- Email: [email protected]
Spatio-temporal carrier dynamics in ultrashort-pulse laser irradiated materials
Nuclear Instruments and Methods in Physics Research A 653 (2011) 72–75
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Spatio-temporal carrier dynamics in ultrashort-pulse laser irradiated materials Tzveta Apostolova Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Bulgaria
a r t i c l e i n f o
a b s t r a c t
Available online 2 February 2011
A space and time-dependent quantum-kinetic theory has been formulated based on previous theoretical approaches to study the spatio-temporal microscopic carrier dynamics in laser excited semiconductor materials that accounts for the effects of inhomogeneous excitation and structural inhomogeneities due to bulk filamentation damage and micro/nano structuring. The approach involves extensive computational effort. & 2011 Elsevier B.V. All rights reserved.
Keywords: Carrier dynamics Ultrashort pulse laser Laser–matter interaction
1. Introduction When dealing with a typical semiconductor-based optoelectronic device irradiated with laser light we consider the created electron–hole plasma (EHP) in the crystal lattice of the chosen material. Carrier generation and recombination, electrical conduction and diffusion determine the behavior of the formed plasma. The photon energy of the laser field is converted and conserved as kinetic and thermal energy of the plasma and thermal energy of the lattice by creation and annihilation of phonons. All the described processes take place on different time and space scales but they should be treated in a self-consistent manner with the appropriate coupled equations. Inhomogeneous excitation, bulk filamentation laser damage, etc. require inclusion of spatial variation in the formalism describing the dynamics of optically generated carriers and lead to space-dependent carrier distributions.
2. Theoretical model When a spatially homogeneous system is excited by a spatially inhomogeneous laser field or a spatially dependent filamentary damage is induced, the dynamical variables become inhomogeneous and off-diagonal density matrices have to be introduced. A mixed momentum and real space representation is most similar to classical distribution function and is best suited for a comparison to semiclassical kinetics described by Boltzmann equation. By using meanfield approximation to the correlation of the electron and hole operators and dipole-coupling approximation to the interaction with the external electromagnetic field, an effective Hamiltonian is obtained in terms of the ascending–descending operators. On the basis of the effective Hamiltonian,the Boltzmann–Bloch equations E-mail address: [email protected] 0168-9002/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2011.01.110
for the description of spatio-temporal dynamics of electrons and holes of inhomogeneously excited semiconductors including the coherent interactions of carriers and the laser light field as well as transport due to spatial gradients and electrostatic forces are obtained. Besides the interaction with the light field other important interactions in the semiconductor such as Coulomb interaction among the carriers giving rise to screening and to thermalization of the nonequilibrium carrier distribution, as well as interaction with phonons leading to an energy exchange between the carriers and the crystal lattice are included. We follow the approach in Ref. [1] but unlike them we treat all the scattering terms explicitly without resorting to relaxation time approximation [2]. We also include terms that lead to transitions between valence and conduction band i.e. impact ionization and Auger recombination [3]. We consider a two-band model of an undoped semiconductor such as GaAs. In the laser–matter interaction process the physical variables that are directly related to observables of the system such as optical polarizations and distribution functions are all single-particle quantities calculated by the density matrix. To describe space-dependent phenomena a Wigner representation of the single-particle density matrix can be used. In Wigner representation the space-dependent distribution functions (intraband density matrices) of electrons and holes and polarization (interband density matrix) are defined as f e ð~ k,~ rÞ ¼
X ~~ y eiq r /c~ ~ q
c~ 1q S, q k2~ k þ 12~
f h ð~ k,~ rÞ ¼
X ~~ y eiq r /d~ ~ q
d~ 1q S q k2~ k þ 12~
and pð~ k,~ rÞ ¼
X ~~ eiq r /d~k þ 1~q c~k þ 1~q S ~ q
2
2
where c~y and d~y ðc~k and d~k Þ denote creation (annihilation) k k operators for electrons and holes with wave vector, respectively, and the brackets denote the expectation value of these operators.
