in semiconductors due an excitonbiexciton additional boundary conditi - PDF document

1 in semiconductors due an exciton-biexcit
in semiconductors due an exciton-biexciton additional boundary conditions and K University, Moscow September 1990) Zh. Eksp. Teor. Fiz. 99,1579-1 optical manifestations optical Stark ( 1991) ] investigated. This pre- consistent microscopic theory in Set. 4. optical Stark biexciton interaction to the dynamic shift level in polari- associated modification spectra of semiconductor. Such elementary excitations result in modification, in exciton resonance region, optical characteristics semiconductor subjected present paper deals with manifes- d t 2M ='/zoti~ (r, t) -2k'pk (r, t) Q(r, t) , =-~M~P~ (r, t) P (r, t) , (1~) 882 Sov. Phys. JETP 72 (5), May 1991 0038-5646/91/050882-10$03.00 @ American Institute and pump can therefore consider separately associated with the pump probe radiation, expression for which can follows i

2 n investigated case a constant equations
n investigated case a constant equations can which must equation with clear physical meaning: d w - + div S+9=0. the magnetic induction vector; energy density in 8 is governed by the parameters 4" energy dissipation With this tional polariton problem, 4n-E(d P/dt) for E found from Eq. ( macroscopic system, some transformations expression for probe radiation energy C S,, = - {[E,'H,]+ H.c.} , Ibn (gal fi $j =-- {[Pa' rot pol +Po* div P, +H.c.) , (8b) '* 4o$M fi Shies = - {[Q,' rot Q~]+Q~' div Q, +H.c.}, (8~) 8o$M where E, , H,, Po, and Qo probe radiation semiconductor in has three S,, the usual Poynting vector, Sex and biexciton Sbiex associated with the translational motion biexcitons, respectively. Characteristically, from the exciton Po +Q, and M+ which reflects only transverse exciton

3 s here) the divergence vanish in express
s here) the divergence vanish in expressions for Experimental studies may involve changes in probe electromagnetic radiation from semiconductor in scattering geometry for the normal incidence (investigated kn~wn~~~~' spatially nonlocal ~(p, w ) in this case related dispersion equation ( 1 ) consid- FIG. 1. Possible scattering geometry for changes in dispersion characteristics CdS probe radiation. the transmitted waves in normal incidence probe radiation on ered here sixth degree p and, therefore, there are generally six different branches of the solutions pi =pi (a) representing eigenwaves a polariton pump probe electromagnetic w different damped the same frequency inside dispersion equation nonphysical because they correspond to spatial faced with additional boundary problem where only the spatial excitons. One should stress here

4 pump-induced changes in elementary excit
pump-induced changes in elementary excitations in semiconductor^.^^^ shall analyze the additional conditions for macroscopic equations based on qualitative mi- croscopic considerations in agreement with are relat- behavior of dipole-inactive biexciton polariza- tion on uo two Maxwellian tangential components conditions describ- biexciton polarizations the boundary. crystal boundary infinitely high potential for excitons the boundary to the Pekar form the additional boundary biexciton polarizations: P (r, t) I .,=O, Q(r, t) I oO=O. (10) Sov. Phys. (5), lvanov and V. V. reflection in separate resonances. w, = 2541 meV its intensity Curve sents the unperturbed exciton reflection spectrum. length limit dispersion curves gated semiconductor. shall discuss greater detail dead layer in probe radiation. physical justification for exciton-

5 free dead layer qualitative idea its fin
free dead layer qualitative idea its finite radius exciton cannot approach the crystal closer In the dead layer and introduce biexciton-free layer LMabiex , where abiex biexciton radius. Figure shows schematically and transmitted in this a, - a, enclosed be- two dead layers unusual. Here, incident probe radiation excites two create, beyond semiconductor in the biexciton-free layer not only activates ton-biexciton mixing, but also induces (by the long-wavelength shift probe electromagnetic the additional accounted for fully one addi- tional boundary condition, which-as before-will Pekar condition i.e., PI,, = 0. corresponding additional boundary condition to the biexciton polarization QI, = 0 exciton polar- its derivative 6' P/d{ at the boundary a,. From the microscopic point spatial derivative 6 where there exciton poten- tial. Th

