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Carrier kinetics for ultrafast optical pulses
LUMINESCENCE JOURN,ALOF
Journal of Luminescence 53 (1992) 187—190
Carrier kinetics for ultrafast optical pulses R. Zimmermann Zentralinstitut für Elektronenphysik, Hausi’ogteiplatz 5-7~0-1086 Berlin, Germany
Using density-matrix theory with inclusion of phonon scattering the kinetics of carriers and polarization in a semiconductor is studied. For ultrafast optical pulses, memory effects (non-Markoff scattering) have to be retained. As well known, they are responsible for non-Lorentzian line shapes; but the carrier thermalization is affected, too.
1. Density matrix for electrons and phonons
apart from the usual electronic density matrix
The availability of ultrafast laser pulses stimulated optical experiments (pump and probe type) which are able to resolve scattering processes in semiconductors on a subpicosecond time scale. It is a challenging task for theory to treat this nonequilibrium, light-field driven system with inclusion of realistic scattering, avoiding the phenomenological introduction of longitudinal and transverse relaxation times. As energy is no longer conserved at very short times, a real time formulation is advised. Non-equilibrium Green’s function technique has been used repeatedly for the problem (see e.g. refs. [1,2] for quite recent applications). An alternative way is density matrix theory which is easier from a conceptual point of view it contains only one time, in contrast to the Green’s functions depending on two times. Recently we have shown how exciton dephasing due to phonon scattering can be formulated on a microscopic level [3,4]. The clue was the introduction of a phonon-assisted density matrix
Pkah
T~(k,q)
=(aqc~qhc~~),
tr~~(t) =iEM~KnIQq~l)KlIQ~Im)
T~(k,q)
=(aqc~_qbc~a)
—
(1)
(C~hCkll). (2) Here, a, b is the band index, and the creation operators for electrons and phonons are defined by C~a and a~,respectively. Coupled equations of motion for p and T are derived from the Hamiltonian containing electrons, Coulomb interaction, phonons, and the coupling to the (classical) light field F(t) which is the driving force of the time dependence. A decoupling of higher expectation values in the Tequation closes the system. By solving the differential equation for T and inserting back into the polarization equation (pk~ ~k), the final result relevant for the linear response in exciton representation (quantum numbers n) is [3] i~1“P 1t =
—
LI
‘~
n~.
+fE~,~(tt )P~~(t) dt
=~~F(t)
—
(3) (h
=
1 throughout) with the memory function
x[N~~exp(—i(e 1~—w0)t)
+(N0+ 1) exp(_i(Eiq+w0)t)J. Correspondence to: Dr. R. Zimmermann, MPG-AG HaIbleitertheorie, Hausvogteiplatz 5-7, 0-1086 Berlin, Germany. 0022-2313/92/$05.00 © 1992
—
(4)
It contains the LO-phonon occupation N0 (exp(w0/kT0) 1)_I in equilibrium at the lattice
Elsevier Science Publishers B.V. All rights reserved
=
—
I
li?. Limmer,na,in
/ ( ‘a,©er 1,/net/cs
for ultraf/isi optical pu/us
temperature kT15, the electron—phonon matrix ciement Mq. and matrix elements of the charge distribution Qq in the exciton. Note the oscillatory time dependence which on Fourier transforming converts into the frequency dependence of the exciton—phonon self-energy. In turn, this gives rise to the non-Lorentzian line shape of the exciton absorption including phonon satellites. Clearly, eq. (4) is much more complicated than a Gaussian time dependence as e.g. assumed in ret.
-
[51. We do not go into nonline ir optics hcrc Our main goal is rather to extend the present approach to the carrier kinetic equation (section 3). Before, we give some remarks on the excitation process which is the link between interhand and intraband kinetics (section 2).
