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The influence of electron-hole-scattering on the gain spectra of highly excited semiconductors
Solid State Communications, Vol. 100, No. 8, pp. 555-559, 1996 Copyright @I996 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/96 $12.00 + .OO
Pergamon
PII:SOO38-1098(96)00494-Z
THE INFLUENCE
OF ELECTRON-HOLE-SCATTERING ON THE GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
S. Hughes, a A. Knorr, a S.W. Koch, a R. Binder, b R. Indik c and J.V. Moloney c a Fachbereich Physik und Zentrum fur Materialwissenschaften, Phillips-Universitat, Renthof 5, D-35032 Marburg, Germany b Optical Sciences Center, University of Arizona, Tucson, AZ 85721, U.S.A. c Department of Mathematics, University of Arizona, Tucson, AZ 85721, U.S.A. (Received 22 January 1996; accepted 14 August 1996 by J Kuhn)
A microscopic treatment of the influence of electron-hole-scattering on the optical dephasing and the lineshape in semiconductor gain media is presented. The calculations incorporate non-diagonal- and diagonalscattering contributions to the optical polarisation. The strong compensation between both contributions leads to gain spectra, which are significantly modified in comparison to those obtained using a pure dephasing approximation. Copyright 01996 Published by Elsevier Science Ltd Keywords: A. semiconductors, B. carrier-carrier optical properties.
The investigation of optical gain in highly excited semiconductor media is of general interest because it is connected to basic Coulomb many-body processes and to the application of semiconductors as devices in optoelectronics, such as amplifiers and lasers. An important goal for the theoretical description is the prediction of threshold density and of the peak gain as a function of frequency. Because of the high carrier densities, carrier-carrier interaction is important for the description and determination of the optical gain; in particular, the dephasing of the polarisation and the lineshape [ 1] plays a crucial role in calculating the gain spectra. In this communication, a self-consistent many body theory on the level of quantum kinetic equations in the Markovian limit for the carrier-carrier scattering processes [2-4] is applied to calculate gain spectra of semiconductor amplifiers. The rigorous inclusion of both, non-diagonal- and diagonal-scattering contributions result in a strong modification of the previously used lineshape results from a pure dephasing approximation (e.g. Lorentzian lineshape). In a two-band semiconductor, each electronhole (e-h) state within a certain bandstructure c,//,(k) with wave number k, contributes to the total optical polarisation: P = 2V-’ Ckdc,(k)P(k),
interaction. B.
where d,,(k)
is the Kane-dipole
matrix
element
d,,(O)/ (1 + (E,(k) + ejl(k))/Egdp) with Egap being the band gap; V is the active volume and the factor of two accounts for the spin summation. Thus, the polarisation transitions, P(k), of each single particle state have to be computed. They are determined by the relevant semiconductor Bloch-equations for the polarisation functions [5]:
Wk)
-
at
= -iAkP(k) - i!%&,(k)
Wk)
+ w,(k) - 1) + at
scatt
(1)
where & is the renormalized energy dispersion for a parabolic two-band semiconductor, Rk is the generalized Rabi frequency, and n,ll,(k) are the occupation numbers for electrons and holes respectively [5,6]. The incoherent contributions, which are corrections to the screened Hartree-Fock equations, take the general form [3]:
Wk)
at
I
= -- Z’(k)
scatt
at
I
d +
aP(k) -iii-
nd’
(2)
where P(k) Id and P(k) )nd contain non-diagonal- and diagonal-contributions (nd,d), of the polarisation
555
GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
556
functions. Here, the diagonal terms can be written as a pure dephasing term. In the following it will be shown that the nun-diagonai terms partially compensate the influence of the diagonal-contributions. Only the consistent treatment yields the correct results. Similar effects are known from the Hartree-Fock equations Eq.(l). It should be pointed out, that after the introduction of the Coulomb interactions, the polarisation functions, P(k), act only as a set of expansion coefficients for the optical polarisation. Thus, even if there are non-diagonal- and diagonal-scattering terms, the optical polarisation which enters into Maxwell’s equation as observed in an experiment, decays through Coulomb-scattering. The Coulomb-scattering terms depend on the e-h density distributions and the polarisation functions for all wave numbers [2-4]: aP(k) a? aP(k) at