6 erefore, in two-layer model proposed add
erefore, in two-layer model proposed additional boundary conditions the transmitted allowing for biexciton-free layer. Two ordinary polariton eigenwaves in the a, - u2. the additional boundary con- electromagnetic energy S,, but it applies also Sex. follows directly from Eq. (8b) exciton energy proposed exci- ton additional boundary conditions fairly cumbersome calculations, obtain an expression for reflection coefficient R, (w) incident on 14), where in the expression ( 15) for the reflection coefficient i? following quantities: the matrix ad Sov. Phys. (5), May 1991 A. L. lvanov and V. V. Panashchenko 886 where q = t - {/us time and ?;;(77) is of five equations can equations for positive- index is cubic nonlinearity on right-hand side the nature excitons in two different also because repulsion between excitons forming e

7 ach additional notation is used given by
ach additional notation is used given by (24):ql =p, q, = k, w, = w, w, = w,. dimensionless time r = wry the approach traditional in nonlinear op- three-wave system not contain polarizations. However, tions in to the polarization components, i.e., exciton compo- wave is one additional and reason for polarization system w and w, the polariton exciton resonance w, incorrect results nonlinear processes with ofjust nonlinear low- electrical susceptibilities, i.e., $,' andXC3). illustrated quite well by (4), from which expression for x =x(w, p, IPk 1') is expanded exciton resonance this series nonlinear susceptibilities x'"', case is x'~). polarization equations in fact explicitly, within for a whole series i.e., problem rigorously. then related slowly varying constraint on v, the duration T, pulses with nondege

8 ner- ate carrier frequencies Iw - w,
ner- ate carrier frequencies Iw - w, I - ' 4~~ in the absence of at- tenuation y"x = yb"" Introducing the p, $, and 5 of positive-frequency soliton envelopes we obtain the nonlinear equations +-fj ,yz sin 0, y=-p2xz sin 0, i=pxy sin 0, xip=-v~x+p, (2x2-y2)x-p,yz cos 0, where 0 = p + $ - 5 amplitude equations three integrals these two motion given allows us fully all possible soliton solutions PI, P,, and p consider only most interesting method adopted C, = 0 out that, in accor- definition generally soliton solutions such three i.e., their amplitudes tend zero for 7- + W. case selected for investigation, Sov. Phys. (5), May 1991 A. L. lvanov and and the given by soliton pulse T, frequency interval soliton solutions, these solutions have the minimum and the the central exact resonance 0. A shi

9 ft of the frequency detuning (OF?, -
ft of the frequency detuning (OF?, - o - a,)--a to the the duration their am- plitude for frequency interval soliton solutions + 1 and, accordance with ) and (42), two solitary soliton solutions vanish the third steady-state nonlinear wave of tensity. Typical allows us expression for v, = v, (o, w,, T,, Therefore, for o,, rS, the second equation algebraic equations for v, consider also time dependence phase-matching angle ~54': calculations were carried o, = 2540 meV, 1 = 0.2 MW/cmZ, v, = 5.54~ lo7 cm/s. curves corre- = 2560meV, (A = - 0.073, x = - 0.385,~~ = 3.42 ps). dotted curves the pulses obtained 0 = 2560.1 meV (A = 0.93, x = - 0.415, T, = 8.94 ps). 9, rad z'q the phase-matching angle ( 1 ) phase shifts between the electric field and the polarizations "dark" calculated for the 7

10 , = 2.1 ps, I= 0.2 MW/cm2, o, = 254
, = 2.1 ps, I= 0.2 MW/cm2, o, = 2540 meV, o = 2560.1 meV (A = 0.51). [ (1-A" ((xZ+ 4Ax+4)]" sh (dz,) tge=-+ A (x2+4Ax+4)'" ch (zl~,) +2A+x (45) phase shift 4, or qbEk,& between the electric the appropriate xi+, y, y$ soliton pulses ) and (42), by the system of equations (26), phase-matching angle. time dependences 6 = 07, 4, . (T) and #,,,Pk (7) We shall conclude mentioning one qualitative difference betwen and the generally accepted nonlinear optics. adopted approach is usually assumed soliton can form only in maximum frequen- dipole-inactive biexciton which obviously as the The authors are deeply grateful Keldysh for many discussions encouragement. They also indebted and A. for many valuable critical comments and 'A. L. V. Keldysh, V. V. 99,641 (1991) *J. J. Phys. Rev. Sov. Phys. (5), lvanov a