‘/ ‘~
~
I
~
F
-
-
Fig. I. Carrier excitation hs a short optical pulse. The mIen5 shown as slashed 5k~ ) curse for (FWI dilterent lM 60sletunings fs). Full curves: ~ = 2)). 3)). 40, 5)) meV (decreasing peak height). source (1, 10. (unction
sitS
2. Carrier generation In the collision-free limit and without Coulomb effects, the density matrix equations simplify to —idlPk —
=
‘0tPkcc
(Ek~ — =
~
L’kJ Pk ~ F( t)( Pk~ (F ~(t )P~ F(t )P~”).
Pk,
)‘
—
(5)
The exciting field is characterized by a midfrequency w 0 and a real amplitude function A(t) centered at t 0. Solving for the polarization linear in the field, we arrive at =
A( t
=
)f
d ‘A( t’ )2 cos(( t
1’
)~) (6)
which represents the source term ~S’k(1)in the diagonal kinetics for the electron distribution Ik p~ (see next section). The detuning is given by -
(Lk( Ek, )‘ (7) and describes the reGonance character of the optical excitation. However, only for sufficiently long pulses the excitation produces nearly monoenergetic electrons at = 0. An often used approximation for the source is given by 5k (t) A (w = ~k) A 2 (t)/A 2 (t = 0). (8) ‘adiab =
—
-‘
which at least accounts for the Fourier limitation of the process: the shorter the pulse, the more spread out in energy are the excited electrons (A(w) is the Fourier transform of A(t)). The functional dependence of eq. (8) is chosen to reproduce I) the time dependence of the energyintegrated source and ii) the energy dependence of the piled-up distribution (if there were no scattering). As shown in fig. I, the dashed curve (8) differs from the full source expression (6). Most remarkable is the evolution of negative features at larger detunings. Here, the electrons which have been excited from the valence band during the pulse rise are later dc-excited, the energy given back to the light field. For an off-resonant excitation helow the fundamental gap, only the high-frequency side feature is active, thus explaining the virtual character of a density which comes and goes with the pulse. It is only the scattering which brings the carriers into real (= long living) existence. The fast character of the optical Stark effect [61is based just on this feature. For resonant excitation treated here we have a co-existence of real and virtual excitation occuring in the center respectively. and at the wings of the frequency spectrum, Including scatter-
R. Zi,nmermann
/
Carrier kinetics for ultrafast optical pulses
ing processes the source term persists somewhat longer if the dephasing time exceeds the pulse duration.
189
a coupled system of differential equations. Solving for the latter we arrive at a generalized Boltzmann equation with memory. We simplify by assuming i) LO-phonons in equilibrium and ii) weak excitation fk(t) << 1,
3. Non-Markoff thermalization Having seen the importance of memory effects for the non-diagonal density matrix one is tempted
= Sk(t)
~tfk(t)
tdt’
+ E2M~f k’
x [cos( Ek~ Ek —
—
‘ii
w 0)( t
to ask for these refinements in the diagonal kinetics. Such non-Markoff kinetic equations have been repeatedly derived in the literature, but to the best of our knowledge no numerical results for optically excited semiconductors have been published till now. We focus on electron—phonon scattering in a single conduction band with optically excited electrons (source term, see preceeding section). In a related work by Lipavsky and coworkers [2] the Green’s function technique is used. Within the Monte Carlo scheme [71,memory effects can be approximately implemented using the collision retardation concept [8]. As described in section 1 the band-diagonal density matrix (electron distribution) and the phonon-assisted intraband density 1~~(k,q) form
40
nergy
~-
/~TT~~
f
(Li
~/ / / ~II ~
~
lO~
b
~-=~ F’-,~
~
/
-
-- -
[~~~N~/z=\ i/ ~re~/.
~
I)
/ /
// /
-~ -
Denc/iIy -
0
-~-~ --- --
-
-
C)
non (p s)
0
04
Fig. 2. Temporal evolution of average energy per carrier and distribution function at resonance fri~ following an optical excitation with 60 fs FWHM and an excess energy of 40 meV. The carriers relax via phonon scattering into equilibrium (kT51 = 15 meV). Full curves: with memory effects, dotted curves; without (Markoff approximation). The total density is shown as dashed curve (arbitrary units).