nd
= cr,d,(k 4
= ri(k,
w,)P(k)
d 4, n,/JP(k
+ q),
(3) (4)
Vol. 100. No. 8
of employing the generalized Markov approximation to the diagonal-scattering terms and not because the non-diagonal-scattering term is neglected (see, for example, Refs. [7,8]); i.e, the changes of the optical polarisation at a given time are only related to the distribution functions at the same time. In this context a phenomenological non-Markovian gain model has also been proposed [9] (‘two-pole approximation’) for calculating semiconductor gain spectra and modelling pulse propagation. The two-pole scheme demonstrates that the phenomenological inclusion of memory effects in the description of dephasing processes, reduces the artifacts in the linear gain spectra brought about by the ~2 dephasing-rate approximation. In this communication, we show that physically reasonable gain spectra are obtained if one takes diagonal- and nondiagonal-scattering into the dephasing of the polarisation, even within the Markov approximation, thus leading to a solution of the lineshape problem on a microscopic basis. The carrier-carrier scattering needed for our microscopic investigations [6] can be calculated from the electron-hole Boltzmann equation [2] :
where r are the scattering rates defined below. Most studies of semiconductor gain spectra assume h,(k) that the dephasing rate of the polarisation in a cer= I-iU,(k, n,~,)[l - n&)1 dt scatt tain wave number state is given by the loss of polarisation (diagonal-contributions) from this state only (pure dephasing model), thus neglecting P(k)l,,d [l]. where ri and I-&, (a = e, h) are the scattering contriThe reason for this procedure is that the non-diagonalbutions in the equations for electron-hole occupations scattering terms are given by a sum over all single for in- and out- scattering, for example polarisations P(k + q). If one assumes that each of them oscillates with its single particle energy and a lIi”,(k, fled = F 1 2W/(q)12 sufficient number of them are coupled in Eq.(4), the k’ .q non-diagonal-scattering should average out due to the h=c,h interference of different phases. On the other hand, .n,,(k+q)[l -nh(k’)]nh(k’ -4) it has been shown recently for certain situations in x6(c,(k) - c,(k + q) + Eh(k’) - Eh(k’ - q)), (6) semiconductor absorbers, that it is important not only to include the diagonal- but also the non-diagonalscattering contributions for each k-state [3,4] because and r&, is obtained by replacing nu/h by 1-nulh. V(q) is the screened Coulomb potential which is treated here of a strong linewidth reduction in absorbers. in a quasi-static approximation [5]. So far, Refs. [3,4] have studied semiconductor abSimilarly, the polarisation function takes the form sorbers. From semiconductor amplifier modelling, it is known that calculations including only the diagonalW(k) = -ri(k, n,/,)f’(k) scattering yield a certain amount of unphysical abat scatt sorption below the gap [l]. This behaviour can be at+ 2 rfddk, q,w,)P(k + q), (7) tributed to the strong influence of high-lying, absorb4 ing k-states, which contribute due to their long range Lorentzian line shape with a width T;~ = F” below the where the total dephasing rate I-1 is one half times the gap. These contributions are significant because they sum over all distribution scattering rates. The addiare weighted by the high density of states. This fact is tional non-diagonal scattering rate, well known as the lineshape problem in semiconductor gain media. The possible failure of the relaxation-rate model P(k) I,,, = T; ‘P(k) in calculating semiconductor gain spectra is assumed however, a consequence
GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
Vol. 100, No. 8 1
c
551
