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Exciton and biexciton correlations for weakly confined semiconductor quantum wires
PERGAMON
Solid State Communications 111 (1999) 741–745
Exciton and biexciton correlations for weakly confined semiconductor quantum wires O.M. Schmitt, L. Ba´nyai, H. Haug Institut fu¨r Theoretische Physik, J.W. Goethe-Universita¨t Frankfurt, Robert-Mayer-Str. 8, D-60054 Frankfurt am Main, Germany Received 22 February 1999; accepted 1 April 1999 by J. Kuhl
Abstract The contributions of excitons and biexcitons to the four-wave mixing (FWM) signal are calculated for semiconductor quantum wells with an additional weak wire confinement. In contrast to the giant oscillator model, in our description the biexciton is formed from two singlet excitons by the Coulomb interactions. The matrix elements of the repulsive and attractive interactions between two excitons are calculated explicitly using 2D quantum well wave functions and quantized center-ofmass envelop functions. By calculating the excitonic polarization in the FWM direction, we obtain coherent exciton–biexciton quantum beats as function of the delay between the two pulses and of their intensity. Quantum beat oscillations are found not only on the rising side of the FWM-signal as in spatially homogeneous x3 -theories, but also on the falling side, as it is observed experimentally. With increasing intensities the quantum beats develop into irregular signals. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Nanostructures; A. Semiconductors; A. Quantum wells; D. Optical properties; E. Time-resolved optical spectroscopies
The interaction between excitons (x) gives rise to coherent nonlinear optical effects, but influences also the dephasing and relaxation kinetics. Both effects can most clearly be studied by femtosecond four-wave mixing (FWM) spectroscopy. The most striking coherent effects are due to the formation of an excitonic molecule (m) by two singlet x with antiparallel spin alignment of the two electrons (e) and therefore also the two holes (h). In FWM experiments one observes x–m quantum beats in semiconductors with sufficiently large x and m binding energies [1–5]. Recently x–m quantum beats have also been observed in InGaAs quantum well wires [6]. The nonlinear optical properties of the bound two-x complex m has been studied mostly in terms of an effective Hamiltonian in which an m is excited from a one-x-state by absorption of a photon [7]. In this
giant oscillator model the m is formed due to the ~ interaction. In a first principle treatment, however, p~ ·A [8], an m is formed by two singlet xs which attract each other in the relative singlet state due to the Coulomb interactions between the two es and the two hs. It has been shown [9] that the latter model describes the measured m-two-photon absorption line shape better than the giant-oscillator strength model. The x–m quantum beats have been explained in the framework of multi-level Bloch equations [2,3] and in the framework of equations of motion for an e–h one and two-particle density matrices [4,10]. However, the x–x and x–m interaction matrix elements have been treated phenomenologically. Up to a thirdorder, a first-principle theory of the x–m nonlinearities [11] has been developed and numerically evaluated at least in 1D.
0038-1098/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00265-3
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O.M. Schmitt et al. / Solid State Communications 111 (1999) 741–745
Here, we will use a Boson picture for x and m and study the amplitude equations of x coherently excited by a laser pulse, coupled, as discussed, via a Coulombic interaction to the amplitude equation of m. The matrix elements of the repulsive x–x interaction matrix and the attractive x–x–m interaction will be evaluated in detail from the Coulomb interactions of the es and hs. With a similar approach [12], we have been able to describe the dephasing by incoherent xs measured in a quantum well wire with weak wire confinement (see also Ref. [13]). Our theory explained the measured x-density dependence and influence of increasing confinement. Surprisingly, an increasing wire confinement is observed to result in a faster dephasing. This unexpected result finds its explanation in the increase of the resulting x–x exchange interaction matrix element with the inverse of the wire width because of the increasing wave function overlap. We will use 2D quantum well x and m wave functions together with quantized center-of-mass x and m envelope functions. For the 2D m wave function