(1 + N0) fk( t’) N0) +cos(Ek Ek + w11)(t X
—
—
—
—
t’)
x(fk,(t’)Nofk(t’)(l +Ng))j. (9) t~is a time before the pulse started. Two assumptions would lead to a kinetic equation without memory: i) the distribution function varies little with time and can be taken out of the integral at I, ii) the integration result can be replaced by a delta function .
sin(
-
Ek
—
Ek
Ek
—
—
Ek
w0)(t
—
t11)
—
(10) Obviously, the second step depends on the weak energy dependence of distribution and matrix element. Consequently, the characteristic memory time (sometimes called collision duration time) is not only determined by the scattering process but by the smoothness of the distribution function, too. For solving eq. (9) as it stands we have discretized on a mesh in energy and time, taking care of a correct integration of the oscillating cosine. Parameters of bulk GaAs have been used, but matrix elements somewhat simplified. We have included apart from the LO-phonon (36 meV) a low-energy phonon (2 meV) in order to mimic the accoustic phonon scattering. In fig. 2 we compare the Markoff assumption (dotted curves) with the full solution (full curves). As initial condition a small equilibrium density (1% of the final one) has been assumed. During the exciting pulse (FWHM = 60 fs) the energy per particle rises to the excess energy, but the following cooling is somewhat slower for the full solution. This is due to a breaking of the energy ,
I OIl
R. Zi,,uner,nann / ( 0,-s-/er l..inc,u-.s fOr ultra fi,si optical pulses
at very short times, and can he related to the collision retardation [8]. Interesting are wavy features appearing in the distribution function which we call plionoti pulsations (here shown as f 4( ) at resonance, ‘k = (0. The period
functions [10] are presently relying on the Markoff approximation but could he improved~’~il~)ng sirniar lines.
is the phonon oscillation 2 ~/w1 the electrons react to a macroscopic lattice distortion set up by the pump pulse. The extension of these perturba— tions ( = memory time) is the shorter the smoother over k the initial distribution is, as revealed jw
References
conservation
i’
—
-
addition il calculations not shown hcrc Further calculations should clarify if the phonon pulsations show up e.g. in the induced transmission change of a delayed probe pulse. This might provide an alternative explanation to the phonon oscillations seen recently in pumpand-probe reflection measurements [91which are presently interpreted as phonon excitation in the time—dependent surface field. In conclusion we state that for nonlinear opties, the density matrix formulation is very useful for the inclusion of memory effects. Advanced numerical investigations starting from Green’s -
-
1l~M. llarlmann. II. Siol7 and R. Zimmcrmann, l’hvs. Stat. SoL (6) 159 (99(1) 35. [2] P. Lipavskv, ES. Khan, A. KaIsova and J.W. Wilkins. Phvs. Rev. II 43 (1991) 665(1. i i[ K LnsrnL im ni, Phys St it Sol (61 I sO (1991)) it 7 [41K. Zimrnermann and M. 1 lartmann. .1. Crvsi. Growth Ill! (199(1) 341.
151
F. l3any:a. 1
H
S. Schmitt-Rink, OS. (hernIa and II. l1aii~. l’hvs. Rcs.
-
Wicht and 11. 1 laug, 7. l’hvs. II 77 (I ~9)
B37 (958) 94
-~
-
17]
P. [ugh. P. Bordone. L. Rcggiani. M. Ricgcr. P. Koccvar and SM. (ioodnick. Phys. Rev. 39 (19S9) 7852. [51l).K. Ferry. AM. Kriman, II. I lida and S. Yamaguchi. Phys. Rev. Letters 67 (1991) 633. [9] (Of’. Cho. W. Kuti and 11. Kurz. Phvs. Res. Lctt 05
(199(1) 764 [I)] U. B/ihnc. 1
-
Sure. RU. EJIbrich and W. Sch4lcr. l’hss.