sech-shaped in time, with a pulse duration of 15 fs (FWHM irradiance). Such a short pulse duration al--. N 3.0x10'"cm" lows an efficient calculation of gain spectra due to the large bandwidth. From the carrier density given by N = 27/-l Ckn,(k) at fixed room temperature, the equilibrium electron-hole occupations can be calculated. In Fig. I(a), we plot the resulting gain spectra versus the detuning of the frequency o with respect to the unrenormalized band edge for a range of carrier densities, where only the rj dephasing term has been accounted for (pure dephasing model). Starting from N = 1 x 1018cm-” to N = 3 x 1018cm-j, the cross-over from absorption to gain is shown. As is well known, the density dependent part of the bandgap renormalization leads to a shift of the gain onset to frequencies well below the zero density band edge. In Fig. l(b), the inclusion of the Ij(k)l,d contribution predicts markedly higher gain curves and different transparency points in comparison to Fig.l(a). Furthermore, for a carrier density of N = 1 x 1018cm-“, gain, and not absorption, is predicted. The reason for this behaviour is that the non-diagonal-scattering improves -0.5 \, the gain by decreasing the influence of the higher kstates, which are not inverted and contribute absorp-10 0 10 20 30 -30 -20 tion to the spectra. Their absorptive contribution is ENERGY, E-EGAP[E,] also strong below the band edge due to their long abFig. 1. (a) Gain spectra calculated by including only sorptive line shapes if only the diagonal-scattering is i)k)d dephasing processes (see text) for the respective taken into account. carrier densities: 1.0 x 1018cm-3 (solid line), 1.5 x To illustrate this, we plot in Fig. 2 the scattering 10’8cm--1 (dotted line) , and 3.0 x 10’*cm-3 (dashed contributions, Eq.(7), to the real (Fig. 2(a)) and imagline). E, is the GaAs exciton binding energy. For de- inary part (Fig.2(b)) of the polarisation. The calcucreasing densities the transition between gain and ablations were performed at the centre of the 15 fs insorption is observed. (b) Same as in Fig. l(a), but with put pulse (t=O), thus giving a snap-shot in time of the the inclusion of the & (ndnon-diagonal-scattering processes. The non-diagonal-scattering results in gain en- scattering contributions. The plotted scattering contrihancement. At 1.0 x 1Oi8cm-3 (solid line) gain instead butions act as source terms, thus increasing (positive contribution) or decreasing (negative contribution) the of absorption is observed. polarisation at a particular k-value. It must be noted . [ [l - 4(k’ - q)ln,(k)n/#‘) that the density of states (- k2) determines the weight of the different one particle states in terms of cal+ [l - n&‘)l[l - n,(k)lnh(k’ -q)] culating the spectra. Consequently (Fig. 2). for high x6(~,(k) - ~,(k + q) + c/Ak’) - Mk’ - q)), (8) wave numbers k above the chemical potential, where the density of states is large, the dominant contribudescribes the rate of polarisation transfer between the tion comes from the real part of both non-diagonalstates k and q due to carrier-carrier scattering. We and diagonal-scattering (Fig. 2(a)). However, the total show that this complex polarisation transfer between scattering contribution is significantly reduced in comdifferent k-states, reduces the optical linewidths in such parison to the diagonal-scattering (compare dashed a way to allow calculations for frequencies below the and dotted line). Therefore the negative source term band gap. For all following numerical computations in the polarisation equation is diminished, indicating of the frequanecy dependent gain spectra, defined by: a reduced absorption into higher k-states, in comparison to the diagonal-scattering only. A partial canG(w) = -2kJmx(w), (9) cellation of non-diagonal- and diagonal-scattering for where x(w) is the calculated susceptibility and kL is high k-values is confirmed also for the imaginary part of the scattering contribution (Fig. 2(b)). This overall the propagation constant of the laser. For numerical convinience, we take input pulses &I(W) which are ,.-I
N=l.O x 10’nc’cm’
-
.
..I...
,.,=,.J s
xl,&,,”
(a)
GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
558
Vol. 100. No. 8
0.6 p
-1.0 . 0
2
4
6
8
10
I2
14
t 1.0
ati
0’
0.5
2 E
0.0
4 *
I
(b)
Diagonal Turn +....a..f.J~_&,p,,d h
-
Full Low Intensity,t=O
I
-20
-10 ENERGY,
........ ‘.,. ....
