Carrier dynamics-induced transient photoexcitation and energy

Carrier dynamics-induced transient photoexcitation and energy deposition in femtosecond-laser irradiated GaAs Tzveta Apostolova*a , Andrey Ionin b, Sergey Kudryashov b, Leonid Seleznev b , Dmitry Sinitsyn b a Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee Blvd, 1784 Sofia, Bulgaria b P.N. Lebedev Physics Institute, Russian Academy of Sciences, 53 Leninskiy prosp., Moscow, Russian Federation ABSTRACT In this work femtosecond laser photo-excitation of GaAs is studied numerically. The transient plasma densities photogenerated during the pumping IR fs-laser pulses were evaluated having in mind experimental data of time-resolved reflectivity measurements of transient bandgap shifts. Theoretical modeling employing quantum kinetic formalism based on a generalized Boltzmann-type equation, including one/multi-photon photo-excitation, Joule heating and free-carrier absorption, interband excitation, impact ionization, Auger recombination of electron-hole plasma, thermal exchange with the lattice, etc. is performed. For the first time the effect of enhancement of ionization by transient bandgap renormalization (BGR) is considered both experimentally and theoretically. The energy spectra of the electron distribution function and the time dependence of the electron density are calculated and the key role of BGR in the transient electron-hole plasma dynamics is pointed out. Keywords: femtosecond laser pulse, electron-hole plasma dynamics, transient band-gap renormalization

1. INTRODUCTION It has been shown [1] that at strong electronic excitation of semiconductors significant narrowing of their dielectric bandgap (up to 50% when about 5% of valence electrons are excited) occurs on a timescale of the electron hole plasma generation, which may result in dramatic increase of efficiencies for non-linear optical processes, especially under nearresonant conditions. The expectation is that ultrafast transient electronic narrowing of semiconductor bandgaps will cause strong abrupt ionization (optical breakdown) of the semiconductor material. Band-gap narrowing in semiconductors is due to the introduction of charge carriers is an important effect. Carriers may be introduced by optical excitation as in the case when electrons are excited by the laser pulse from the top of the valence band to the bottom of the conduction band in a direct-gap material such as GaAs, leaving holes behind. Intense shortpulsed laser irradiation of GaAs leads to the formation of high carrier (electron) densities in excess of 1021 cm-3. In semiconductors the laser light is absorbed by the electrons excited to the conduction band by the absorption of single or multiple photons. At the same time laser energy heats up the conduction electrons, producing electrons with very high kinetic energy. Some of the high energy electrons collide with the bound electrons leading to further ionization. Consequently the number density of conduction electrons together with the electron temperature increase dramatically. The deposited energy quickly, within femtoseconds is equilibrated among the electrons and more slowly, is transferred to the atomic vibrations. The optical properties of these highly excited semiconductors are very strongly influenced by many-particle effects. Thus the effective mass approximation is used in a theoretical treatment applicable when the carriers have relaxed to the extrema of the bands. The gap narrowing itself can be determined by calculating the quasiparticle self-energy in a screened-interaction (e.g., GW approximation [2,3]) perturbation theoretic treatment [4,5]. In all these works the dynamical interplay between photo-induced formation of electron-hole plasma and instantaneous change of optical and electronic properties of the photo-excited materials was not taken into account, i.e. how the plasma parameter (plasma density) influences the narrowing of the band-gap and vice versa. We use the theoretical framework offered in reference [6] to account for the renormalization of the bandgap with increasing excited electron density. The generation and transient behavior of the electron density in the conduction band are described by the theoretical model developed by us in reference [7]. XVIII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers, edited by Tanja Dreischuh, Petar A. Atanasov, Nikola V. Sabotinov, Proc. of SPIE Vol. 7751, 77511J · © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.880977 Proc. of SPIE Vol. 7751 77511J-1 Downloaded From: http://spiedigitallibrary.org/ on 10/16/2013 Terms of Use: http://spiedl.org/terms

In the present work we demonstrate the effect of bandgap narrowing on the electron plasma dynamics and vice versa. The cumulative transient electronic bandgap narrowing directly affects the one/multiphoton absorption, the impact ionization processes and the Auger recombination process and this affects the conduction electron density.