T. Apostolova / Nuclear Instruments and Methods in Physics Research A 653 (2011) 72–75
The single-particle Hamiltonian describing the free carrier interacting with a classical light field as well as the free phonons is given by X y X X H0 ¼ e~ek c~ c~k þ e~hk d~y d~k þ ‘ o~q b~yq b~q k
~ k
~ k
k
~ q
X y ~cv ð~ ½m kÞ ~ E ~q ðtÞc~y
dy 1~ þ ~cv ð~ kÞ q ~ k þ 12~ k þ 2q
~ k, ~ q
m
~ E ~q ðtÞdy ~
cy 1~ q ~ k þ 12~ k þ 2q ð1Þ
kÞ is the component in the direction of the laser field where mð~ polarization of the interband optical dipole matrix element between the electron state jc,~ kS and the hole state jv,~ kS. The field is represented by two counterpropagating waves and the positive frequency component is given by y y ~ E ð~ r ,tÞ ¼ 12ð~ E ð~ r ,tÞeiK z ziot þ ~ E ð~ r ,tÞeiK z ziot Þ
ð2Þ
and is expanded in a Fourier series X 0 X y y ~ ~ ~ E ~q ðtÞeið~q ~r otÞ ¼ E ~q ðtÞei~q ~r : E ð~ r ,tÞ ¼
ð3Þ
~ q
~ q
In the absence of an external light field the electron states are eigenstates of an ideal periodic lattice. Deviations from this idealized periodicity due to lattice vibrations lead to a coupling of the different electronic states. This interaction is described by the carrier–phonon Hamiltonian. X HIcp ¼ C~q ½c~y ~ b~q c~k c~y b~yq c~k þ ~q d~y ~ b~q d~k þ d~y b~yq d~k þ ~q , ð4Þ k þq
~ k, ~ ku, ~ q
k
k þq
k
where C~q is the electron–phonon coupling constant for interaction with optical phonons, er ð1Þ and er ð0Þ are the relative static and optical dielectric constant, respectively, e0 is the absolute dielectric constant of the vacuum, ‘ oLO is the optical phonon energy and V is the normalization volume. The charged carriers interact via the Coulomb potential V~q and the Hamiltonian describing carrier–carrier interaction processes conserving the number of particles per band is given by: X 1 y y 1 HIcc ¼ V~q c~ c~ c~ku þ ~q c~k~q þ d~y d~y d~ku þ ~q d~k~q c~y dy ~ d~ku þ ~q c~k~q : k ku 2 k ku 2 k ku ~~ k, ku, ~ q
ð5Þ The carrier–carrier Hamiltonian can be separated into a mean field (Haretree–Fock) Hcc HF part and a remaining part depending on two-particle correlations Hcc corr. The effective single-particle Hamiltonian is Heff ¼H0 +Hcc HF. The correlation part of the carrier– carrier interaction Hamiltonian gives two phenomena: scattering processes between the carriers and the screening of the bare Coulomb interaction. The part of the perturbation Hamiltonian that yields impact ionization and its inverse process, Auger recombination is given by [3,4] X HIccðcvÞ ¼ ½Me ðqÞc~y ~ c~y ~ dy ~ c~k þ Me ðqÞc~y d~ku c~ku~q c~y ~ k þ q kuq ku
~ k,~ ku, ~ q
þ
X
k
½Mh ðqÞd~y
~ k,~ ku, ~ q
k þq
dy ~ cy ~ d~k þ Mh ðqÞd~y c~ku d~ku~q d~k þ ~q k þ~ q ~ kuq ku k
ð6Þ
Me ðqÞ ¼ V~q g~q , where V~q ¼ e2 =V e0 er q2 is the Coulomb potential and g~q is the interband-transition form factor.
73
functions of electrons and holes and for interband polarization. Being Wigner distributions these quantities are functions of space and momentum but there is a big difference of time scales between the momentum space and the real space dynamics. Scattering and dephasing processes lead to fast relaxation of the microscopic variables towards their local quasi-equilibrium values on a femtosecond time-scale while the spatial transport happens on a much slower time-scale (10 ps to ns). Because of a typical separation of time scales between the ~ k-space and ~ r -space dynamics, the influence of spatial gradients on the k-space dynamics is often negligible. However, some of the scattering terms in the equations of motion for the distribution functions conserve the density of carriers and therefore the density is not influenced by the fast relaxation processes and its spatial transport cannot be neglected. In the equation of motion for the polarization no conserved quantities exist and thus the spatial transport of polarization is usually not important. In principle, the complete set of equations required is, therefore, the Maxwell– Boltzmann–Bloch–Poisson equations for the nonequilibrium distribution functions f a ð~ k,~ r Þ, interband polarization pð~ k,~ r Þ, electric potential Fð~ r Þ, and the laser field ~ Eð~ r ,tÞ, with ~ k and ~ r being the two-dimensional (2D) vectors in reciprocal (momentum) space and real space, respectively. Keeping the first-order spatial derivatives of the distribution functions and neglecting any spatial transport of polarization, the equations of motion for electron and hole distribution functions are given by the generalized Boltzmann equations for two-band model including the coherent interband contributions. @ a~ 1 @ea ð~ k,~ r Þ @f a ð~ k,~ r ,tÞ 1 @ @f a ð~ k,~ r ,tÞ f ðk,~ ½dea ð~ r ,tÞ þ k,~ r Þ þ qFðrÞ @t @r @k ‘ @k ‘ @r