0 E-EGAP
10
&I
Fig. 3. Enlarged gain spectra showing the difference between the calculations with and without nondiagonal-scattering for a total carrier density of N = 1.5 x 10’8cm-3. The neglect of the non-diagonalscattering yields strong unphysical absorption below the band edge.
detail the difference between the calculations with and without non-diagonal-scattering. In this plot, it can be recognized that the unphysical absorption below the .... J _, .o k gap is obtained if the non-diagonal-scattering terms IO 4 6 R I2 14 0 2 are neglected. WAVE NUMBER [a,,“] In conclusion, we have demonstrated that for a consistent determination of the optical dephasing in inFig. 2. Real (a) and imaginary (b) part of the scattering verted semiconductors, non-diagonal- and diagonalcontributions -P(k) ld and P(k) lnd which enter in the scattering in the polarisation equation must be inpolarisation equation, for excitation with a 15 fs sechcluded in the analysis. The consistent treatment of the shaped pulse with a carrier frequency at E = Egap; scattering is shown to substantially modify the gain (N = 1.5 x lOi*cm-s). The plot is a temporal snapshot at time, I = 0 fs. The solid line represents the usual spectrum both in shape and magnitude. P(k) Id-scattering and the dotted line corresponds to Acknowledgements-SH thanks the Comission of the -P(k)I,d-scattering. The dashed line depicts the the European Communities for an HMC European sum of the two, characterizing the overall dephasing Fellowship. This work was supported by the “Quanof the optical polarisation. a,, is the Bohr radius for tum Coherence in Semiconductors Program” of the bulk GaAs (=135 A). Deutsche Forschungsgemeinschaft (Germany). CPUtime at the HLRZ Jiilich is acknowledged. reduction in the total scattering contributions is also obtained for lower wave numbers k, but the cancelREFERENCES lation is not quite so pronounced. However the corresponding change of the scattering contributions at 1. Chow, W.W., Koch, S.W. and Sargent III, M., low k-states does not strongly influence the calculated Semiconductor-Laser Physics, Springer, Berlin, spectra, because the density of states is smaller at low 1994. k-numbers. We conclude that the change in the scat2. Lindberg, M. and Koch, S.W., Phys. Rev. B, B 38, tering contributions for high k-values drastically in1988, 3342. fluences the calculated gain spectra. This interpretaRappen, T., Peter, U.-G., Wegener, M. and Schafer, 3. tion can be confirmed by numerically switching off the W., Phys. Rev. B, 49, 1994, 10 774. higher k-states. In the standard Lorentzian lineshape 4. Rossi, F., Haas, S. and Kuhn, T., Phys. Rev. Lett., model, this reduction of high k-absorption transitions 72, 1994, 152. would not be obtained. 5. for a textbook discussion see, Haug, H. and Koch, The consequences of this behaviour for the gain S.W., Quantum Theory of the Optical and Elecspectra, are demonstrated clearly in the case of the cartronic Properties of Semiconductors, World Scirier density N = 1.5 x 10i8cm-3. Fig. 3 shows in more -0.5
Vol. 100, No. 8
GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
entific, Singapore, 3rd edn, 1994, and references therein. 6. In the following numerical calculations all material parameters are chosen corresponding close to bulk GaAs: a mass ratio of me/Mb = 713, where me./, is the electron/hole mass; an exciton binding energy E, of 4.2 meV; and a dipole moment of d,,(O) = 3e A. 7. Yamanishi, M. and Lee, Y., IEEEJ Quantum Electron., QE-14, 1987, 367. 8. Ohtoshi, T. and Yamanishi, M., IEEE J Quantum Electron., QE-27, 199 1, 46. 9. lndik, R.A., Mlejnek, M., Moloney, J.V., Binder, R., Hughes, S. Knorr, A. and Koch, S.W. accepted for publication in Phys Rev. A.
559
10. Note, that polarisation scattering [3] which turns out to be important for the situation of nonlinear excitation such as four-wave-mixing (FWM) close to the band edge can be neglected in our case, because we treat the situation of linear optics where products of polarisations, such as the polarisation scattering are higher-order terms. In this paper we are interested only in the principle effects of the interplay of non-diagonal- and diagonal-scattering, therefore exchange contributions of the Coulomb potential to the lineshape are neglected for numerical convinience.
