’

(intraband)

(interband)

(1)

D. H. AUSTONet al.

148

where p.., is the momentum matrix element, f(En) are the carrier distribution functions, E,(k) is the energy of the nth energy band, and m is the free electron mass. The real part is separated into an intraband, or freeelectron like component, and an interband component representing the contribution from transitions between energy bands. If the Laplacian operator in (1) is evaluated at the band minima, the usual Drude expression results for the optical polarizability of a free electron gas having an effective mass m*. When the optical frequency is much less than the interband transitions frequencies, this term provides an adequate description. However when the optical frequency is comparable to the interband transition frequencies, as in the case that interests us here, the interband terms must be included. The complete evaluation of expressions (1) and (2) is an extremely difficult task requiring a detailed knowledge of the band structure. We will not pursue this direction, since our intention is to understand the basic physical mechanisms responsible for the experimental results, and not to achieve perfect numerical agreement. Consequently an approximate analysis will suffice at this time. To gain further insight into the qualitative features of c(w), we consider the case of a direct gap semiconductor in which the contribution from a single isotropic valence and conduction band dominate. If we further assume that the carriers are in quasi-equilibrium at the band edge (i.e. low carrier temperature), and are not exactly in resonance with w, We find the incremental change in c(o) due to optical pumping is: 4me*

E,)

and l :“’ = the unperturbed dielectric function, and the subscripts u and c refer to the valence and conduction bands. At other critical points such as M, saddle points, this type of factorization is not valid. If, however, the distribution functions are slowly varying in the region of the singular point, we can write the following approximate expression MO)

= -e2(o)(wXl- f(E”i) + f(E,,)l

(5)

where Eui and E, are the valence and conduction band energies of the jth critical point which can be determined from band structure calculations such as the recent work of Chelikowsky and Cohen[81. If the frequency domain is partitioned into regions where E&) can be uniquely attributed to a specific critical point, we can write A&J) = -e2(0)(M - f(E&)) + f(E,(o))J where E, and EC are piecewise continuous functions of frequency whose particular form depends on the critical points. At the carrier densities encountered in the experiment of typically lOI to 10aOcmm3,the rate of carrier-carrier scattering is estimated to be much greater than the electron-phonon scattering rates[9]. Hence we can approximate the carrier distribution functions by Boltzman distributions with a characteristic temperature T, i.e. during the period of interest, the carriers are in equilibrium with each other but not with the lattice.

’

k=-;;;i;;IpJ *

fc[Ec(o)] = &

where m* = reduced effective mass, h,, = band gap = E,, n,p = density of electrons and holes. The matrix element lpi’ = fiwsm’/4m* from k 3p perturbation theory[6] was used in (3). This expression also reduces to the classical Drude result when ho *E,, and produces a characteristic negative contribution to e(o). If, however, &I > E,, the sign of he(w) reverses and is positive. A useful approximation for the evaluation of the expression (2) for the imaginary part of the dielectric function is the method of critical points171 whereby the dominant contributions are attributed to specific points in the Brillouin zone, ki, where the joint density of states function has singularities. These occur where: grad, (E&J - E,&)) = 0.

Ado) = -c,‘o’(o)ll -f(EJ

+f(E,)}

where -E,)

e-(Ec(w)‘kT) for electrons

1_ f ” [E ( )] = P e--(EcbW) for holes. ” OJ (N”) The normalization factors (N,) and (NJ are determined by the conditions.

I I

pc(Ec)jc(Ec) dE, = n = total number of electrons

in conduction bands p,(E,)[l - jJEJ1 dE. = P = total number of holes in valence bands

where pDand pc are the density of states functions. The initial temperature of the carrier distribution functions is determined by the photon energy of the excitation pulse tie, -E, = (E,) t (EC) = I

For the particular case of an A4,,(minimum) singularity such as occurs at a direct band edge, an explicit factorization of the imaginary part of the dielectric function is possible:

E, = -(m*Im,)(frw

EC = E,+(m*/m,)(fio-

(4)

I

+I I

Ep,(E) e-(E’kT) dE P,(E) e

-(e/tT) dE

EpJE) e-‘H“T’dE p,(E) ewcElk7)dE

which for tw_ = 4 eV gives kTi = 0.62 eV. We have made a semi-emperical calculation of the reflectivity of GaAs as a function of the nonequilibrium

