Band structure of femtosecond-laser-pulse excited GaAs

0038-1098/94$6.00+.00 Pergamon Press Ltd

Solid State Communications, Vol. 89, No. 2, pp. 119-122, 1994. Printed in Great Britain.

BAND

STRUCTURE

OF FEMTOSECOND-LASER-PULSE

EXCITED

GaAs

Don H. Kim, H. Ehrenreich and E. Runge Physics Department and Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 (Received 25 October 1993 by M. Cardona)

A major contribution to the change in the band structure of highly excited GaAs is shown to be electronic in origin and due to the change in the screened pseudopotential and exchange-correlation effects. For 10% electron excitation, the fundamental gap, which becomes indirect as in GaP, vanishes, and the bonding-antibonding gap at X is reduced by almost one half.

changes in excited semiconductors have been concerned mainly with exchange-correlation effects, assuming that the Hartree term and screened ionic potential contribution do not change appreciably. For the high carrier densities of interest here, enhanced screening of the effective ion potential produces very substantial effects. By contrast, changes of the Hartree term are expected to remain small as the contribution of the newly created electrons is largely canceled by that of the holes. They will be neglected here. The self-energy correction due to exchangecorrelation effects are treated within the framework of earlier work on BGR’Ol Il. Because of the limited achievable accuracy of local pseudopotentids and in order to best present the physics associated with screening of the effective ion potential, a rather simple model for the dielectric function will be used. The form factors w(G) entering a pseudo-potential band structure can be visualized as a bare ionic potential screened by an appropriate dielectric function e(G)12. The change of c(G) is assumed responsible for the modification of the pseudopotentials in an electronically highly excited material. Thus we replace v(G) by

Recent femtosecond laser melting experiments’ suggest that the bonding-antibonding gap of GaAs shrinks bv more than 50% in less than 300 fs for fluences of the These results have been explained in o;der 1 kJ/m’. terms of the structural transformation resulting from lattice instability ‘. However, we believe that certain aspects of these experiments, especially in the short pump-probe time delay regime when the lattice remains substantially ordered, should be interpreted in terms of band structure changes due to electronic excitations induced by the intense femtosecond laser field. This electronic state will be shown to result from (1) substantial modifications of the pseudopotential and (2) band gap renormalization (BGR), resulting from the generation of an electron-hole plasma. The first effect is described in terms of the changes in the screening of a bare local pseudopotential utilizing a simple model dielectric function. The second effect has been investigated extensively in the past2 for carrier densities much lower ( 1016 - 10” cm-“) than the carrier density produced by intense femtosecond laser pulses (- 10 2 cms3). The band structure of the highly excited state is calculated by using only well known ground state properties assuming perfect tetrahedral order. Purely electronic mechanisms will be shown to account for most of the gap shrinkage. Lattice distortions are involved as well, particularly in the continuing gap decrease for pump-probe delay times of 1 ps and longer, and, without doubt in the vanishing of second harmonic generation even earlier3T 4. Initially these changes manifest themselves in modified phonon properties, such as soft TA phonons51 ‘1 7, and finally in structural rearrangements. The electronic structure of a highly excited semiconductor is best discussed in terms of three separate contributions to the quasi-particle energy of electrons in crystals: the screened ionic potential, the Hartree term. and the exchange-correlation term’. These contributions are diagramaticallv represented in Fie. 1. Vertex corrections to The exchange-cbrrelation teri are neglected. This leaves the screened exchange (sx) and Coulomb hole (ch) contributions8y ‘. Previous treatments of band structure

w*(G) =

w(G)[e(G)/c*(G)] .