Rev. B 41 (1990) 7540.
Journal of Luminescence 53 (1992) 187—190
Carrier kinetics for ultrafast optical pulses R. Zimmermann Zentralinstitut für Elektronenphysik, Hausi’ogteiplatz 5-7~0-1086 Berlin, Germany
Using density-matrix theory with inclusion of phonon scattering the kinetics of carriers and polarization in a semiconductor is studied. For ultrafast optical pulses, memory effects (non-Markoff scattering) have to be retained. As well known, they are responsible for non-Lorentzian line shapes; but the carrier thermalization is affected, too.
1. Density matrix for electrons and phonons
apart from the usual electronic density matrix
The availability of ultrafast laser pulses stimulated optical experiments (pump and probe type) which are able to resolve scattering processes in semiconductors on a subpicosecond time scale. It is a challenging task for theory to treat this nonequilibrium, light-field driven system with inclusion of realistic scattering, avoiding the phenomenological introduction of longitudinal and transverse relaxation times. As energy is no longer conserved at very short times, a real time formulation is advised. Non-equilibrium Green’s function technique has been used repeatedly for the problem (see e.g. refs. [1,2] for quite recent applications). An alternative way is density matrix theory which is easier from a conceptual point of view it contains only one time, in contrast to the Green’s functions depending on two times. Recently we have shown how exciton dephasing due to phonon scattering can be formulated on a microscopic level [3,4]. The clue was the introduction of a phonon-assisted density matrix
Pkah
T~(k,q)
=(aqc~qhc~~),
tr~~(t) =iEM~KnIQq~l)KlIQ~Im)
T~(k,q)
=(aqc~_qbc~a)
—
(1)
(C~hCkll). (2) Here, a, b is the band index, and the creation operators for electrons and phonons are defined by C~a and a~,respectively. Coupled equations of motion for p and T are derived from the Hamiltonian containing electrons, Coulomb interaction, phonons, and the coupling to the (classical) light field F(t) which is the driving force of the time dependence. A decoupling of higher expectation values in the Tequation closes the system. By solving the differential equation for T and inserting back into the polarization equation (pk~ ~k), the final result relevant for the linear response in exciton representation (quantum numbers n) is [3] i~1“P 1t =
—
LI
‘~
n~.
+fE~,~(tt )P~~(t) dt
=~~F(t)
—
(3) (h
=
1 throughout) with the memory function
x[N~~exp(—i(e 1~—w0)t)
+(N0+ 1) exp(_i(Eiq+w0)t)J. Correspondence to: Dr. R. Zimmermann, MPG-AG HaIbleitertheorie, Hausvogteiplatz 5-7, 0-1086 Berlin, Germany. 0022-2313/92/$05.00 © 1992
—
(4)
It contains the LO-phonon occupation N0 (exp(w0/kT0) 1)_I in equilibrium at the lattice
Elsevier Science Publishers B.V. All rights reserved
=
—
I
li?. Limmer,na,in
/ ( ‘a,©er 1,/net/cs
for ultraf/isi optical pu/us
temperature kT15, the electron—phonon matrix ciement Mq. and matrix elements of the charge distribution Qq in the exciton. Note the oscillatory time dependence which on Fourier transforming converts into the frequency dependence of the exciton—phonon self-energy. In turn, this gives rise to the non-Lorentzian line shape of the exciton absorption including phonon satellites. Clearly, eq. (4) is much more complicated than a Gaussian time dependence as e.g. assumed in ret.
-
[51. We do not go into nonline ir optics hcrc Our main goal is rather to extend the present approach to the carrier kinetic equation (section 3). Before, we give some remarks on the excitation process which is the link between interhand and intraband kinetics (section 2).