Pergamon
PII:SOO38-1098(96)00494-Z
THE INFLUENCE
OF ELECTRON-HOLE-SCATTERING ON THE GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
S. Hughes, a A. Knorr, a S.W. Koch, a R. Binder, b R. Indik c and J.V. Moloney c a Fachbereich Physik und Zentrum fur Materialwissenschaften, Phillips-Universitat, Renthof 5, D-35032 Marburg, Germany b Optical Sciences Center, University of Arizona, Tucson, AZ 85721, U.S.A. c Department of Mathematics, University of Arizona, Tucson, AZ 85721, U.S.A. (Received 22 January 1996; accepted 14 August 1996 by J Kuhn)
A microscopic treatment of the influence of electron-hole-scattering on the optical dephasing and the lineshape in semiconductor gain media is presented. The calculations incorporate non-diagonal- and diagonalscattering contributions to the optical polarisation. The strong compensation between both contributions leads to gain spectra, which are significantly modified in comparison to those obtained using a pure dephasing approximation. Copyright 01996 Published by Elsevier Science Ltd Keywords: A. semiconductors, B. carrier-carrier optical properties.
The investigation of optical gain in highly excited semiconductor media is of general interest because it is connected to basic Coulomb many-body processes and to the application of semiconductors as devices in optoelectronics, such as amplifiers and lasers. An important goal for the theoretical description is the prediction of threshold density and of the peak gain as a function of frequency. Because of the high carrier densities, carrier-carrier interaction is important for the description and determination of the optical gain; in particular, the dephasing of the polarisation and the lineshape [ 1] plays a crucial role in calculating the gain spectra. In this communication, a self-consistent many body theory on the level of quantum kinetic equations in the Markovian limit for the carrier-carrier scattering processes [2-4] is applied to calculate gain spectra of semiconductor amplifiers. The rigorous inclusion of both, non-diagonal- and diagonal-scattering contributions result in a strong modification of the previously used lineshape results from a pure dephasing approximation (e.g. Lorentzian lineshape). In a two-band semiconductor, each electronhole (e-h) state within a certain bandstructure c,//,(k) with wave number k, contributes to the total optical polarisation: P = 2V-’ Ckdc,(k)P(k),
interaction. B.
where d,,(k)
is the Kane-dipole
matrix
element
d,,(O)/ (1 + (E,(k) + ejl(k))/Egdp) with Egap being the band gap; V is the active volume and the factor of two accounts for the spin summation. Thus, the polarisation transitions, P(k), of each single particle state have to be computed. They are determined by the relevant semiconductor Bloch-equations for the polarisation functions [5]:
Wk)
-
at
= -iAkP(k) - i!%&,(k)
Wk)
+ w,(k) - 1) + at
scatt
(1)
where & is the renormalized energy dispersion for a parabolic two-band semiconductor, Rk is the generalized Rabi frequency, and n,ll,(k) are the occupation numbers for electrons and holes respectively [5,6]. The incoherent contributions, which are corrections to the screened Hartree-Fock equations, take the general form [3]:
Wk)
at
I
= -- Z’(k)
scatt
at
I
d +
aP(k) -iii-
nd’
(2)
where P(k) Id and P(k) )nd contain non-diagonal- and diagonal-contributions (nd,d), of the polarisation
555
GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
556
functions. Here, the diagonal terms can be written as a pure dephasing term. In the following it will be shown that the nun-diagonai terms partially compensate the influence of the diagonal-contributions. Only the consistent treatment yields the correct results. Similar effects are known from the Hartree-Fock equations Eq.(l). It should be pointed out, that after the introduction of the Coulomb interactions, the polarisation functions, P(k), act only as a set of expansion coefficients for the optical polarisation. Thus, even if there are non-diagonal- and diagonal-scattering terms, the optical polarisation which enters into Maxwell’s equation as observed in an experiment, decays through Coulomb-scattering. The Coulomb-scattering terms depend on the e-h density distributions and the polarisation functions for all wave numbers [2-4]: aP(k) a? aP(k) at