(1)

Here e(q) is the ground-state dielectric function of GaAs, q = G, a reciprocal lattice vector, and e*(q) is the dielectric function of the laser-excited GaAs. (The asterisk(*) will be used to refer to excited state properties throughout.) Because the plasma is very dense we assume an approximate zero-temperature formalism to be reasonable. Other subtle effects such as that of bond charges on pseudopotential form factors 13yl4 are ignored in evaluating v*(G), even though they may be included in the v(G) of Eq. (1). For e(q) we use a numerically calculated dielectric function based on the two band Penn model15, thus replacing, for the evaluation of the dielectric function only, the complicated semiconductor band structures by two spherically symmetric bands separated by an average gap Eg. This gap is associated with the bonding-antibonding gap at X in k space16. With Eg = 4.4eV the numeri119

120

FEMTOSECOND-LASER-PULSE

a)

b)

C. In practice, about three or four iterations are sufficient to reach self-consistency. The excited state band structure calculated using Walter-Cohen pseudopotential form factors 14 screened self-consistently by a 10% excited carrier distribution is shown in Fig. 2. This carrier density n is typical of the fetrltosecond laser experiments under discussion. The gap at X, Ex, is seen to shrink by almost 1 eV. This large effect can be inferred from the perturbation theory result 17, 18

c)

Figure 1: Diagrammatic representation of the quasiparticle energy as a sum of a) the screened ion potential, b) the Hartree term and c) the exchange correlation contribution neglecting vertex corrections.

cally evaluated e(q) for GaAs is in remarkable agreement with the results of Walter and Cohen 14 calculated from empirical pseudopotentials. Both dielectric functions are significantly different from Penn's approximate analytical expression 15 which exhibits a maximum at small but finite q. A dielectric function similar to the one used by Biswas and Ambegaokar6 in their calculation of the phonon spectrum of electronically excited silicon has been used for C(q). They use a weighted average of the free electron-like polarization of excited electrons and holes and the semiconductor-like polarization of the remaining valence electrons:

e* (q,n, Eg * ) •

=

l+2[ey(q;n)-l]+[e(q)-l] n ° - n

. (2)

no

Here n is the density of e-h pairs; no is the total valence electron density; e(q) is the dielectric function for the Penn model and es,(q; n) is the Lindhard dielectric function. The factor of 2 accounts for the contributions of both electrons and holes. With this choice of e*, v* in Eq. (1) correctly reduces to v for n ~ 0. The excited state dielectric function C(q) still depends on the average excited state gap E~ through e(q). E~ must be determined self-consistently by requiring that the e*(G; n, E~) involved in v*(G) leads to a band structure having the same X-gap E$ as that used to determine 5.0

Vol. 89, No. 2

. . . . . . . . -.---'". . . . . . - - - ' ~

(vS(111))2 + ( v A ( l l l ) ) 2] Ex = 2 vS(220) + - ~ - _ - ~ 0 ~ 1 ) ) / - - ~ m j

,

(3)

where v s and v a are symmetric and anti-symmetric form factors respectively. Since vS(220) is very small compared to the second term, the gap at X of the excited system, according to Eq. (1), is approximately E ] [ e ( l l l ) / C ( l l l ) ] 2 Ex. The present calculations for excited GaAs yield E~c/Ex = 0.77, whereas [ e ( l l l ) / C ( l l l ) ] 2 = 0.72, in good agreement. The gap changes at F and L are seen to be much smaller. These gaps are determined mainly by form factors having larger G. For these G, e(G)/e*(G) is far closer to unity. We now turn to the self-energy effects shown by the diagrams of Fig. lb and lc. When the concentration of excited carriers is smaller than about 1020 cm -3 the excited holes and electrons reside close to the band edge. The parts of the band structure occupied by excited carriers occupy small regions of the Brillouin zone (BZ) having well defined effective masses. In the case of carrier concentration on the order of 1022 cm -3 the carrier occupation extends over most of the BZ. Since the final result will be seen to be relatively insensitive to the effective mass, we shall describe the occupied states as degenerate multiple valleys with degeneracy factors v~ and Vh for electrons and holes respectively. The renormalized energy of the i-th band is given by e i k = ~ik + E,(k,elk)

,

(4)

where elk is the unperturbed energy of the i-th band and the self-energy Ei(k, e l k ) = E~=(k,e/k) + ~ch(k,e/k) is a sum of screened exchange (sx) and coulomb hole (ch) many particle contributions. Within the RPA and the parabolic band approximation 10, 8 E~=(k,e/k) = - ~ Vs(q, elk -- elk+q)f/(elk+q)

,

(5)

q

~ m

and

0.0 ~

5"].~h(k, e i k ) =

Vs(q, hw + i~) eik

hw

l

-5.0

(6)