‘/ ‘~
~
I
~
F
-
-
Fig. I. Carrier excitation hs a short optical pulse. The mIen5 shown as slashed 5k~ ) curse for (FWI dilterent lM 60sletunings fs). Full curves: ~ = 2)). 3)). 40, 5)) meV (decreasing peak height). source (1, 10. (unction
sitS
2. Carrier generation In the collision-free limit and without Coulomb effects, the density matrix equations simplify to —idlPk —
=
‘0tPkcc
(Ek~ — =
~
L’kJ Pk ~ F( t)( Pk~ (F ~(t )P~ F(t )P~”).
Pk,
)‘
—
(5)
The exciting field is characterized by a midfrequency w 0 and a real amplitude function A(t) centered at t 0. Solving for the polarization linear in the field, we arrive at =
A( t
=
)f
d ‘A( t’ )2 cos(( t
1’
)~) (6)
which represents the source term ~S’k(1)in the diagonal kinetics for the electron distribution Ik p~ (see next section). The detuning is given by -
(Lk( Ek, )‘ (7) and describes the reGonance character of the optical excitation. However, only for sufficiently long pulses the excitation produces nearly monoenergetic electrons at = 0. An often used approximation for the source is given by 5k (t) A (w = ~k) A 2 (t)/A 2 (t = 0). (8) ‘adiab =
—
-‘
which at least accounts for the Fourier limitation of the process: the shorter the pulse, the more spread out in energy are the excited electrons (A(w) is the Fourier transform of A(t)). The functional dependence of eq. (8) is chosen to reproduce I) the time dependence of the energyintegrated source and ii) the energy dependence of the piled-up distribution (if there were no scattering). As shown in fig. I, the dashed curve (8) differs from the full source expression (6). Most remarkable is the evolution of negative features at larger detunings. Here, the electrons which have been excited from the valence band during the pulse rise are later dc-excited, the energy given back to the light field. For an off-resonant excitation helow the fundamental gap, only the high-frequency side feature is active, thus explaining the virtual character of a density which comes and goes with the pulse. It is only the scattering which brings the carriers into real (= long living) existence. The fast character of the optical Stark effect [61is based just on this feature. For resonant excitation treated here we have a co-existence of real and virtual excitation occuring in the center respectively. and at the wings of the frequency spectrum, Including scatter-
R. Zi,nmermann
/
Carrier kinetics for ultrafast optical pulses
ing processes the source term persists somewhat longer if the dephasing time exceeds the pulse duration.
189
a coupled system of differential equations. Solving for the latter we arrive at a generalized Boltzmann equation with memory. We simplify by assuming i) LO-phonons in equilibrium and ii) weak excitation fk(t) << 1,
3. Non-Markoff thermalization Having seen the importance of memory effects for the non-diagonal density matrix one is tempted
= Sk(t)
~tfk(t)
tdt’
+ E2M~f k’
x [cos( Ek~ Ek —
—
‘ii
w 0)( t
to ask for these refinements in the diagonal kinetics. Such non-Markoff kinetic equations have been repeatedly derived in the literature, but to the best of our knowledge no numerical results for optically excited semiconductors have been published till now. We focus on electron—phonon scattering in a single conduction band with optically excited electrons (source term, see preceeding section). In a related work by Lipavsky and coworkers [2] the Green’s function technique is used. Within the Monte Carlo scheme [71,memory effects can be approximately implemented using the collision retardation concept [8]. As described in section 1 the band-diagonal density matrix (electron distribution) and the phonon-assisted intraband density 1~~(k,q) form
40
nergy
~-
/~TT~~
f
(Li
~/ / / ~II ~
~
lO~
b
~-=~ F’-,~
~
/
-
-- -
[~~~N~/z=\ i/ ~re~/.
~
I)
/ /
// /
-~ -
Denc/iIy -
0
-~-~ --- --
-
-
C)
non (p s)
0
04
Fig. 2. Temporal evolution of average energy per carrier and distribution function at resonance fri~ following an optical excitation with 60 fs FWHM and an excess energy of 40 meV. The carriers relax via phonon scattering into equilibrium (kT51 = 15 meV). Full curves: with memory effects, dotted curves; without (Markoff approximation). The total density is shown as dashed curve (arbitrary units).