nd
= cr,d,(k 4
= ri(k,
w,)P(k)
d 4, n,/JP(k
+ q),
(3) (4)
Vol. 100. No. 8
of employing the generalized Markov approximation to the diagonal-scattering terms and not because the non-diagonal-scattering term is neglected (see, for example, Refs. [7,8]); i.e, the changes of the optical polarisation at a given time are only related to the distribution functions at the same time. In this context a phenomenological non-Markovian gain model has also been proposed [9] (‘two-pole approximation’) for calculating semiconductor gain spectra and modelling pulse propagation. The two-pole scheme demonstrates that the phenomenological inclusion of memory effects in the description of dephasing processes, reduces the artifacts in the linear gain spectra brought about by the ~2 dephasing-rate approximation. In this communication, we show that physically reasonable gain spectra are obtained if one takes diagonal- and nondiagonal-scattering into the dephasing of the polarisation, even within the Markov approximation, thus leading to a solution of the lineshape problem on a microscopic basis. The carrier-carrier scattering needed for our microscopic investigations [6] can be calculated from the electron-hole Boltzmann equation [2] :
where r are the scattering rates defined below. Most studies of semiconductor gain spectra assume h,(k) that the dephasing rate of the polarisation in a cer= I-iU,(k, n,~,)[l - n&)1 dt scatt tain wave number state is given by the loss of polarisation (diagonal-contributions) from this state only (pure dephasing model), thus neglecting P(k)l,,d [l]. where ri and I-&, (a = e, h) are the scattering contriThe reason for this procedure is that the non-diagonalbutions in the equations for electron-hole occupations scattering terms are given by a sum over all single for in- and out- scattering, for example polarisations P(k + q). If one assumes that each of them oscillates with its single particle energy and a lIi”,(k, fled = F 1 2W/(q)12 sufficient number of them are coupled in Eq.(4), the k’ .q non-diagonal-scattering should average out due to the h=c,h interference of different phases. On the other hand, .n,,(k+q)[l -nh(k’)]nh(k’ -4) it has been shown recently for certain situations in x6(c,(k) - c,(k + q) + Eh(k’) - Eh(k’ - q)), (6) semiconductor absorbers, that it is important not only to include the diagonal- but also the non-diagonalscattering contributions for each k-state [3,4] because and r&, is obtained by replacing nu/h by 1-nulh. V(q) is the screened Coulomb potential which is treated here of a strong linewidth reduction in absorbers. in a quasi-static approximation [5]. So far, Refs. [3,4] have studied semiconductor abSimilarly, the polarisation function takes the form sorbers. From semiconductor amplifier modelling, it is known that calculations including only the diagonalW(k) = -ri(k, n,/,)f’(k) scattering yield a certain amount of unphysical abat scatt sorption below the gap [l]. This behaviour can be at+ 2 rfddk, q,w,)P(k + q), (7) tributed to the strong influence of high-lying, absorb4 ing k-states, which contribute due to their long range Lorentzian line shape with a width T;~ = F” below the where the total dephasing rate I-1 is one half times the gap. These contributions are significant because they sum over all distribution scattering rates. The addiare weighted by the high density of states. This fact is tional non-diagonal scattering rate, well known as the lineshape problem in semiconductor gain media. The possible failure of the relaxation-rate model P(k) I,,, = T; ‘P(k) in calculating semiconductor gain spectra is assumed however, a consequence
GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
Vol. 100, No. 8 1
c
551
