Solid State Communications, Vol. 104, No. 2, pp. 107-I 12, 1997 @ 1997 Elsevier Science Ltd

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NON-MARKOVIAN MEMORY EFFECTS IN GaN LASERS

Steve Hughes

Department of Physics, University of Tokyo, 7-3-l Hongo, Bunkyo-Ku, Tokyo 113, Japan

(Received 31 March 1997; accepted 7 May 1997 by R. T Phillips)

Non-Markovian carrier-carrier memory effects calculated for the wide- bandgap GaN optical amplifier are shown to be important for time scales of up to 200fs and 5Ofs for the holes and electrons, respectively. Ultrafast carrier-carrier scattering processes, which occur within a rel- atively small energy range, manifest in correlation times which are ap- proximately double those calculated for GaAs. Implications for non- linear saturation and optical gain studies of GaN lasers are discussed. @ 1997 Elsevier Science Ltd

Keywords: A. semiconductors, B. optical properties, B. carrier-carrier interaction.

The physics of the group-ZZZ Nitrides like AlN and GaN is currently coming out of its infancy, and has been receiving a great deal of attention lately, primar- ily because of their application for short-wavelength laser diodes. Subsequently, and as a consequence of the growth technology of GaN and related devices, high-efficiency light-emitting diodes have been devel- oped [ 1,2].

An important goal in optimizing GaN diode lasers involves understanding the microscopic interactions that are responsible for gain and stimulated emission. Recently, theoretical studies of GaN have focused on optical gain calculations [3-51, band structure inves- tigations [6], and carrier-carrier scattering within the Markov approximation [7]. In the latter work it was found that GaN laser devices yield exceptionally large carrier-carrier scattering rates (significantly larger than those, for example, of GaAs) in part due to the large excitonic binding energies and in part due to the effective mass properties.

On very short times, relatively speaking, the semi- conductor many-body system has a memory of its ear- lier states - which can manifest in a substantial mod- ification of the dynamical dephasing processes. Ex- perimental signatures of memory effects in semicon- ductors are well known, for example, in reproducing the Urbach tail absorption and in modelling ultra- fast non-Markovian relaxation processes [8]. Further, in the field of semiconductor lasers, the role of non-

Markovian relaxation has been employed to explain the apparent inconsistency of modelling gain spectra with a Lorentzian profile for the interband transition lineshape (since Lorentzian lineshapes yield a certain amount of unphysical absorption below the renor- malized band gap - which is not obtained in exper- iments). Consequently, for modelling electronic pro- cesses within the ultrafast timescale regime, one re- quires a kinetic theory beyond the usual Boltzmann kinetics with energy conservation for each of its col- lisions. In this sense, Yamanishi et al. [9] successfully demonstrated that non-Markovian processes modify the lineshape of semiconductor lasers in such a man- ner as to calculate no considerable absorption below the renormalized band gap.

Recently, a phenomenological memory model has also been proposed [lo] (‘two-pole approximation’) for calculating non-Markovian lineshapes and modelling pulse propagation. The two-pole scheme demonstrates that the phenomenolical inclusion of memory effects in the description of dephasing processes, reduces the artifacts in the local linear gain spectra brought about by the dephasing-rate approximation. Additionaliy, in the nonlinear regime, the inclusion of memory effects were seen to significantly affect the dynamical pulse reshaping processes; in particular, the threshold for the saturation of the gain becomes significantly reduced - even for excitation by 15Ofs optical pulses.

On the other hand, as shown in Ref. [l 11, if one

108 NON-MARKOVIAN MEMORY EFFECTS IN GaN LASERS Vol. 104, No. 2

includes the proper on-diagonal and off-diagonal (Vertex corrections) scattering for the semiconductor polarisation equations, then the lineshape obtained at typical gain plasma densities deviates considerably from Lorentzian, and no absorption below the renor- malized band gap is calculated because of the reduced influence of the higher wave number (absorbing) states - even within the Markov approximation. These modified dephasing processes are also important in explaining a number of experimental measurements for the semiconductor absorber [12, 131. This how- ever does not imply that non-Markovian relaxation processes are not important for estimating the linear gain spectra (or nonlinear processes) of semiconduc- tor lasers. It depends very much on the experimental conditions, the relative timescale of the probing pulse, and on the optical amplifier under investigation.