(1 + N0) fk( t’) N0) +cos(Ek Ek + w11)(t X
—
—
—
—
t’)
x(fk,(t’)Nofk(t’)(l +Ng))j. (9) t~is a time before the pulse started. Two assumptions would lead to a kinetic equation without memory: i) the distribution function varies little with time and can be taken out of the integral at I, ii) the integration result can be replaced by a delta function .
sin(
-
Ek
—
Ek
Ek
—
—
Ek
w0)(t
—
t11)
—
(10) Obviously, the second step depends on the weak energy dependence of distribution and matrix element. Consequently, the characteristic memory time (sometimes called collision duration time) is not only determined by the scattering process but by the smoothness of the distribution function, too. For solving eq. (9) as it stands we have discretized on a mesh in energy and time, taking care of a correct integration of the oscillating cosine. Parameters of bulk GaAs have been used, but matrix elements somewhat simplified. We have included apart from the LO-phonon (36 meV) a low-energy phonon (2 meV) in order to mimic the accoustic phonon scattering. In fig. 2 we compare the Markoff assumption (dotted curves) with the full solution (full curves). As initial condition a small equilibrium density (1% of the final one) has been assumed. During the exciting pulse (FWHM = 60 fs) the energy per particle rises to the excess energy, but the following cooling is somewhat slower for the full solution. This is due to a breaking of the energy ,
I OIl
R. Zi,,uner,nann / ( 0,-s-/er l..inc,u-.s fOr ultra fi,si optical pulses
at very short times, and can he related to the collision retardation [8]. Interesting are wavy features appearing in the distribution function which we call plionoti pulsations (here shown as f 4( ) at resonance, ‘k = (0. The period
functions [10] are presently relying on the Markoff approximation but could he improved~’~il~)ng sirniar lines.
is the phonon oscillation 2 ~/w1 the electrons react to a macroscopic lattice distortion set up by the pump pulse. The extension of these perturba— tions ( = memory time) is the shorter the smoother over k the initial distribution is, as revealed jw
References
conservation
i’
—
-
addition il calculations not shown hcrc Further calculations should clarify if the phonon pulsations show up e.g. in the induced transmission change of a delayed probe pulse. This might provide an alternative explanation to the phonon oscillations seen recently in pumpand-probe reflection measurements [91which are presently interpreted as phonon excitation in the time—dependent surface field. In conclusion we state that for nonlinear opties, the density matrix formulation is very useful for the inclusion of memory effects. Advanced numerical investigations starting from Green’s -
-
1l~M. llarlmann. II. Siol7 and R. Zimmcrmann, l’hvs. Stat. SoL (6) 159 (99(1) 35. [2] P. Lipavskv, ES. Khan, A. KaIsova and J.W. Wilkins. Phvs. Rev. II 43 (1991) 665(1. i i[ K LnsrnL im ni, Phys St it Sol (61 I sO (1991)) it 7 [41K. Zimrnermann and M. 1 lartmann. .1. Crvsi. Growth Ill! (199(1) 341.
151
F. l3any:a. 1
H
S. Schmitt-Rink, OS. (hernIa and II. l1aii~. l’hvs. Rcs.
-
Wicht and 11. 1 laug, 7. l’hvs. II 77 (I ~9)
B37 (958) 94
-~
-
17]
P. [ugh. P. Bordone. L. Rcggiani. M. Ricgcr. P. Koccvar and SM. (ioodnick. Phys. Rev. 39 (19S9) 7852. [51l).K. Ferry. AM. Kriman, II. I lida and S. Yamaguchi. Phys. Rev. Letters 67 (1991) 633. [9] (Of’. Cho. W. Kuti and 11. Kurz. Phvs. Res. Lctt 05
(199(1) 764 [I)] U. B/ihnc. 1
-
Sure. RU. EJIbrich and W. Sch4lcr. l’hss.
Rev. B 41 (1990) 7540.