sech-shaped in time, with a pulse duration of 15 fs (FWHM irradiance). Such a short pulse duration al--. N 3.0x10'"cm" lows an efficient calculation of gain spectra due to the large bandwidth. From the carrier density given by N = 27/-l Ckn,(k) at fixed room temperature, the equilibrium electron-hole occupations can be calculated. In Fig. I(a), we plot the resulting gain spectra versus the detuning of the frequency o with respect to the unrenormalized band edge for a range of carrier densities, where only the rj dephasing term has been accounted for (pure dephasing model). Starting from N = 1 x 1018cm-” to N = 3 x 1018cm-j, the cross-over from absorption to gain is shown. As is well known, the density dependent part of the bandgap renormalization leads to a shift of the gain onset to frequencies well below the zero density band edge. In Fig. l(b), the inclusion of the Ij(k)l,d contribution predicts markedly higher gain curves and different transparency points in comparison to Fig.l(a). Furthermore, for a carrier density of N = 1 x 1018cm-“, gain, and not absorption, is predicted. The reason for this behaviour is that the non-diagonal-scattering improves -0.5 \, the gain by decreasing the influence of the higher kstates, which are not inverted and contribute absorp-10 0 10 20 30 -30 -20 tion to the spectra. Their absorptive contribution is ENERGY, E-EGAP[E,] also strong below the band edge due to their long abFig. 1. (a) Gain spectra calculated by including only sorptive line shapes if only the diagonal-scattering is i)k)d dephasing processes (see text) for the respective taken into account. carrier densities: 1.0 x 1018cm-3 (solid line), 1.5 x To illustrate this, we plot in Fig. 2 the scattering 10’8cm--1 (dotted line) , and 3.0 x 10’*cm-3 (dashed contributions, Eq.(7), to the real (Fig. 2(a)) and imagline). E, is the GaAs exciton binding energy. For de- inary part (Fig.2(b)) of the polarisation. The calcucreasing densities the transition between gain and ablations were performed at the centre of the 15 fs insorption is observed. (b) Same as in Fig. l(a), but with put pulse (t=O), thus giving a snap-shot in time of the the inclusion of the & (ndnon-diagonal-scattering processes. The non-diagonal-scattering results in gain en- scattering contributions. The plotted scattering contrihancement. At 1.0 x 1Oi8cm-3 (solid line) gain instead butions act as source terms, thus increasing (positive contribution) or decreasing (negative contribution) the of absorption is observed. polarisation at a particular k-value. It must be noted . [ [l - 4(k’ - q)ln,(k)n/#‘) that the density of states (- k2) determines the weight of the different one particle states in terms of cal+ [l - n&‘)l[l - n,(k)lnh(k’ -q)] culating the spectra. Consequently (Fig. 2). for high x6(~,(k) - ~,(k + q) + c/Ak’) - Mk’ - q)), (8) wave numbers k above the chemical potential, where the density of states is large, the dominant contribudescribes the rate of polarisation transfer between the tion comes from the real part of both non-diagonalstates k and q due to carrier-carrier scattering. We and diagonal-scattering (Fig. 2(a)). However, the total show that this complex polarisation transfer between scattering contribution is significantly reduced in comdifferent k-states, reduces the optical linewidths in such parison to the diagonal-scattering (compare dashed a way to allow calculations for frequencies below the and dotted line). Therefore the negative source term band gap. For all following numerical computations in the polarisation equation is diminished, indicating of the frequanecy dependent gain spectra, defined by: a reduced absorption into higher k-states, in comparison to the diagonal-scattering only. A partial canG(w) = -2kJmx(w), (9) cellation of non-diagonal- and diagonal-scattering for where x(w) is the calculated susceptibility and kL is high k-values is confirmed also for the imaginary part of the scattering contribution (Fig. 2(b)). This overall the propagation constant of the laser. For numerical convinience, we take input pulses &I(W) which are ,.-I
N=l.O x 10’nc’cm’
-
.
..I...
,.,=,.J s
xl,&,,”
(a)
GAIN SPECTRA OF HIGHLY EXCITED SEMICONDUCTORS
558
Vol. 100. No. 8
0.6 p
-1.0 . 0
2
4
6
8
10
I2
14
t 1.0
ati
0’
0.5
2 E
0.0
4 *
I
(b)
Diagonal Turn +....a..f.J~_&,p,,d h
-
Full Low Intensity,t=O
I
-20
-10 ENERGY,
........ ‘.,. ....