@ ¼ Ra ð~ k,~ r Þ þ f a ð~ k,~ r Þcol : @t
ð7Þ
The lowest order contribution to the polarization is included, where the spatial coordinate enters only as a parameter and locally the dynamics coincide with those of the inhomogeneous case and there are no transport effects. This lowest order picture is sufficient to describe pump-probe experiments in which filamentation is observed. @ ~ i pðk,~ r ,tÞ ¼ ½ee ð~ k,~ r ,tÞ þ eh ð~ k,~ r ,tÞpð~ k,~ r ,tÞ @t ‘ k,~ r Þ½f e ð~ k,~ r ,tÞ þf h ð~ k,~ r ,tÞ1 þ iOð~
@ ~ pðk,~ r Þcol @t
ea ð~ k,~ r Þ ¼ ea ð~ kÞ þ qa FðrÞ þ dea ð~ k,~ rÞ
ð8Þ
2 e ð~ kÞ ¼ ‘ k2 =2ma is the single-particle energy, X 1X s s dea ð~ k,~ r Þ ¼ f a ð~ k,~ r ÞV~ku þ ½V V~ku~k ~ k 2 ~ ~ku~k ~ e,h
ku
ku
is the renormalization of the single-particle carrier energy due to exchange interaction, V~qs is the screened Coulomb potential. The electrostatic potential due to the Hartree terms in the mean field Hamiltonian satisfies the Poisson equation. The generation rate in the Eq. (7) is given as follows: Ra ð~ k,~ r Þ ¼ i½Oð~ k,~ r Þp ð~ k,~ r ÞO ð~ k,~ r Þpð~ k,~ r Þ
ð9Þ
where Oð~ k,~ r Þ is the renormalized Rabi frequency defined by 3. Generalized Boltzmann–Bloch equations By using Heisenberg’s equations of motion, the equations of motion for the single-particle density matrices in Wigner representation can be derived. The effective single-particle Hamiltonian Heff gives a closed set of equations for the distribution
‘ Oð~ k,~ r Þ ¼ mð~ kÞ~ Eð~ r ,tÞ þ
X s pð~ ku,~ r ÞV~k : ~ ku
ð10Þ
~ ku
The second term in the above expression is the internal field responsible for Coulomb enhancement.
74
T. Apostolova / Nuclear Instruments and Methods in Physics Research A 653 (2011) 72–75
4. Scattering processes Within a semiclassical picture when scattering processes are described in terms of scattering rates, the scattering contributions to the equation of motion have the structure of the Boltzmann collision terms. The electron–phonon scattering rates are obtained from Fermi’s golden rule and quadratic or higher order terms and terms involving simultaneous electron–phonon and hole–phonon interaction have been neglected in the polarization and in the carrier–phonon Hamiltonian. Incoherent scattering processes appear for the first time in second-order contributions [1,3,5,6]. Collisional contributions in Eqs. (1) and (2) lead to relaxation in the carrier distributions and decay in the interband polarization: @ a~ @ @ @ f ðk,~ r Þcol ¼ f a ð~ k,~ r Þaa þ f a ð~ k,~ r Þeh þ f a ð~ k,~ r ÞLO @t @t @t @t
q
ð13Þ
kq , k
where the scattering matrices are given by 2p X X s 2 e,h Þ a a ee,h Þf a ð~ W~e,hð~aa jV~q j dðe~ ~ þ e~ku e~ku ku,~ rÞ ~ ~ ¼ þ~ q k
ku þ ~ q ,~ r Þ ½1f a ð~ ¼ W~e,hðLOÞ ~~ kq , k
2p
‘ þ
ð14Þ
jCqe,h j2 ðNq þ1Þdðe~e,h ~ e~e,h þ ‘ oq Þ kq
2p
‘
e
The Boltzmann scattering matrices
‘ oq Þ:
ð15Þ
W~e,h~ ~ kq , k
for electrons and holes
are real quantities and the scattering matrices W~p
k~ q ,~ k
in the
equation of motion for the polarization are complex and the real part is related to W~e,h~ ~ according to kq , k
e e ½1f e ð~ k~ q ,~ r Þ þW~k, f e ð~ k~ q ,~ rÞ ReðW~p ~ ~ Þ ¼ 12W~k ~ ~ q ,~ k k~ q kq , k
þ 12W~qh~k,~k ½1f h ð~ q ~ k,~ r Þ þWh ~k,~q ~k f h ð~ q ~ k,~ r Þ: The real part of
W~p ~ ~ kq , k
ð17Þ
@ e~ 2p X f ðk,~ r ÞehðrecÞ ¼ jV j2 g dðee 2e~ek~q e~hk2~q EG Þ @t ‘ ~ ~q ~q ~k q
k,~ r Þ ½f e ð~ k~ q ,~ r Þ2 f h ð~ k2~ q ,~ rÞ ½1f e ð~ 2p X 2 e e h jV~q j g~q dðe~k þ ~q 2e~k e~k~q EG Þ þ
‘
~ q
ð18Þ
g~q 2ðme =m0 Þ.
