Photon-induced localization in optically absorbing materials

15 March 1999 PHYSICS

LETTERS

A

Physics Letters A 253 ( 1999) 93-97

ELSEWIER

Photon-induced localization in optically absorbing materials L. Mantese a,‘, K.A. Bell ‘, D.E. Aspnes ‘, U. Rossow b a Department

of Physics. h Institut fir

North

Carolina

Physik,

Stute University,

TlJ-llmenurr.

D-98484

Raleigh, Ilmenuu.

NC 27695-8202,

USA

Germany

Received 25 November 1998; accepted for publication IS December Communicated by V.M. Agranovich

1998

Abstract We show that components of surface- and interface-related optical spectra that are related to derivatives of their bulk dielectric functions are due to a dynamic photon-induced localization of the initial and final states. Localization is described by correlation effects that arise from the finite penetration depth of light in optically absorbing materials, and lead to a substantially different perspective of optical absorption than that given by conventional theory. @ 1999 Elsevier Science

B.V. PACS:

73; 73.20.r;

Keywords;

73.20.At Localization; Surface optical absorption

The accurate calculation and interpretation of optical spectra related to surfaces and interfaces remains a difficult and challenging theoretical goal. General formulations of the problem are available [ l-31, but these require extensive numerical computation and thus are not conducive toward elucidating basic mechanisms. This is of concern since a number of optical spectroscopies, including for example reflectancedifference (-anisotropy) spectroscopy (RDS/RAS) [ 4,5] and second harmonic generation (SHG) [ 6-81, are now used extensively to probe the dielectric responses of surfaces and interfaces. While agreement between theory and experiment has improved, the full diagnostic power of these techniques will not be realized until a more complete understanding of the origin of these spectra has been achieved.

’ Currently at Department of Physics, at Austin, Austin, Texas 78712, USA.

The University of Texas

Here, we consider components of surface- and interface-related spectra that are related to energy derivatives of the bulk dielectric function E/,, and to apparent values of bulk critical point energies E,, and broadening parameters r where differences AE, and AT of the order of a few meV have been measured relative to values obtained from reference spectra. Derivative components are common, appearing for example in RD spectra of hydrogen- and oxygenterminated vicinal (001 )Si surfaces [9-l 1 ] and in differences among ellipsometrically determined spectra for crystalline (c-) Si [ 121. Relatively large values of AE, have been reported in SHG [6,8] and modulation spectroscopy data [ 131. We have previously noted [ 11,141 that such differences are inconsistent with conventional theory, which ignores final-state correlations [ 1.51, since the energies of the Bloch functions from which Eg originates are determined by their semi-infinite extent into the bulk and hence

0375-9601/99/$ see front matter @ 1999 Elsevier Science B.V. All rights reserved PIf SO37S-9601(98)00953-O

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et trl. /Phvsics

cannot be influenced by surface effects. Therefore, we concluded that the observation of such effects is direct evidence that the excited valence/conduction band states are localized. Further, since spatial attenuation of a Bloch function is not possible in the static case, this localization must be a dynamic effect arising from the finite penetration depth of the photon [ I I 1. The present work provides a quantitative approach to address these ideas and yields an unconventional perspective of optical absorption. Specifically, we show directly from first-order time-dependent perturbation theory that the absence of translational invariance normal to the surface within the material in which the photon is absorbed gives rise to ( 1 ) final-state localization, such that at the beginning of excitation the envelopes of the excited state wave packets match that of the electric field of the photon, as expected from energy-transfer arguments, and (2) nonlocal polarization, since such packets must propagate with a group velocity ~1~~ = dw( k)/dk normal to the surface. This evolution continues until either the excitation fills the entire crystal or the excited state scatters. The fact that we observe wave packet behavior indicates finite lifetime effects, which will be shown to enter through constraints placed on 11,~.(3) Since the energy of a wave packet is its expectation value of the Hamiltonian, E,, becomes a weighted average of bulk and surface potentials. Thus, differences in apparent values of E,? due to surfaces and interfaces are immediately explained. An example of derivative components in a surfacerelated optical spectrum is given in Fig. I [ I I 1. The solid line is RD data obtained on a ( 1 13)Si surface prepared by a standard RCA clean with no tinal HF dip, leaving the surface terminated by a suboxide. These data show structure near the (E& El ) complex and the E2 critical point energies of c-Si at 3.4 and 4.2 eV, respectively. The dashed lines were calculated assuming that the spectra arise from small apparent differences AE,q and AT in the E,c and r of the individual critical points. The (E& El) and E2 structures were fit separately with a combination of firstand second-energy derivatives of oh as described elsewhere ( I I 1. Since RDS measures the difference between normal-incidence rehectances of light linearly polarized along the two principal axes in the surface plane and since these data were obtained at 10 ms intcrvals on a single chemically etched surface, these