In this work, the non-Markovian carrier-carrier quantum kinetics for GaN are shown to be impor- tant for timescales of up to 200fs and 50fs for the holes and electrons, respectively. We have specialized our analysis to the carrier-carrier scattering, since for high densities, the dephasing and relaxation processes in laser-excited semiconductors are dominated by carrier-carrier scattering. The calculated correlation times for GaN are shown to be significantly longer than those for GaAs and as a consequence - memory effects in GaN laser devices are expected to be much more important. As in Ref. [7], we treat the interaction between the conduction band and heavy-hole valence band only. Material and band structure parameters are taken from the ab initio calculations presented in Ref. [3], which yield a mass ratio, m/,h/me, of 9:l; an excitonic Bohr radius, a0 of 32.8 A; and an excitonic binding energy of 22meV

Memory effects can be accounted for self-consistently by employing the appropriate many-body quantum- kinetic equations. However, as a first attempt, one may estimate the correlation (memory) time - via. the uncertainty principle - as being the inverse en- ergy transfer in the carrier-carrier scattering [14]. Assuming that the average energy transfer is the Fermi energy, then for GaAs, the correlation time for electrons is approximately 4fs, while for GaN, the the correlation time is approximately 12fs; these val- ues were estimated assuming an incoherent density N = 4 x 10i8cm-’ at room temperature and material parameters as discussed below. The detailed calcula- tions to follow confirm this trend.

For our theoretical approach we assume a two- band bulk semiconductor where each electron-hole (e-h) state within a certain bandstructure with a wave number k, contributes to the total carrier density:

N = 2V-’ & fi, where fi is the electron or hole carrier distribution and the factor of two accounts for the spin summation. The choice of treating the heavy- hole interaction only is a reasonable assumption for determining the scattering rates since on one hand, the mass of the light hole is so small that its influence can be neglected; on the other hand the properties of the CH (crystal field split-off hole) band are very similar to those of the heavy-hole valence band which enter into the quantum kinetic calculations.

The carrier-carrier scattering is calculated by a di- rect numerical integration of the electron-hole quan- tum kinetic equation [ 151:

afk" at scatt

= r$,ia,ck,f)[l -f$l -r,&tkf)f;, (1)

where I& and f,,,,, (a = e, h) are the expressions for in and out scattering, for example, in excitonic units [16] :

1 2@(q)f;+q(T)[1 - f$+@l

f; (T) cos (W’[E,(k) + cb(k’ + q) - e,(k + q)-

e,(k’)lO - T)} dT (2)

and T& is obtained by replacing f by 1 - f. The Coulomb potential, W(q), is treated in a quasi-static screening model (single plasmon-pole) [17] and E, is the excitonic binding energy. The time integral incor- porates the effects of non-Markovian relaxation on the motion of particles between collisions.

By assuming that the carrier distributions are suffi- ciently close to the quasiequilibrium, one can write

where Tt (t ) is the sum of the in and out scattering con- tributions which can be obtained from Eq. (2), and

fk O” is the Fermi-Dirac distribution of the electrons (a = e) or holes (a = h). Within the Markov approxi- mation, the cosine term in Eq. (2) can be replaced by a delta function which assumes a perfect energy con- servation on a transition between two quantum states in the conduction and valence bands.

For a comparison to the original semiconductor laser work on non-Markovian lineshapes presented in Ref. [9], we reformulate the calculation of the scatter- ing rates as follows:

I1 P(Z) = P(o) - 11 -p - T)r;(T)P(t - T)dT

a 0

Vol. 104, No. 2 NON-MARKOVIAN MEMORY EFFECTS IN GaN LASERS 109

= P(0) exp -iU.‘(r) + Lb(t)) , i I

(4)

where P(t) is the decaying polarisation (real contri- bution) from the electron and hole scattering. This approach, of course, is in a very oversimplified form, since on one hand it treats the lineshape at one k point only (delta function pulse), and on the other hand, it ignores the in-scattering contributions (Vertex correc- tions) [I 11. In spite of the above limitations, however, the above analysis is a useful formulism firstly to com- pare with previous works on the damping function [2L(t) = Le(t) +Lh(t)] and secondly for obtaining the memory effects for the intraband carrier-carrier scat- tering. Since we are interested in the carrier-carrier rates, we calculate the real part of the polarisation only (cosine contribution).