I/~

Here V(q) = 4re2/%q 2 and Vs(q, hw) = V(q)/e(q,w) are the bare and screened interaction, eo is the dielectric constant of the material, and fi(eik) and g(¢) are Fermi- and Bose occupation functions, respectively. A simple expression for ~i(k, elk ) is obtained by using the single-plasmon-pole approximation 8' 19.

pseudopotential n 1 8.10

cm

band structure

-10.0

k

F

X

,-X(q,w+i~) = 1 +w~z[(w+i6) 2 -w~2] -1 Figure 2: Excited state band structure for GaAs with n = 10% excited e-h pairs, based on Walter-Cohen form factors screened self-consistently as described in the text (solid line). Also plotted is the Walter-Cohen band structure of the ground state (dashed line).

,

(7)

where wp = (4~re2/,o) 1/2 Ei(nl/rni) t/2 is the plasma frequency. With the choice %2 = w~ + ~q2/~2, Eq. (7) satisfies requisite sum rules. The inverse screening length = l(4re~/,o)E~0m/0ml '/2 reduces at zero tempera-

Table I.

Dependence of self-energy (in eV) on degeneracy v~ of conduction band valleys; me = 0.4m and mh = m are used. Table II.

Dependence of self-energy (in eV) on me

for a model with a single representative conduction band minimum; mh = m is assumed.

0.5m 1.0m 1.bm

-0.40 -0.44 -0.49

-0.62 -0.63 -0.65

-0.52 -0.55 -0.58

8 3 1

ture to the Thomas-Fermi value given by ~ F = (4e2/ rh2eo) 7~i mikFi with kFi = (3~r2ni)l/3; /gl is the quasiFermi level. This approximation, together with the quasistatic approximation (eik+q -- e i k = 0) 10' 20, yields E~h(k, e i k ) = --e2~/2eo

,

(8)

which is independent of k and the band index i, and ~ = ( k = 0) = -(2e~/Tr~o)[k~ - ~tan-l(kF~/~)]

, (9)

at the band edge assuming fi = O(kf. -- ]k]). E~=(k) falls off rather rapidly as k approaches the unoccupied states between valleys. Thus contributions related to different band minima add up independently to good approximation. The quasistatic approximation yields results only weakly dependent on band structure details. The results (8) and (9) depend only on t¢ and kf,. The band structure dependence appears in the masses mi's and total degeneracies v~ and t'h of conduction band and valence band maxima, respectively. The uppermost heavy va,'",

n=O

3.0 2.0 >

(D

LU

121

FEMTOSECOND-LASER-PULSE

Vol. 89, No. 2

1.0 --~,.,. 0.0 ~ ' , , , ,

incl. BGR

" "~'~ .... n=l.8,10~.cm -3

-0.49 -0.43 -0.39

-0.52 -0.51 -0.60

-0.59 -0.54 -0.52

lence bands are approximately degenerate (uh = 2), are close to parabolic for n = 10% band filling, and have mh ~ m, as is to be expected. The light valence band is neglected. The multiple conduction valley schemes 2 for calculating E have shortcomings because individual electron valleys are filled and connected to other valleys for the present electron densities. We have therefore performed several calculations that assume (1) a single representative conduction band minimum for different density of states masses rne and (2) multiple independent valleys with m~ = 0.4m for different u~. The results are shown in Tables I and II respectively. The total self-energy is seen to be relatively insensitive to the degeneracy of the valleys within the parabolic band approximation as has been previously pointed out by Vashishta and Kalia21 for the lower carrier density regime. The dot-dashed curve in Fig. 3 shows the total energy shift obtained when the pseudopotential and self-energy shifts are added for the conduction valley model, consisting of three equivalent minima centered at X with mass me = 0.4m and n = 10%. The band filling corresponds to a quasi Fermi energy of about 3 eV. The independent Coulomb hole contribution is about 0.43 eV per band. The self-energy shifts contract the valenceconduction band gap by about 0.9 eV along the entire F - X direction. Since the conduction band minimum at X lies slightly below the valence band maximum at F for 10% valence band excitation, GaAs has become a metal. The bonding-antibonding gap has decreased from 4.4 eV to 2.4 eV, in at least partial accord with the requirements set by the experiments of Ref. 1.