Letters A 253 (1999)

93-97

(113)

Si

I-

__ t

v

RD data

..........Model

calculations

’

I

I

-1’

2

3

5

4 E (ev)

Fig.

I, Solid

ide-rerminated tions

line:

( I 13)Si

using first-

described

normal

in Ref.

incidence

surface.

and second-energy

I I1 I.

The

RD spectrum

Dashed line: best-fit derivative

(E:,,El )

complex

of a suboxrepresenta-

spectra of E,, as and El

regions

were fit separately.

derivatives cannot be due to differences in instrumentation, sample preparation, temperature, and/or bulk strain. Instead, they must originate from a fundamental difference among apparent bulk critical point energies that depends on the direction of the polarization of the light used to obtain these data and on the surface termination of the bulk lattice. The existence of different apparent values of E, for the same bulk critical point is direct evidence of localization, as we now discuss. First, although these spectra are surface-related, the derivative-type structures occur at bulk critical point energies, implying that the relevant states are strongly connected to the bulk yet are somehow affected by the surface. We begin by examining whether a surface potential can modify the energies of semi-infinite states in the static case. To estimate the surface effect on the energy of such states we consider a one-dimensional model consisting of an infinitely deep quantum well from -a < i: < a with a S-function potential, L’,dS( z), at the center. In practice a is of the order of laboratory dimensions and d of Angstroms. Thus this model, which simulates two back-to-back crystals each of thickness a and surface potential L’,d8( z ) /2, retains the essential physics for an energy calculation and can be solved analytically. If v, = 0 the normalized wave functions are 5 cos( E z ) and -& sin( $$z ) for tz odd and even, respectively. For v, $ 0 the sine functions are unaffected, while the cosines develop a first-derivative dis-

L. Muntese et al. /Physics

Letters A 253 (1999) 93-97

continuity (cusp) at z = 0, a general characteristic of one-dimensional Green-function problems of this type. This change of slope is equivalent to a change of the effective width of the well. To lowest order in d/a the cosine energies become E,, =

n2r21i2

Vd

where mp is the free electron mass. Thus as a + co the effect of the surface on the cosine energies also vanishes. Alternatively, if energy differences are observed then a must be finite, i.e., the wave functions must be localized even if the surface potential is too weak to create a bound state [ 111. The question then arises as to how this localization occurs. Since the unperturbed Hamiltonian contains no mechanism for localization, we consider the photon perturbation H’( r, t), which in absorbing materials introduces a characteristic length L = 1 /K = 1/2a due to the finite penetration depth of light. Here, K and (Y are the extinction and absorption coefficients, respectively. To illustrate the physics of localization through an analytic solution we again suppose an infinitely deep quantum well extending from -a < i 5 a, and let H’(r,r) = -qExLe-“/“I cos(wr)u(t). Here, H’(r, t) represents a wave of field amplitude E, linear polarization n, and penetration depth L propagating along i and arriving at the surface at t = 0, as indicated by the unit step function, u(t). Our initial unperturbed wave function then takes the form t&,(r, t) = pn,(x, y) 9 e-i(w~~+w~lr, where pPn,( X, _v) is a transverse Bloch function of energy E,,, = fiiw,,,, k,, = nn/a, and tiw,, = ti2ki/2m,. This state evolves in time under the action of the photon field as

cos(knz)

_ e-i(w.,,,+w)f

X

[

~dll’