We characterize inverted GaN by an electron-hole plasma at room temperature and having a total inco- herent carrier density of N = 4 x 1018cm-3, where a light mode with a certain propagation constant k = k(w) experiences gain below (Es < hw < p) the transparency point. We have chosen a k point near the electron-hole transparency regime. The calculated damping is shown in Fig. l(a) for GaN, where the dashed line depicts the contribution from the holes [L”(t)], and the solid line depicts the contribution from the electrons [Le(t)]. As can be recognized, the damp- ing is proportional to t2 for small t (<2Ofs), and pro- portional to t for large t (> 1 OOfs) - in qualitative agree- ment with the results shown in Ref. [9]. However, the deviation from the Lorentzian result (Markov approx- imation) continues up to the respective times of 200fs and 5Ofs for the electrons and holes. At a picosec- ond, the intraband scattering rates are approximately 10Op~-~ and 25~s~’ for the holes and electrons, re- spectively; these very fast carrier-carrier times are con- sistent with the GaN quantum well values presented elsewhere [7].

Next, we turn our attention to GaAs, where we assume the following parameters: electron/hole mass ratio, m,/mh = 713; excitonic binding energy, Ex = 4.2meV; and excitonic Bohr radius, as = 135A. Fig. l(b) shows the calculated damping for the incoher- ent carrier density of 4 x 10%rn-3, where the dashed line depicts the contribution from the holes, and the solid line depicts the contribution from the electrons. In this case, deviation from the linear decay continues up to the respective times of 1OOfs and 20fs for the electrons and holes. At a picosecond, the intraband

L

I

1o-5 ‘. 10’ 2 5 loo 2 5 10’ 2 5 lo2 2 5 10'

fi lOA V

(a)

Time (fs)

Fig. 1. (a) Calculated damping function due to intra- band carrier-carrier scattering between two quantum states near the electron-hole transparency point for GaN. The incoherent density is 4 x 10%m-3 at room temperature; the wave numbers are in units of ai’. (b) As in Fig. l(a) for GaAs.

scattering times are approximately 40~s~’ and lops-’ for the holes and electrons, respectively; these values are consistent with the Markovian calculations shown in Ref. [18].

One can attempt to fit the non-Markovian damping [L’(t)] by the following simple stochastic correlation function [ 191:

I

F”(t) = r{ 1 -exp(-$) dT J [ 0 1 t+T,"eX&-)-T: , 1 (5)

where l$ is the Markovian (time-independent) scatter- ing rate, I”[ 1 - exp(-r /T;)] is the assumed response of the intraband scattering rates (a = e, h), and T," is the correlation (memory) time for the intraband relax- ation rates. In Figs. 2(a) and 2(b) we attempt to fit the

110 NON-MARKOVIAN MEMORY EFFECTS IN GaN LASERS Vol. 104, No. 2

1 O3

Time (fs)

Fig. 2. (a) A fit to the damping response shown in Fig l(a) by employing the formula: Fa(t) = r[[t + -r;exp(-t/r;) - T,"] (see text). (b) As in Fig. 2(a) for GaAs.

previous calculations by using the above formula for GaN and GaAs. As can be seen, Eq. (5) reproduces the non-Markovian dynamics very well indeed. In the case of GaN [Fig. 2(a)], correlation times of 4fs and 15fs are extracted for the electrons and holes, respec- tively. While for GaAs, respective correlation times of 2.5fs and 8fs are obtained for the electrons and holes. Consequently, one we can conclude that memory ef- fects in GaN lasers persist up to much longer times (approximately double) than those for GaAs laser de- vices, since the typical energy exchange is much less. On the other hand, the GaN optical amplifier has much faster non-Markovian scattering times in com- parison to those of GaAs, which is a manifestation of the comparitively very large excitonic binding ener- gies; this increase in the carrier-carrier rates is antic- ipated for GaN lasers, in a somewhat similar fashion to an increase in the band-gap-renormalization [7]. In- deed, recently, carrier-carrier collision times of 15fs were estimated in order to explain Raman scattering

,

.

.’

.C,_._ ._._*_._. -_.__.