6. Numerical procedures in progress Based on typical length and time scales approximations are made with the aim of obtaining numerically tractable system of equations.The microscopic dynamics of the distribution function and the nonlinear polarization are governed by the equations of motion (1) and (2). A full kinetic treatment of the scattering processes will be performed by using, e.g. Monte Carlo simulations. Generalized Monte Carlo methods taking into account phase relations between different type of carriers (polarization effects), interaction of carriers with external coherent inhomogeneous electromagnetic field (generation effects), and the correlation and renormalization effects associated with carrier–carrier interaction can be utilized [7].
k
jCqe,h j2 Nq dð ~e,h~ ~e,h kq k
e
~ q
k,~ r Þ½1f e ð~ k~ q ,~ r Þ2 ½1f h ð~ k2~ q ,~ r Þ f ð~
with
k,~ r Þð1f e,h ð~ k~ q ,~ r ÞÞ W~e,h~ ~ f e,h ð~
kq
‘
e
ð12Þ
The first term on the RHS of Eq. (3) depicts the scattering processes arising from the correlation part of the carrier–carrier Hamiltonian and the third term arises from the carrier–phonon Hamiltonian X @ a~ f ðk,~ r Þaa=LO ¼ ½W~e,h~ ~ f e,h ð~ k~ q ,~ r Þð1f e,h ð~ k,~ r ÞÞ k, kq @t ~
a ¼ e,h ~ ku
q
½1f ð~ k,~ r Þ2 f e ð~ k þ~ q ,~ r Þ½1f h ð~ k~ q ,~ r Þ 2p X 2 e e jV~q j g~q dð2ej~k~q j e~k þ ehj~k2~q j þ EG Þ þ Neff e
k,~ r Þ2 ½1f e ð~ k þ~ q ,~ r Þf h ð~ k~ q ,~ rÞ ½f e ð~
q
‘
@ e~ 2p X f ðk,~ r ÞehðimpÞ ¼ Neff jV j2 g dð2e~ek eej~k þ ~q j þ ehj~k~q tj þEG Þ @t ‘ ~ ~q ~q
ð11Þ
X p @ ~ pðk,~ r Þcol ¼ ½W~ ~ ~ pð~ k~ q ,~ r ÞW~p ~ ~ pð~ k,~ r Þ: k, kq kq , k @t ~
kq , k
These contributions are derived using second-order perturbation theory such as Coulomb scattering [3] which is very important to conduction-electron dynamics and the change of electron density.
ð16Þ
describes scattering processes leading to a
dephasing of the polarization and the imaginary part describes second-order contributions to the band-gap renormalization.
5. Impact ionization and Auger recombination Since we interested in the processes of laser damage and filamentation in the semiconductor material, we include the ð@=@tÞf a ð~ k,~ r Þeh terms that lead to transitions between valence and conduction bands, i.e. impact ionization term and Auger recombination term [3]. Impact ionization and Auger recombination are second-order two-particle Coulomb scattering processes (proportional to Coulomb scattering). In a case when we have a homogeneous system (material) that is either homogeneously or inhomogeneously excited the Coulomb matrix elements depend on the momentum transfer ~ q only, i.e. pjV~q j2 .
7. Conclusion A microscopic quantum-kinetic theory based on density matrix formalism [8] is formulated to describe the processes of short pulse laser interaction with materials such as semiconductors accounting for arbitrary spatial inhomogeneities in the excitation conditions and other spatial phenomena such as filamentation of tightly focused femtosecond laser pulses, structural modification and catastrophic optical damage. A system of Boltzmann–Bloch transport equations are established that include both space and momentum dependence of the electron and hole distribution functions and the polarization. Microscopic electron–phonon and electron–electron scattering terms as well as scattering terms that lead to transitions between valence and conduction bands, i.e. impact ionization and recombination terms, are included explicitly in the equations. Since some of the scattering processes take place on the same time scale as the laser pulse duration, it is of importance to capture the transient relaxation of the excited carriers as well as the transient energy exchange between carriers and phonons. For this purpose explicit treatment of the scattering processes are considered instead of resorting to relaxation timed approximation. The formulated theory describes the spatio-temporal dynamics of electrons and holes including the coherent interactions of carriers and the laser light field as well as transport due to spatial gradients and electrostatic forces. The laser intensities and pulse durations for which our theory is applicable are IL ¼ 1 1015 21 1017 W=m2 , tL ¼ 100 fs.
T. Apostolova / Nuclear Instruments and Methods in Physics Research A 653 (2011) 72–75
References [1] O. Hess, T. Kuhn, Phys. Rev. A 54 (1996) 3347. [2] J. Li, C.Z. Ning, Phys. Rev. A 66 (2002) 023802. [3] T. Apostolova, D. Huang, P.M. Alsing, J. McIver, D.A. Cardimona, Phys. Rev. B 66 (2002) 075208.
[4] [5] [6] [7] [8]
W. Quade, E. Scholl, Phys. Rev. B 50 (1994) 7398. O. Hess, T. Kuhn, Prog. Quantum Electron. 20 (1996) 85. T. Kuhn, F. Rossi, Phys. Rev. B 46 (1992) 7496. S. Haas, F. Rossi, Phys. Rev. B 53 (1992) 12855. M. Lindberg, S.W. Koch, Phys. Rev. B 38 (1988).