we use the variational form of Kleinman [14]. We do not use the x-polariton and the bi-polariton picture [15], because the x-coherence length in the etched quantum wires is shorter than a photon wavelength. Our theory is not limited to third order, which seems important, because most semiconductor FWM experiments are already at the lowest possible intensities beyond that order. In particular, we will consider rectangular quantum well wires, with a very strong quantum well confinement in z direction (Lz ! a0 ), where a0 is the 2D x Bohr radius, and a comparably weak wire confinement in x-direction with Lx . a0 . The remaining ydirection remains totally free, as we do not attempt to treat spatial disorder along the wire. We will make a fairly simple ansatz for the x and m states in order to have analytic expressions, with which we can calculate the repulsive x–x and the attractive x–x–m matrix elements. In the rectangular quantum well wire one can construct the x wave function from an unperturbed quasi-2D wave function for the relative motion in the x–y plane times a center-of-mass envelop function which is quantized by the extra wire confinement potential. Such a center-of-mass motion quantization is appropriate in the weak confinement regime [16], it is based on an systematic expansion in
powers of a0 =Lx . For simplicity we assume infinitely high wire barrier potentials. The x wave function has the form: ÿ ~ c2D Cn;k ~r e ; ~rh xn;k R r ; 1 1s ~ where ~ me ~re 1 mh ~rh R M and ~r ~r e 2 ~rh are 2D center-of-mass and relative coordinates of the e–h pair. The wave function for the r is the unperturbed 1s–2D– e–h relative motion c2D 1s ~ x wave function / e2r=a0 , while the center-of-mass wave function is a product of a plane wave in the wire direction y and a particle-in-a-box wave function perpendicular to the wire: ~ p1 eikRy p2 sinkn Rx ; xn;k R Lx Ly where the center-of-mass subbands are given by kn np=Lx and 0 # Rx # Lx . For the m wave function we use ÿ ÿ ~ fm ~r e1 ; ~rh1 ; ~re2 ; ~rh2 : FN;K ~r e1 ; ~rh1 ; ~re2 ; ~rh2 xN;K R 2 ~ is the m center-of-mass envelop funcHere xN;K R ~ me ~re1 1 ~re2 1 mh ~rh1 1 ~rh2 = 2me tion with R 12mh . For the wave function of the relative motion we use the variational ansatz [14] 1 2 k s 1s bk fm e 4a0 1 2 cosh t1 2 t2 C 4a0 r × k h2;h1 e2krrh2;h1 =2a0 1 le2ktrh2;h1 =2a0 3 2a0 with the five variational parameters k; b; r; l; t. s and t are determined by the particle distances ~re;h ~r e 2 ~r h si rei ;h1 1 rei ;h2 ; ti rei ;h1 2 rei ;h2 ;
4
and C is the normalization constant. The lowest m binding energy with respect to the 2x level is eb2D 0:146ER . It is obtained with the coefficients k 2:51, b 0:66, r 0:43, l 0:42, t 0:28, ER is the 2D x Rydberg. While the wire length Ly vanishes when finally the limit Ly ! ∞ is performed, the wire width has a fixed value Lx . The quantization of the center-of-mass motion results in many x and m subbands. The
O.M. Schmitt et al. / Solid State Communications 111 (1999) 741–745
resulting x and m energy dispersions are (with mx me 1 mh ) "2 k2 " 2 np 2 exn;k ex1s 1 1 5 2mx 2mx Lx
emN;K
"2 K 2 "2 e 1 1 4mx 4mx m
Np Lx
2
:
6
In this situation the usual m binding energy ebm 2Ex 2 Em is replaced by a whole spectrum of binding energies ebn1 ;n2 ;N exn1 1 exn2 2 em N . For the lowest subband x and m one gets, e.g. a growing m binding energy with decreasing wire width 3 "2 p 2 eb1;1;1 eb2D 1 : 4 mx L x We consider in the following two types of singlet x with the polarization s^ , for the e spin orientations se "; # and correspondingly sh #; ". Note, that as a simplification we do not use the fully antisymmetric spin singlet state. We assume that the x and m operators b and c, respectively, obey Bose commutation relations, which holds if x densities are not too high (nx a20 p 1) and the x and m binding energies are sufficiently large. We further assume linearly polarized coherent light pulses, so that both xs are excited equally. We split the Hamiltonian in the following form H Hx 1 Hm 1 Hx;x 1 Hx;m 1 Hx;E :
7
With the quantum number n n; k, the free x and m Hamiltonians are X x † X m † e n bn ; s bn ; s ; Hm en c n c n : 8 Hx n
n;s^
The repulsive x–x interaction Hamiltonian between two singlet xs with the same e spin alignment is 1 X w b† b† b b : 9 Hx;x 4 {n };s n1 ;n2 ;n3 ;n4 n1 ;s n2 ;s n3 ;s n4 ;s i
We couple the m to the x system similarly by the attractive Coulombic interaction between two xs in the relative singlet state X vn;n1;n2 c†n bn1;1 bn2;2 1 h:c: ; 10 Hx;m n;n1;n2
where h.c. denotes the Hermitian conjugate of the
743