--------___--

Time (fs)

Fig. 3. Non-Markovian intraband carrier-carrier rates obtained for the GaN and GaAs optical amplifier.

spectra at the non-equilibrium plasma density of N = 3 x 10i8cmW3 for bulk GaN [20].

Finally, Fig. 3 depicts the non-Markovian intraband carrier-carrier rates - corresponding to the above de- tailed calculations - for the GaN and GaAs optical amplifier. For GaN, the electron scattering rates (solid line) and hole scattering rates (short-dashed line) take the following form:

rAaN(t) = ri 1 - exp(--&) [ I

[

t r&+&) = r; 1 - exp(-15fs) . 1

And for GaAs (the long-dashed and chain

rAaAs(t) = r; [

1 - exp(- &)I

r&As(t) = rj i - exp(-&) . [ I

(6)

(7)

line):

(8)

(9)

These finite correlation times are caused by the absence of a strict energy conservation in the non-Markovian time regime, whereby memory effects extend over the

Vol. 104, No. 2 NON-MARKOVIAN MEMORY EFFECTS IN GaN LASERS 111

time interval TV and significantly affect the intraband scattering times. To m-iterate however, the averages of these carrier-carrier scattering times are not equiva- lent to the dephasing times of the optical polarisation, since it has been shown elsewhere that by calculating the proper quantum kinetic equations, the dephasing times resulting from carrier-carrier scattering alone are larger than the (average) intraband carrier-carrier scattering times themselves (pure-dephasing approxi- mation); this is particulary true for the high wave num- bers where they can differ substantially [l 11.

The behaviour of semiconductor optical amplifiers and related devices, is, of course, determined by a range of microscopic processes. However the incoher- ent manifestations of carrier-carrier scattering are par- ticularly important for describing the experimentally observed nonlinear optical effects including, for exam- ple, spectral hole burning, gain saturation, and tem- poral pulse reshaping - such as those obtained from excitation-probe [21] and four-wave-mixing [22] mea- surements. By employing a phenomenological model for the non-Markovian relaxation processes, Indik et al. [lo] showed that finite memory times, even those as short as 3fs, will significantly affect the computed linear gain spectra and pulse reshaping processes dur- ing the propagation of a pulse though a semiconduc- tor amplifier. In addition to yielding no calculated ab- sorption below the renormalized band gap - in agree- ment with experimental measurements - a strong en- hancement of the gain peak occurs. Moreover, longer memory times of, say 2Ofs, cause an even greater gain enhancement and a shift in the predicted transparency point of the semiconductor amplifier. In the nonlin- ear regime, it was further shown that memory effects inhibit gain saturation and decrease the carrier heat- ing - because carrier heating results mainly from ab- sorption into noninverted momentum states; in this respect, longer correlation times will reduce the tran- sient heating. Gain enhancement, and a reduction of the transient heating, can be interpreted as a result of the violation of energy conservation on time scales shorter than the correlation time - and will become more important for optical amplifiers that yield longer correlation times such at GaN.

In conclusion, ultrafast correlation effects for GaN and GaAs lasers have been obtained, quantitatively, by calculating the intraband carrier-carrier rates from first principle - at typical lasing plasma densi- ties - without invoking the Markov approximation. A simple stochastic memory function was found to fit the detailed non-Markovian dynamics very well, and subsequently, correlation times were obtained for the electron and hole scattering. These lifetimes

are important for predicting the dynamical properties of laser diodes from the microscopic carrier-carrier scattering parameters and, for aiding the more sim- ple phenomonological semiconductor laser theories to incorporate the effects of memory. The effects of non-Markovian relaxation on the linear and nonlin- ear optical properties of semiconductor lasers were discussed. The carrier-carrier correlation times for GaN were found to be 4fs and 15fs for the electrons and holes, respectively. These correlation times were found to be significantly longer (approximately dou- ble) than those calculated for GaAs at an equivalent plasma density. It has been discussed that these intra- band scattering times do not necessarily imply that the optical polarisation decays on such timescales, and a proper quantitative analysis of the optical de- phasing, including the role of the Vertex corrections, will hopefully be performed in the near future.

Acknowledgements-The author thanks the the Japanese Society for the Promotion of Science [JSPS] for a Post-Doctoral Fellowship.

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