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Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Spatio-temporal carrier dynamics in ultrashort-pulse laser irradiated materials Tzveta Apostolova Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Bulgaria
a r t i c l e i n f o
a b s t r a c t
Available online 2 February 2011
A space and time-dependent quantum-kinetic theory has been formulated based on previous theoretical approaches to study the spatio-temporal microscopic carrier dynamics in laser excited semiconductor materials that accounts for the effects of inhomogeneous excitation and structural inhomogeneities due to bulk filamentation damage and micro/nano structuring. The approach involves extensive computational effort. & 2011 Elsevier B.V. All rights reserved.
Keywords: Carrier dynamics Ultrashort pulse laser Laser–matter interaction
1. Introduction When dealing with a typical semiconductor-based optoelectronic device irradiated with laser light we consider the created electron–hole plasma (EHP) in the crystal lattice of the chosen material. Carrier generation and recombination, electrical conduction and diffusion determine the behavior of the formed plasma. The photon energy of the laser field is converted and conserved as kinetic and thermal energy of the plasma and thermal energy of the lattice by creation and annihilation of phonons. All the described processes take place on different time and space scales but they should be treated in a self-consistent manner with the appropriate coupled equations. Inhomogeneous excitation, bulk filamentation laser damage, etc. require inclusion of spatial variation in the formalism describing the dynamics of optically generated carriers and lead to space-dependent carrier distributions.
2. Theoretical model When a spatially homogeneous system is excited by a spatially inhomogeneous laser field or a spatially dependent filamentary damage is induced, the dynamical variables become inhomogeneous and off-diagonal density matrices have to be introduced. A mixed momentum and real space representation is most similar to classical distribution function and is best suited for a comparison to semiclassical kinetics described by Boltzmann equation. By using meanfield approximation to the correlation of the electron and hole operators and dipole-coupling approximation to the interaction with the external electromagnetic field, an effective Hamiltonian is obtained in terms of the ascending–descending operators. On the basis of the effective Hamiltonian,the Boltzmann–Bloch equations E-mail address: [email protected] 0168-9002/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2011.01.110
for the description of spatio-temporal dynamics of electrons and holes of inhomogeneously excited semiconductors including the coherent interactions of carriers and the laser light field as well as transport due to spatial gradients and electrostatic forces are obtained. Besides the interaction with the light field other important interactions in the semiconductor such as Coulomb interaction among the carriers giving rise to screening and to thermalization of the nonequilibrium carrier distribution, as well as interaction with phonons leading to an energy exchange between the carriers and the crystal lattice are included. We follow the approach in Ref. [1] but unlike them we treat all the scattering terms explicitly without resorting to relaxation time approximation [2]. We also include terms that lead to transitions between valence and conduction band i.e. impact ionization and Auger recombination [3]. We consider a two-band model of an undoped semiconductor such as GaAs. In the laser–matter interaction process the physical variables that are directly related to observables of the system such as optical polarizations and distribution functions are all single-particle quantities calculated by the density matrix. To describe space-dependent phenomena a Wigner representation of the single-particle density matrix can be used. In Wigner representation the space-dependent distribution functions (intraband density matrices) of electrons and holes and polarization (interband density matrix) are defined as f e ð~ k,~ rÞ ¼
X ~~ y eiq r /c~ ~ q
c~ 1q S, q k2~ k þ 12~
f h ð~ k,~ rÞ ¼
X ~~ y eiq r /d~ ~ q
d~ 1q S q k2~ k þ 12~
and pð~ k,~ rÞ ¼
X ~~ eiq r /d~k þ 1~q c~k þ 1~q S ~ q
2
2
where c~y and d~y ðc~k and d~k Þ denote creation (annihilation) k k operators for electrons and holes with wave vector, respectively, and the brackets denote the expectation value of these operators.