-1.0 -2.0

F

X

Figure 3: Excited state band structure of GaAs for n = 10% carrier concentration calculated from screened pseudopotentials (solid line) and including self-energy corrections (dash-dotted line). Walter-Cohen band structure given for comparison (dashed line). Only the lowermost conduction band and the uppermost valence band are shown. Self-energy corrections are calculated with degeneracies u¢ = 3, Uh = 2 and masses me = 0.4m, mh = 1.Ore.

Acknowledgment - We thank E. Mazur, Y. Siegal and E. Glezer for stimulating this work. We are grateful to them for making the results of Ref. 1 available prior to publication. We also thank N. Bloembergen and P.M. Young for helpful discussions. D.H.K. thanks the Harvard Materials Research Laboratory, Contract No. DMR89-20490, for a grant under the Research Experience for Undergraduates Program during the summer of 1993 and M. Cardona and C.H. Grein for the pseudopotential program. This work was additionally supported by the U.S. Joint Services Electronic Program (JSEP) through the U.S. Office of Naval Research (ONR) Contract No. N00014-89-J-1023 and by the U.S. Advanced Research Projects Agency (ARPA) through ONR Contract No. N00014-93-1-0549.

References [1] Y. Siegal, E. Glezer, and E. Mazur, preprint and private communication.

[3] H.W.K. Tom, G.D. Aumiller and C.H. Brito-Cruz, Phys. Rev. Lett. 60, 1438 (1988).

[2] H. Kalt and M. Rinker, Phys. Rcv. B 45, 1139 (1992), and references therein,

[4] P. Saeta, J.-K. Wang, Y. Siegal, N. Bloembergen and E. Mazur, Phys. Rev. Lett. 67, 1023 (1991).

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[5] J.A. Van Vechten, R. Tsu, and F.W. Saris, Phys. Left. 74A, 422 (1979). [6] R.Biswas and V. Ambegaokar, Phys. Rev. B 26, 1980 (1982). [7] P. Stampfli and K.H. Bennemann, Phys. Rev. B 42, 7163 (1991). [8] L. Hedin and S. Lundqvist, in Solid State Physics, (F. Seitz, D. Turnbull and H. Ehrenreich, Eds.) Vol. 23, p. 1. Academic Press, New York (1969). [9] L. Hedin, Phys. Rev. 139, A796 (1965). [10] H. Hang and S. Schmitt-Rink, Prog. Quant. Electron. 9, 3 (1984); R. Zimmermann, Many-Particle Theory of Highly Excited Semiconductors. Teubner, Leipzig (1988). [11] A. Selloni, S. Modesti, and M. Capizzi, Phys. Rev. B 30, 821 (1984). [12] M.L. Cohen and V. Heine, in Solid State Physics, (F. Seitz, D. Turnbull and H. Ehrenreich, Eds.) Vol. 24, p. 38. Academic Press, New York (1970).

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[13] J.C. Phillips, Phys. Rev. 166, 832 (1968). [14] J.P. Walter and M.L. Cohen, Phys. Rev. B 2, 1821 (1970). [15] D.R. Penn, Phys. Rev. 128, 2093 (1962); G. Srinivasan, Phys. Rev. 178, 1244 (1969); see also Ref. 17.

[16] V. Heine and R.O. Jones, J. Phys. C 2, 719 (1969). [17] M. Cardona in Atomic Structure and Properties of Solids (E. Burstein, Ed.) p. 514. Academic Press, New York (1972). [18] V. Heine and D. Weaire, in Solid State Physics, (F. Se]tz, D. Turnbull and H. Ehrenreich, Eds.) Vol. 24, p. 249. Academic Press, New York (1970). [19] B.I. Lundqvist, Phys. Kondens. Mat. 6, 193 (1967); ibid. 6, 206 (1967); A.W. Overhauser, Phys. Rev. B 3, 1888 (1971). [20] R. Zimmermann, M. Roesler, and V.M. Asnin, phys. ' star. sol. (b) 107, 579 (1981). [21] P. Vashishta and R.K. Kalia, Phys. Rev. B 25, 6492 (1982).