-

w,,,,, -

0

(1)

s,,+YY 2

= cprr,(X> .Y)

e-iwv,t,,,l

9.5

where w,,,, = o,,, + w,, and wIII~I1’ = w,,,t + w,,~ are the eigenenergies of the initial and final states, respectively, Ak = k,,! - k,,, and x,!,/,, = ((P~,~(x.Y)IxI~,~,(x,~)). For t -+ 0 the envelope of the final-state wave packet maps identically onto the electric field of the photon that created it or alternatively the probability amplitude [A$[’ of the final state tracks the local intensity I z [El’ as required by energy conservation. An analytic expression for A#,,,,, for reasonably large t can be obtained by the following procedure. We interpret Eq. (3) in the effective mass approximation, supposing that mn and m’n’ refer to valence and conduction bands, respectively, and that o,,,J,,I - w,,,,, is finite even when k,! = k, (vertical transitions in the reduced-zone scheme). We next suppose the two-band model, thereby eliminating the sum over m’. We then convert the sum to an integral, linearize the final-state band structure as w,,, (k) = O,,JO+ rl,Ak, where W,,IOis the conduction band energy for k,! = k,,, define an energy difference Aw by w,,,I,,I - w,,,, - w = Ao + L!,Ak and perform a standard contour integration. Considering for simplicity energy-conserving transitions Aw = 0 only, the second non-resonant term in Eq. (3) can be discarded, and the part of A@,,,,, relevant for absorption becomes

e-i(w,,+w,,)r

J;;

where f(

i,

t)

=

]

_

ewK17xlcosh(rcz)

=e --K’7’sinh( Klz,l) Using first-order time-dependent perturbation theory and assuming that L < a, the coefficients ai:’ are given by

for 1~1 < /:,J, for Ii1 > Iz,~I,

where r+ = dw/ak is the group velocity of the packet and zR = u,t is the distance that a point on the packet travels in time t. Thus the excited state takes the form

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L. Mantese et ul. /Physic.~ Levers A 253 (1999) 93-97

of the resorzant wave function multiplied by an amplitude modulation factor, which represents the facts that it ( I ) is launched at t = 0 by the photon; (2) propagates at a group velocity uK, and (3) is continuously renewed by the photon until the state scatters. For L + 0, f(: ) reduces to a unit-step function whose leading edge propagates with speed uK. For L ----t00, f(; ) becomes independent of Z, reducing to the standard situation of uniform absorption. Obviously, the price paid for creating a localized state is that it must propagate. Other aspects can also be noted. First, the spread Ak of wave vectors about k,, needed to construct the wave packet Ati,,,,, originates from momentum conservation for a single-frequency wave train of finite penetration depth. Second, in the absence of scattering the packet would evolve to fill the entire well and therefore be described by a single component k, at a single frequency o,,, reducing to the standard result. However, this situation is never realized in practice since the first scattering event occurs on the average at a time r = /i/T, placing a limit on the distance that the front can extend and therefore on the overall width of the packet in frequency space. Third, broadening appears as a natural consequence of energy conservation for a packet with a finite lifetime (integration over Aw # 0, not done here) and does not need to be introduced as an ad hoc turn-on parameter for the photon. Fourth, the existence of propagation implies that the induced polarization is strictly local only for vanishingly small times, when the amplitude of A@,,,,,follows that of the photon field. Whether the inherent nonlocality is significant for a particular problem depends on the distance li~:,~/f relative to the actual spatial extent L of the source. Finally, and most important to our present purposes, the energy of A#,1,,, is not tiw,,, that of the resonant component, but rather is given by the expectation value

(5) and thus is a weighted average of bulk and surface potentials that depends on the overlap between the potentials and Agrttn. Consequently, the apparent energy and lifetime of the final state will depend on various factors including, through selection rules, the polarization state of light inducing the transitions. Thus the appearance of derivative lineshapes in RD data can be understood.