149

Picosecondspectroscopyof semiconductors

carrier temperature using the approach outlined in the previous sections. The experimental data of Philip and Ehrenreich[lO] was used for the equilibrium r,“‘(~) spectrum. The density of states functions and the critical point energies were obtained from the band structure calculations of Chelikowsky and Cohen[8]. The perturbed AtI(o) spectrum has been calculated by a KramersKranig integration of A&). Both Ar, and he* are then used to calculate the incremental change in reflectivity. The result for GaAs is shown in Fii. 2, where the change in reflectivity due to optical pumping is plotted as a function of carrier temperature. The excitation and probing wavelengths match the experimental results depicted in Fig. 1. Also the sum of the mean carrier energies (E,,) t (E,) is also plotted vs temperature. The dashed line represents the free electron gas case: (E,) t (EC)= 3kT. As seen in the figure, the reflectivity reverses sign at a carrier temperature of approximately 0.17eV. The sum of the mean electron and hole energies of this point is approximately 1.0eV. The initial electron and hole energies in the experiment were equal to o, -0, = 2.56 eV. Our analysis of the experiment suggests that the hot carrier distribution lost approximately 1.5eV of energy in the first 2 ps following excitation. This rate of energy loss is consistent with estimates of optic and acoustic phonon emission rates [2]. Our use of the Kramers-Kriinig integral to derive BE, from AQ*omits the intraband contribution to the real part since the range of integration spans the optical spectrum

.Oi

and omits the low frequency collision-induced absorption spectrum of the free carrier plasma. When the carriers are in thermal equilibrium with the lattice, the free electron contribution .can be accounted for by the straightforward inclusion of the Drude term in eqn (1). This reduces the reflectivity at low carrier temperatures by a factor of two. This result is then in good agreement with the approximate expression (3) for the two-band model which accounts for both intraband and interband contributions. At higher temperatures, the Drude term is expected to be small due to the cancellation of positive and negative contributions to the effective mass (note: the uniform integration of the effective mass tensor over the entire Brillouin zone is zero). Our conclusions about the energy relaxation rates are in sharp contrast to the results of Smirl, Matter, Elci, and Scully[l11 who used picosecond optical pulses to measure hot carrier relaxation rates in germanium by absorption saturation. They deduced energy relaxation times of approximately 25 ps at room temperature and 1OOps at 105°K. Although the experimental conditions were quite different, one would not expect to observe such long relaxation times even in nonpolar semiconductors[2]. The interpretation of the results of Smirl et al. is complicated by the possible influence of free carrier absorption and enhanced indirect transitions by Coulomb scatteringfl21 at high excitation which can cause an apparent relaxation effect by introducing a minimum in the absorption vs density relationship.

.I CARRIER

I TEMPERATURE

IO

(eV I

Fig. 2. Calculated reflectivity modulation due to non-equitibrium electron and hole distributions of temperature T. Excitation and probing wavelengths are same as in Fig. 1. Carrier density: n = p = lOI cm-‘. The sum of the mean electron and hole energies are also plotted vs carrier temperature. The dashed line shows the classical free electron means energy of 3/2kT per carrier.

D. H. AUSTON et al.

150

Although our discussion has focussed on the interpretation of a single experiment, other experimental evidence tends to co&m our basic model. For example, in CdSe, a comparable sign reversal of AR(t) was observed when excited at 4eV and probed at 2eV, but no sign reversal was observed when the excitation and probing photons energies were both 2 eV. The details of these experiments and their analysis will appear as a subsequent publication. Acknowledgemenrs-The

authors are grateful to J. R. Chelikowsky for supplying additional density of states data, and to D. Aspnes for a critical reading of the manuscript.

REFERENCB 1. For recent reviews see the volume: Ullrashorf Optical Pulses (Edited by S. Shapiro). Springer-Vedag, New York (1977). 2. For a review see K. Seeger, EIccrronic Materials (Edited by

N. B. Hannayand U. Colombo), Chap. 5. Plenum Press, New York (1973). 3. See, for example, the review papers in this issue by J. Shah and R. G. Ulbrich. 4. E. P. Ippen and C. V. Shank, Appl. Phys. Lett 27,488 (1975). 5. H. Ehrenreich, in Optical Properties of Solids (Edited by J. Taut). Academic Press,New York(1966). 6. J. Callaway, Quantum Theory of the Solid Stare, Part A, Chapt. 4. AcademicPress, New York (1974);L. M. Narducci et ai., Phys. Rev. B 14,2508 (1976). 7. See for examole F. Bassini and G. Pastori-Paravicini. Electronic Stares-and Oprical Transitions in Solids. Pergamon, New York (1975). 8. J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14, 556 (1976). 9. R. Stratton, Proc. Roy. Sot. Lond. A246.406 (1958). 10. H. R. Philip and H. Ehrenreich, Phys. Rev. 129, 1550(1963). 11. A. L. Smirl, J. C. Matter, J. C. Matter, A. Elci and M. 0. Scully, Optics Comm. 16, 118(1976). 12. S. McAfee and D. H. ,Auston, in preparation.