T. Apostolova / Nuclear Instruments and Methods in Physics Research A 653 (2011) 72–75
The single-particle Hamiltonian describing the free carrier interacting with a classical light field as well as the free phonons is given by X y X X H0 ¼ e~ek c~ c~k þ e~hk d~y d~k þ ‘ o~q b~yq b~q k
~ k
~ k
k
~ q
X y ~cv ð~ ½m kÞ ~ E ~q ðtÞc~y
dy 1~ þ ~cv ð~ kÞ q ~ k þ 12~ k þ 2q
~ k, ~ q
m
~ E ~q ðtÞdy ~
cy 1~ q ~ k þ 12~ k þ 2q ð1Þ
kÞ is the component in the direction of the laser field where mð~ polarization of the interband optical dipole matrix element between the electron state jc,~ kS and the hole state jv,~ kS. The field is represented by two counterpropagating waves and the positive frequency component is given by y y ~ E ð~ r ,tÞ ¼ 12ð~ E ð~ r ,tÞeiK z ziot þ ~ E ð~ r ,tÞeiK z ziot Þ
ð2Þ
and is expanded in a Fourier series X 0 X y y ~ ~ ~ E ~q ðtÞeið~q ~r otÞ ¼ E ~q ðtÞei~q ~r : E ð~ r ,tÞ ¼
ð3Þ
~ q
~ q
In the absence of an external light field the electron states are eigenstates of an ideal periodic lattice. Deviations from this idealized periodicity due to lattice vibrations lead to a coupling of the different electronic states. This interaction is described by the carrier–phonon Hamiltonian. X HIcp ¼ C~q ½c~y ~ b~q c~k c~y b~yq c~k þ ~q d~y ~ b~q d~k þ d~y b~yq d~k þ ~q , ð4Þ k þq
~ k, ~ ku, ~ q
k
k þq
k
where C~q is the electron–phonon coupling constant for interaction with optical phonons, er ð1Þ and er ð0Þ are the relative static and optical dielectric constant, respectively, e0 is the absolute dielectric constant of the vacuum, ‘ oLO is the optical phonon energy and V is the normalization volume. The charged carriers interact via the Coulomb potential V~q and the Hamiltonian describing carrier–carrier interaction processes conserving the number of particles per band is given by: X 1 y y 1 HIcc ¼ V~q c~ c~ c~ku þ ~q c~k~q þ d~y d~y d~ku þ ~q d~k~q c~y dy ~ d~ku þ ~q c~k~q : k ku 2 k ku 2 k ku ~~ k, ku, ~ q
ð5Þ The carrier–carrier Hamiltonian can be separated into a mean field (Haretree–Fock) Hcc HF part and a remaining part depending on two-particle correlations Hcc corr. The effective single-particle Hamiltonian is Heff ¼H0 +Hcc HF. The correlation part of the carrier– carrier interaction Hamiltonian gives two phenomena: scattering processes between the carriers and the screening of the bare Coulomb interaction. The part of the perturbation Hamiltonian that yields impact ionization and its inverse process, Auger recombination is given by [3,4] X HIccðcvÞ ¼ ½Me ðqÞc~y ~ c~y ~ dy ~ c~k þ Me ðqÞc~y d~ku c~ku~q c~y ~ k þ q kuq ku
~ k,~ ku, ~ q
þ
X
k
½Mh ðqÞd~y
~ k,~ ku, ~ q
k þq
dy ~ cy ~ d~k þ Mh ðqÞd~y c~ku d~ku~q d~k þ ~q k þ~ q ~ kuq ku k
ð6Þ
Me ðqÞ ¼ V~q g~q , where V~q ¼ e2 =V e0 er q2 is the Coulomb potential and g~q is the interband-transition form factor.
73
functions of electrons and holes and for interband polarization. Being Wigner distributions these quantities are functions of space and momentum but there is a big difference of time scales between the momentum space and the real space dynamics. Scattering and dephasing processes lead to fast relaxation of the microscopic variables towards their local quasi-equilibrium values on a femtosecond time-scale while the spatial transport happens on a much slower time-scale (10 ps to ns). Because of a typical separation of time scales between the ~ k-space and ~ r -space dynamics, the influence of spatial gradients on the k-space dynamics is often negligible. However, some of the scattering terms in the equations of motion for the distribution functions conserve the density of carriers and therefore the density is not influenced by the fast relaxation processes and its spatial transport cannot be neglected. In the equation of motion for the polarization no conserved quantities exist and thus the spatial transport of polarization is usually not important. In principle, the complete set of equations required is, therefore, the Maxwell– Boltzmann–Bloch–Poisson equations for the nonequilibrium distribution functions f a ð~ k,~ r Þ, interband polarization pð~ k,~ r Þ, electric potential Fð~ r Þ, and the laser field ~ Eð~ r ,tÞ, with ~ k and ~ r being the two-dimensional (2D) vectors in reciprocal (momentum) space and real space, respectively. Keeping the first-order spatial derivatives of the distribution functions and neglecting any spatial transport of polarization, the equations of motion for electron and hole distribution functions are given by the generalized Boltzmann equations for two-band model including the coherent interband contributions. @ a~ 1 @ea ð~ k,~ r Þ @f a ð~ k,~ r ,tÞ 1 @ @f a ð~ k,~ r ,tÞ f ðk,~ ½dea ð~ r ,tÞ þ k,~ r Þ þ qFðrÞ @t @r @k ‘ @k ‘ @r