2. THEORETICAL FRAMEWORK 2.1 Photo-excited electrons in semiconductors irradiated by short laser pulses A quantum kinetic theory for laser-induced damage in semiconductors based on a generalized Boltzmann type equation is derived previously and modified in this paper to include free-carrier absorption. We consider a pulsed laser radiation incident on GaAs semiconductor at finite lattice temperature and with temperature dependent bandgap energy EG . The total Hamiltonian of the electron-phonon system exposed to a pulsed laser field in second quantization [7] is considered. Then the solution of the Schrodinger equation containing the vector potential for an electron in an intense laser field is found and from there the modified electron creation and annihilation operators are obtained. Using Fermi’s Golden Rule we obtain a dynamic equation by including only the electron phonon interaction within the first order perturbation theory.

) ∑ C (e qr.Ε(t )

∂ er 2π nk = 1 − nker ∂t h

(

[

(

2

v qλ

v qλ

r

m*Ω 2L

)

2

)

(

)(

× nker − qr N qvphλ δ Ekr − Ekr − qr − hωqvλ − hΩ L + nker + qr N qph + 1 δ Ekr − Ekr + qr + hωqvλ + hΩ L − nker

2π h

(

2 rr v C e q ∑ qλ .Ε(t ) m*Ω2L v qλ

[(1 − n )N δ (E e r r k +q

ph v qλ

r r k +q

)]

)

2

) (

)(

)(

− Ekr − hωqvλ − hΩ L + 1 − nker − qr N qvphλ + 1 δ Ekr − qr − Ekr + hωqvλ + hΩ L

)]

In the diffusive limit hω q 1 is included the rates of the above equation read:

τ 1 V (E , t ) = VT (E ) + σ (Ω L )Ε 02 L + AT ( E ) p σ (Ω L )Ε 02 L , 3 3 τ 2 D(E , t ) = DT (E ) + σ (Ω L )Ε 02 L E + VT ( E ) p σ (Ω L )Ε 02 L , where 3 3

σ c (Ω L ) = e 2τ e me* (1 + Ω 2Lτ p2 ) is the Drude ac-conductivity.

When free carrier absorption of photons is included the rates of the Fokker-Planck type equation are given by:

V (E , t ) = VT (E ) + VF (E ) , D(E , t ) = DT (E ) + DF (E ) and A(E , t ) = AT (E ) + AF (E ) , where for example

Proc. of SPIE Vol. 7751 77511J-2 Downloaded From: http://spiedigitallibrary.org/ on 10/16/2013 Terms of Use: http://spiedl.org/terms

VT (Ekr ) =

[

(

2π h

∑λ C λ

2

v q

v q

hωqvλ

) (

)(

× N qvphλ δ Ekr − Ekr − qr − hωqvλ − N qvphλ + 1 δ Ekr − Ekr + qr + hωqvλ

)]

(e qr.Εr (t ) m Ω ) × N (E − E )[δ (E − E − hω + hΩ ) + δ (E − E − hω − hΩ )] + (N + 1)(E − E )[δ (E − E + hω + hΩ ) + δ (E − E + hω − hΩ )]

VF =

2π h

ph v qλ

∑λ C λ

2

*

v q

v q

r k

k −q

k

ph v qλ

2

r r k −q

r k

k −q

k

2 L

v qλ

r k

L

r r k +q

v qλ

L

r r k −q

r k

v qλ

r r k +q

L

v qλ

L

Equation (1) contains analytical expressions for all the source and sink terms up to second order perturbation theory, including stimulated interband electron transitions due to a single or multiphoton absorption, impact ionization due to Coulomb interaction between electrons and holes, and non-radiative recombination due to a phonon-mediated interaction. The coefficients of equation (1) already derived for GaAs are detailed in [7]. The terms that are expected to be affected by the bandgap renormalization are the source terms - one photon absorption and impact as well as the sink term – non-radiative (Auger) recombination. These terms are given explicitly by the following expressions: one-photon absorption

S abs ∝

2

Fk ≈

⎡ 2π 2 Fk π 2 Fk ⎢ 2 h ⎢⎣ hΩ L − Eke − Ekh − EG + 4 Fk

(

e 2Ε02 L m0Ω L

)

⎤ ⎥ with 2 ⎥⎦

⎡⎛ m0 ⎞ EG (EG + Δ 0 ) ⎤ ⎢⎜⎜ * − 1⎟⎟ ⎥, ⎠ 2(EG + 2Δ 0 3) ⎦ ⎣⎝ me

impact ionization

= (1) 'Ic

(2)

+ E, t) - 2)fe(Ee t)

n3D\,/meEk /

EoEr(0))

mm0E e 3D

4mm0

\2

e2

e2 Ic)

(i

4EZ\

E)

)2

EoEr(0)

and non-radiative recombination


(rec)