0 E-EGAP
10
&I
Fig. 3. Enlarged gain spectra showing the difference between the calculations with and without nondiagonal-scattering for a total carrier density of N = 1.5 x 10’8cm-3. The neglect of the non-diagonalscattering yields strong unphysical absorption below the band edge.
detail the difference between the calculations with and without non-diagonal-scattering. In this plot, it can be recognized that the unphysical absorption below the .... J _, .o k gap is obtained if the non-diagonal-scattering terms IO 4 6 R I2 14 0 2 are neglected. WAVE NUMBER [a,,“] In conclusion, we have demonstrated that for a consistent determination of the optical dephasing in inFig. 2. Real (a) and imaginary (b) part of the scattering verted semiconductors, non-diagonal- and diagonalcontributions -P(k) ld and P(k) lnd which enter in the scattering in the polarisation equation must be inpolarisation equation, for excitation with a 15 fs sechcluded in the analysis. The consistent treatment of the shaped pulse with a carrier frequency at E = Egap; scattering is shown to substantially modify the gain (N = 1.5 x lOi*cm-s). The plot is a temporal snapshot at time, I = 0 fs. The solid line represents the usual spectrum both in shape and magnitude. P(k) Id-scattering and the dotted line corresponds to Acknowledgements-SH thanks the Comission of the -P(k)I,d-scattering. The dashed line depicts the the European Communities for an HMC European sum of the two, characterizing the overall dephasing Fellowship. This work was supported by the “Quanof the optical polarisation. a,, is the Bohr radius for tum Coherence in Semiconductors Program” of the bulk GaAs (=135 A). Deutsche Forschungsgemeinschaft (Germany). CPUtime at the HLRZ Jiilich is acknowledged. reduction in the total scattering contributions is also obtained for lower wave numbers k, but the cancelREFERENCES lation is not quite so pronounced. However the corresponding change of the scattering contributions at 1. Chow, W.W., Koch, S.W. and Sargent III, M., low k-states does not strongly influence the calculated Semiconductor-Laser Physics, Springer, Berlin, spectra, because the density of states is smaller at low 1994. k-numbers. We conclude that the change in the scat2. Lindberg, M. and Koch, S.W., Phys. Rev. B, B 38, tering contributions for high k-values drastically in1988, 3342. fluences the calculated gain spectra. This interpretaRappen, T., Peter, U.-G., Wegener, M. and Schafer, 3. tion can be confirmed by numerically switching off the W., Phys. Rev. B, 49, 1994, 10 774. higher k-states. In the standard Lorentzian lineshape 4. Rossi, F., Haas, S. and Kuhn, T., Phys. Rev. Lett., model, this reduction of high k-absorption transitions 72, 1994, 152. would not be obtained. 5. for a textbook discussion see, Haug, H. and Koch, The consequences of this behaviour for the gain S.W., Quantum Theory of the Optical and Elecspectra, are demonstrated clearly in the case of the cartronic Properties of Semiconductors, World Scirier density N = 1.5 x 10i8cm-3. Fig. 3 shows in more -0.5
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entific, Singapore, 3rd edn, 1994, and references therein. 6. In the following numerical calculations all material parameters are chosen corresponding close to bulk GaAs: a mass ratio of me/Mb = 713, where me./, is the electron/hole mass; an exciton binding energy E, of 4.2 meV; and a dipole moment of d,,(O) = 3e A. 7. Yamanishi, M. and Lee, Y., IEEEJ Quantum Electron., QE-14, 1987, 367. 8. Ohtoshi, T. and Yamanishi, M., IEEE J Quantum Electron., QE-27, 199 1, 46. 9. lndik, R.A., Mlejnek, M., Moloney, J.V., Binder, R., Hughes, S. Knorr, A. and Koch, S.W. accepted for publication in Phys Rev. A.
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10. Note, that polarisation scattering [3] which turns out to be important for the situation of nonlinear excitation such as four-wave-mixing (FWM) close to the band edge can be neglected in our case, because we treat the situation of linear optics where products of polarisations, such as the polarisation scattering are higher-order terms. In this paper we are interested only in the principle effects of the interplay of non-diagonal- and diagonal-scattering, therefore exchange contributions of the Coulomb potential to the lineshape are neglected for numerical convinience.