@ ¼ Ra ð~ k,~ r Þ þ f a ð~ k,~ r Þcol : @t
ð7Þ
The lowest order contribution to the polarization is included, where the spatial coordinate enters only as a parameter and locally the dynamics coincide with those of the inhomogeneous case and there are no transport effects. This lowest order picture is sufficient to describe pump-probe experiments in which filamentation is observed. @ ~ i pðk,~ r ,tÞ ¼ ½ee ð~ k,~ r ,tÞ þ eh ð~ k,~ r ,tÞpð~ k,~ r ,tÞ @t ‘ k,~ r Þ½f e ð~ k,~ r ,tÞ þf h ð~ k,~ r ,tÞ1 þ iOð~
@ ~ pðk,~ r Þcol @t
ea ð~ k,~ r Þ ¼ ea ð~ kÞ þ qa FðrÞ þ dea ð~ k,~ rÞ
ð8Þ
2 e ð~ kÞ ¼ ‘ k2 =2ma is the single-particle energy, X 1X s s dea ð~ k,~ r Þ ¼ f a ð~ k,~ r ÞV~ku þ ½V V~ku~k ~ k 2 ~ ~ku~k ~ e,h
ku
ku
is the renormalization of the single-particle carrier energy due to exchange interaction, V~qs is the screened Coulomb potential. The electrostatic potential due to the Hartree terms in the mean field Hamiltonian satisfies the Poisson equation. The generation rate in the Eq. (7) is given as follows: Ra ð~ k,~ r Þ ¼ i½Oð~ k,~ r Þp ð~ k,~ r ÞO ð~ k,~ r Þpð~ k,~ r Þ
ð9Þ
where Oð~ k,~ r Þ is the renormalized Rabi frequency defined by 3. Generalized Boltzmann–Bloch equations By using Heisenberg’s equations of motion, the equations of motion for the single-particle density matrices in Wigner representation can be derived. The effective single-particle Hamiltonian Heff gives a closed set of equations for the distribution
‘ Oð~ k,~ r Þ ¼ mð~ kÞ~ Eð~ r ,tÞ þ
X s pð~ ku,~ r ÞV~k : ~ ku
ð10Þ
~ ku
The second term in the above expression is the internal field responsible for Coulomb enhancement.
74
T. Apostolova / Nuclear Instruments and Methods in Physics Research A 653 (2011) 72–75
4. Scattering processes Within a semiclassical picture when scattering processes are described in terms of scattering rates, the scattering contributions to the equation of motion have the structure of the Boltzmann collision terms. The electron–phonon scattering rates are obtained from Fermi’s golden rule and quadratic or higher order terms and terms involving simultaneous electron–phonon and hole–phonon interaction have been neglected in the polarization and in the carrier–phonon Hamiltonian. Incoherent scattering processes appear for the first time in second-order contributions [1,3,5,6]. Collisional contributions in Eqs. (1) and (2) lead to relaxation in the carrier distributions and decay in the interband polarization: @ a~ @ @ @ f ðk,~ r Þcol ¼ f a ð~ k,~ r Þaa þ f a ð~ k,~ r Þeh þ f a ð~ k,~ r ÞLO @t @t @t @t
q
ð13Þ
kq , k
where the scattering matrices are given by 2p X X s 2 e,h Þ a a ee,h Þf a ð~ W~e,hð~aa jV~q j dðe~ ~ þ e~ku e~ku ku,~ rÞ ~ ~ ¼ þ~ q k
ku þ ~ q ,~ r Þ ½1f a ð~ ¼ W~e,hðLOÞ ~~ kq , k
2p
‘ þ
ð14Þ
jCqe,h j2 ðNq þ1Þdðe~e,h ~ e~e,h þ ‘ oq Þ kq
2p
‘
e
The Boltzmann scattering matrices
‘ oq Þ:
ð15Þ
W~e,h~ ~ kq , k
for electrons and holes
are real quantities and the scattering matrices W~p
k~ q ,~ k
in the
equation of motion for the polarization are complex and the real part is related to W~e,h~ ~ according to kq , k
e e ½1f e ð~ k~ q ,~ r Þ þW~k, f e ð~ k~ q ,~ rÞ ReðW~p ~ ~ Þ ¼ 12W~k ~ ~ q ,~ k k~ q kq , k
þ 12W~qh~k,~k ½1f h ð~ q ~ k,~ r Þ þWh ~k,~q ~k f h ð~ q ~ k,~ r Þ: The real part of
W~p ~ ~ kq , k
ð17Þ
@ e~ 2p X f ðk,~ r ÞehðrecÞ ¼ jV j2 g dðee 2e~ek~q e~hk2~q EG Þ @t ‘ ~ ~q ~q ~k q
k,~ r Þ ½f e ð~ k~ q ,~ r Þ2 f h ð~ k2~ q ,~ rÞ ½1f e ð~ 2p X 2 e e h jV~q j g~q dðe~k þ ~q 2e~k e~k~q EG Þ þ
‘
~ q
ð18Þ
g~q 2ðme =m0 Þ.