- E)/2, t])2 - 142[fe(EZ, t)]2

(1) Rk

n3Dn3Dame

(2) Rk

:: n3Dn3Dame mm0EC0

/

1

- E)

2mm0EZC0

1

(

2mEZEc

4-2

2mn

)3/2

2hwLo

x

+E)/2]2 - (hwLo)2

3E\ / 2mh2 E ) x mkBT)

O(EZEc) 2

3/2

2hcoLo {[(Eff

21

+E)/2]2 - (hwLo) j

2.2 Band-gap renormalization In semiconductors an absorbed photon takes a valence electron from a bonding state to an anti-bonding state and the excited electron in the conduction band and the hole in the valence band are charged quasiparticles that can move freely through the crystal. The long-range Coulomb interaction and the mobility of the excitations render the optical properties of the semiconductor under intense excitation a real many-body problem. The strength of the Coulomb interaction is different for different semiconductors. In semiconductors with narrow energy band-gap, the Coulomb forces are screened strongly so that many-body effects are not so pronounced. Only with increasing gap, the strength of the Coulomb forces increases because of the reduced screening. In narrow-gap semiconductors a simplified picture is justified in which the effects of the Coulomb interaction between optically generated electrons and holes are not taken into account into the determination of optical response. In such a simplified description, the only difference with conventional linear response theory is that a finite concentration of electrons and holes is assumed that is maintained by the pumping action of the applied laser beam. A direct measure of the strength of the Coulomb forces is the binding energy of an individual electron-hole pair – an exciton. While it is small for narrow-gap semiconductors, it is large in wide-gap semiconductors. Since under strong optical excitation a finite concentration of electron-hole pairs is created, these charged and freely moving particles contribute rather efficiently to the screening of the Coulomb forces in addition to the background screening referred to above. Exchange and correlation effects lead to lowering of the single particle energies and therefore to a shrinkage of the band-gap. We use the many-body derivation given in ref [6] and the resulting dependence of the quasiparticle bandgap (for a direct gap) on the excited electron density and incorporate it into our calculation. The total number of valence electrons in the GaAs material is taken to be 1.8×1023 cm-3.

3. RESULTS AND DISCUSSION For GaAs semiconductor sample (all parameters given in [7]), pulse duration τ L = 100 fs we show in figures (1) and (2) comparison of the electron energy distribution functions as functions of the scaled electron energy Eke EG with and without the band-gap renormalization effect, at two times relative to the laser pulse and for peak intensity and peak laser intensity of I L = 1×1016 W m 2 .At scaled time t = −1.25 in figure (1) the effect of band-gap renormalization as could be expected is not seen at all. At t = 1 it can be seen in figure (2) that close to the edge of the conduction band the effect of the one-photon absorption term is enhanced and for higher energies the overall effect of the band-gap narrowing is manifested in a decrease of the peak of the electron distribution function. This might be explained with the fact that the non-radiative recombination sink term is more strongly affected by the band-gap narrowing effect than the source terms – one- photon absorption and impact ionization. The same tendency of course is confirmed by figure (3) where for reference we have also shown the behavior of conduction electron density when the bangap is reduced to 1.32eV. Vice versa, in figure (4) it is demonstrated how the dynamical formation of excited electrons in the conduction band influences the narrowing of the band-gap.


Fig. 1. Electron energy distribution function f ( Eke ) as a function of

the

scaled

electron

energy Eke EG at

scaled

time

Fig. 2. Electron energy distribution function f ( Eke ) as a function of the scaled electron energy Eke EG at scaled time t

τL =1

t τ L = −1.25

Fig. 3. Electron density as a function of t τ L for two different values of the laser intensity and the intensity profile

Fig. 4. Electron bandgap energy density as a function of t τ L for two different values of the laser intensity and the intensity profile

4. CONCLUSIONS For the first time the effect of enhancement of internal photo-ionization in GaAs due to transient cumulative electronic bandgap renormalization (BGR) is theoretically considered. However, the calculated energy spectra of the electron distribution function and the time dependence of the electron density indicate significant negative, rather than positive feedback of BGR in the transient electron-hole plasma dynamics because of strong BGR-induced enhancement of nonradiative recombination rate.


ACKNOWLEDGEMENTS Authors thank Inter-academic exchange program between Russian and Bulgarian Academies of Sciences for exchange grants. One of the authors (SIK) is thankful to Russian Foundation for Basic Research for partial support of this work (grant No. 10-08-00941-a). This effort (T.A.) is sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-08-1-3080.

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