The above discussion has been given entirely in terms of the final conduction state generated from the initial unlocalized valence state +$‘~,, (r, t). In fact, the creation of a localized excited electron implies the simultaneous creation of a correspondingly localized excited hole state that propagates with the group velocity of the hole packet. Thus an expression equivalent to Eq. (4) exists for the initial hole state as well. Also, the use of actual Bloch rather than cosine standing waves results in packets that attempt to propagate our of the solid as well as in, thereby describing photoemission. Further, since in the present formalism propagation is a direct consequence of excitation, two of the three steps in the so-called three-step model of photoemission are seen to be aspects of the same process. We now consider estimates appropriate to the (EA, El ) complex of c-Si. Here, the penetration depth of the photon field, as opposed to the intensity, is about 200& From band structure calculations we estimate that for excitations along the [ 11 l] directions, and a surface potential of 1 eV, u,? = dw/dk, z I eV/( fifirr/a,,), where a, = 5.43 8, is the lattice constant. Given a broadening parameter of about 40 meV for the transition, we find therefore that us7 x 25 A. Therefore, the local assumption is justified in this case and we estimate the effect of the surface potential on the energy of Al//,,,, as being of the order of I eV * 4/200 g 20 meV. The difference between final states excited with different polarizations is expected to be a fraction of that, appropriate with the results shown in Fig. 1. Other implications may also be noted. First, these effects will increase as the penetration depth of light decreases, consistent with the larger derivative signals obtained for the E2 transition and also for the highly excited material involved in SHG ]6-81. Second, slab calculations [ 31, which incorporate spatial confinement as an unwelcome consequence of limited computing power, may actually provide a more accurate representation of surface-related phenomena than calculations that take the entire bulk into account. Such calculations may be simplified by representing the slabs with an exponential attenuation factor normal to the surface. Third, it may be necessary to treat optical absorption in strongly absorbing materials as an inherently nonlocal phenomenon, since in principle the polarization can propagate an appreciable frac-

L. Mantese

et 01. /Physics

tion of the penetration depth of light. Finally, we have discussed only energy consequences and have ignored contributions such as surface-amplitude effects which have recently received considerable attention [ 16,171. These arise by imposing the true surface potential and boundary conditions on the exact wave functions and result in a more complicated spatial profile for optical absorption near the surface than a simple exponential, one that would have to be obtained self-consistently. However, a spatial dependence would still exist, and therefore localization and propagation would still be aspects of this more general solution. The details are beyond the scope of the present paper and will be discussed elsewhere. It is a pleasure to acknowledge support of this work by the Office of Naval Research under contract No. N00014-93- l-0255, and of further support of U.R. by the Alexander von Humbolt Foundation.

References [ II P.J. Feibelman, Phys. Rev. B 12 (1975) 1319. (2 1 P.J. Feibelman, Phys. Rev. B 14 ( 1976) 762. 13 1 F. Manghi, R. Del Sole, A. Selloni. E. Molinari, B 41 (1990) 9935.

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141 D.E. Aspnes, J.P. Harbison. A.A. Studna, L.T. Florez, J. Vat. Sci. Technol. A 6 (1988) 1327. [SJ T. Yasuda. K. Kimura, S. Miwa. L.H. Kuo, A. Ohtake, C.G. Jin, K. Tanaka, T. Yao, J. Vat. Sci. Technol. B I5 ( 1997) 1212. 161 W. Daum, H.-J. Krause, U. Reichel. H. Ibach, Phys Rev. Lett. 71 (1993) 1234. (71 J.I. Dadap, B. Doris, Q, Deng. M.C. Downer, J.K. Lowell, A.C. Diebold. Appl. Phys. Lett. 64 ( 1994) 2139. 181 2. Xu, X.F. Hu, D. Lim. J.G. Ekerdt. M.C. Downer. J. Vat. Sci. Technol. B 15 (1997) 1059. 191 L. Mantese. U. Rossow. D.E. Aspnes, Appl. Surf. Sci. 107 (1996) 35. I101 U. Rossow, L. Mantese, D.E. Aspnes. J. Vat. Sci. Technol. B 14 (1996) 3070. IllI L. Mantese. K.A. Bell, U. Rossow. D.E. Aspnes, J. Vat. Sci. Technol. B 15 (1997) 1196. 1121 K.A. Bell, L. Mantese. U. Rossow. D.E. Aspnes, J. Vat. Sci. Technol. B 15 (1997) 1205. 1131 J.T. Fitch, C.H. Bjorkman, G. Lucovsky. F.H. Pollak. X. Yin, J. Vat. Sci. Technol. B 7 (1989) 775. 1141 U. Rossow, L. Mantese, D.E. Aspnes, in: Proc. 23rd Int. Conference on the Physics of Semiconductors, M. Scheffler. R. Zimmerman, eds. (World Scientific. Singapore, 1996) p. 831. 1151See, for example, V.I. Gavrilenko. F. Bechstedt. Phys. Rev. B 54 (1996) 13416. It61 K. Uwai, N. Kobayashi, Phys. Rev. Lett. 78 (1997) 959. 1171 R.Eryigit, I.P. Herman, Phys. Rev. B 56 ( 1997) 9263; B 56 (1997) 9263.