6. Numerical procedures in progress Based on typical length and time scales approximations are made with the aim of obtaining numerically tractable system of equations.The microscopic dynamics of the distribution function and the nonlinear polarization are governed by the equations of motion (1) and (2). A full kinetic treatment of the scattering processes will be performed by using, e.g. Monte Carlo simulations. Generalized Monte Carlo methods taking into account phase relations between different type of carriers (polarization effects), interaction of carriers with external coherent inhomogeneous electromagnetic field (generation effects), and the correlation and renormalization effects associated with carrier–carrier interaction can be utilized [7].
k
jCqe,h j2 Nq dð ~e,h~ ~e,h kq k
e
~ q
k,~ r Þ½1f e ð~ k~ q ,~ r Þ2 ½1f h ð~ k2~ q ,~ r Þ f ð~
with
k,~ r Þð1f e,h ð~ k~ q ,~ r ÞÞ W~e,h~ ~ f e,h ð~
kq
‘
e
ð12Þ
The first term on the RHS of Eq. (3) depicts the scattering processes arising from the correlation part of the carrier–carrier Hamiltonian and the third term arises from the carrier–phonon Hamiltonian X @ a~ f ðk,~ r Þaa=LO ¼ ½W~e,h~ ~ f e,h ð~ k~ q ,~ r Þð1f e,h ð~ k,~ r ÞÞ k, kq @t ~
a ¼ e,h ~ ku
q
½1f ð~ k,~ r Þ2 f e ð~ k þ~ q ,~ r Þ½1f h ð~ k~ q ,~ r Þ 2p X 2 e e jV~q j g~q dð2ej~k~q j e~k þ ehj~k2~q j þ EG Þ þ Neff e
k,~ r Þ2 ½1f e ð~ k þ~ q ,~ r Þf h ð~ k~ q ,~ rÞ ½f e ð~
q
‘
@ e~ 2p X f ðk,~ r ÞehðimpÞ ¼ Neff jV j2 g dð2e~ek eej~k þ ~q j þ ehj~k~q tj þEG Þ @t ‘ ~ ~q ~q
ð11Þ
X p @ ~ pðk,~ r Þcol ¼ ½W~ ~ ~ pð~ k~ q ,~ r ÞW~p ~ ~ pð~ k,~ r Þ: k, kq kq , k @t ~
kq , k
These contributions are derived using second-order perturbation theory such as Coulomb scattering [3] which is very important to conduction-electron dynamics and the change of electron density.
ð16Þ
describes scattering processes leading to a
dephasing of the polarization and the imaginary part describes second-order contributions to the band-gap renormalization.
5. Impact ionization and Auger recombination Since we interested in the processes of laser damage and filamentation in the semiconductor material, we include the ð@=@tÞf a ð~ k,~ r Þeh terms that lead to transitions between valence and conduction bands, i.e. impact ionization term and Auger recombination term [3]. Impact ionization and Auger recombination are second-order two-particle Coulomb scattering processes (proportional to Coulomb scattering). In a case when we have a homogeneous system (material) that is either homogeneously or inhomogeneously excited the Coulomb matrix elements depend on the momentum transfer ~ q only, i.e. pjV~q j2 .
7. Conclusion A microscopic quantum-kinetic theory based on density matrix formalism [8] is formulated to describe the processes of short pulse laser interaction with materials such as semiconductors accounting for arbitrary spatial inhomogeneities in the excitation conditions and other spatial phenomena such as filamentation of tightly focused femtosecond laser pulses, structural modification and catastrophic optical damage. A system of Boltzmann–Bloch transport equations are established that include both space and momentum dependence of the electron and hole distribution functions and the polarization. Microscopic electron–phonon and electron–electron scattering terms as well as scattering terms that lead to transitions between valence and conduction bands, i.e. impact ionization and recombination terms, are included explicitly in the equations. Since some of the scattering processes take place on the same time scale as the laser pulse duration, it is of importance to capture the transient relaxation of the excited carriers as well as the transient energy exchange between carriers and phonons. For this purpose explicit treatment of the scattering processes are considered instead of resorting to relaxation timed approximation. The formulated theory describes the spatio-temporal dynamics of electrons and holes including the coherent interactions of carriers and the laser light field as well as transport due to spatial gradients and electrostatic forces. The laser intensities and pulse durations for which our theory is applicable are IL ¼ 1 1015 21 1017 W=m2 , tL ¼ 100 fs.
T. Apostolova / Nuclear Instruments and Methods in Physics Research A 653 (2011) 72–75
References [1] O. Hess, T. Kuhn, Phys. Rev. A 54 (1996) 3347. [2] J. Li, C.Z. Ning, Phys. Rev. A 66 (2002) 023802. [3] T. Apostolova, D. Huang, P.M. Alsing, J. McIver, D.A. Cardimona, Phys. Rev. B 66 (2002) 075208.
[4] [5] [6] [7] [8]
W. Quade, E. Scholl, Phys. Rev. B 50 (1994) 7398. O. Hess, T. Kuhn, Prog. Quantum Electron. 20 (1996) 85. T. Kuhn, F. Rossi, Phys. Rev. B 46 (1992) 7496. S. Haas, F. Rossi, Phys. Rev. B 53 (1992) 12855. M. Lindberg, S.W. Koch, Phys. Rev. B 38 (1988).
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