preceding term. The coupling of the xs with k 0 to the linearly polarized coherent light field E t is given by 1X p p Hx;E 2 g E tbn;s 1 h:c:; 11 2 n;s n;s with the x optical matrix element gn;s gn . The matrix elements g, w and v are calculated within the underlying e–h picture. For the x-dipole matrix element one obtains with the interband dipole matrix element dcv Z gn d2 re dvc cn;k ~re ; ~re dk;0 q dvc Lx Ly
2
p3=2 a
0n
1 1 21 n11 dk;0 :
12
In the considered weak-confinement limit only the exchange contributes to the x–x interaction [12] Z Z Z Z wn1 ;n2 ;n3 ;n4 2 d2 re1 d 2 re2 d2 rh1 d2 rh2 × Cn1 ~re1 ; ~rh1 Cn2 ~r e2 ; ~rh2 Cn3 ~re2 ; ~rh1 Cn4 ~re1 ; ~rh2 0 1 X X V ~rj1 2 ~rj2 2 V ~r j1 2 ~ri±j2 A; 13 ×@ je;h
i;je;h
which yields [12] a wn1 ;n2 ;n3 ;n4 Cx 0 ER dk1 1k2 ;k3 1k4 an1 ;n2 ;n3 ;n4 : Lx
14
where a describes the inter-subband selection rules: 1X an1 ;n2 ;n3 ;n4 d 1 dn1 1n2 ;^n3 1^n4 8 ^ n1 2n2 ;^n3 7n4 2 dn1 2n2 ;^n3 ^n4 2 dn1 1n2 ;^n3 7n4 :
15
The remaining six dimensional integral denoted by Cx has to be calculated numerically by Monte Carlo integration [12]. An identical procedure for the x–m coupling yields in leading order of a0 =Lx Y Z vN;K;n1 ;k1 ;n2 ;k2 2 exn1 1 exn2 2 em d2 rjl N je;h;l1;2
× Fpm ~re1 ; ~rh1 ; ~r e2 ; ~rh2 C re1;h1 C re2;h2 :
16
The resulting integrals have the same structure as
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O.M. Schmitt et al. / Solid State Communications 111 (1999) 741–745
amplitudes, the Heisenberg equations yield d i X x 2 iDn kbn;^ tl 2 v 0 kc lkb† 0 l dt " N;n 0 N;n ;n N n ;7
100 4 3 2 1
10
1
-4
-2
0
τ (ps)
2
4
6
8
Fig. 1. Calculated time-integrated FWM signals versus delay time for the following parameters: Curve 1: Pulse strength 0.004 p, wire width Lx 29 nm, x density nx a20 3:3 × 1025 ; curve 2: 0.01 p, 29 nm, 2:2 × 1024 ; curve 3: 0.01 p, 43 nm, 2:3 × 1024 ; curve 4:0.02 p, 100 nm, 1:02 × 1023 .
those of the repulsive x–x exchange interaction, however with different selection rules b and a different coupling constant Cmx : vN;K;n1 ;k1 ;n2 ;k2 Cmx dK2k1 2k2 bN;n1 ;n2
17
with bN;n1 ;n2
" 1X 1 1 21N7n1 7n2 11 1 2 dN;^n1 ^n2 2 4 ^ N 7 n1 7 n2
1 1 212 N7n1 ^n2 21 2 1 2 dN;^n1 7n2 N 7 n1 ^ n2
# :
18
Again the interaction strength Cmx is calculated by Monte Carlo integration. In Ref. [12] we calculated the dephasing of the coherent x amplitude due to scattering with thermal xs excited by a prepulse. Here, we concentrate on the coherent effects of the x–x and x–m couplings, which will give rise to x–m quantum beats as observed in the time-integrated FWM. The coherent light pulses excite a coherent x amplitude kbn;k0;s l kbn;s l for both polarizations which in turn will drive a coherent m amplitude Considering only coherent kcN;K0 l kcN l.
2
i X w kb† lkb lkb l 2" n1 ;n2 ;n3 n;n1 ;n2 ;n3 n1 ;^ n2 ;^ n3 ;^
1
i kb l gn E0 t 2 n;^ ; 2" T2;x
19
and d i X 0 kb 2 iDm v lkb 0 l N kcN tl 2 dt " n;n 0 N;n;n n;1 n ;2 2
kcN l ; T2;m
20
where the detunings are given for x by Dxn "v 2 exn and for m DmN 2"v 2 emN . Similar equations in other geometries and approximation schemes have been treated also in Refs. [4,8,10]. A special feature of these equations is that the m amplitude is not driven directly by the light field, but is only excited via the x–x–m interaction. In order to get realistic FWM signals we added to these coherent amplitude equations phenomenological dephasing terms with T2;x and T2;m 12 T2;x . For the numerical evaluation we took the material parameters of the etched quantum well wires investigated in Ref. [12]. We assume Gaussian pulses with a FWHM pulse width of 150 fs, tuned to the lowest 2Dx level. For a 29 nm wire, e.g. the central frequency of the pulses is detuned by Dx1 22:22 meV below the lowest x resonance. The two colinearly polarized pulses of equal strength are traveling in the directions k1 and k2 with a delay t. We calculate the x-polarization, i.e. the x amplitude in the FWM direction 2k2 2 k1 by a projection technique described, e.g. in Ref. [17]. In Fig. 1 the calculated time-integrated FWM signals are shown versus the delay time t for various wire widths Rand various excitation intensities ∞ measured by 1 2 ∞ dtdcv E0 t xp in fractions of a p-pulse. The x dephasing time has chosen as T2;x 8ps. Curve 1 for Lx 29 nm corresponds to an extremely low excitation. With the width of only 29 nm the level splitting is so large that only the lowest subband of x and of m are excited. The system
O.M. Schmitt et al. / Solid State Communications 111 (1999) 741–745
should behave similarly as a 2D quantum well. For such an extremely low excitation one reproduces indeed the result of the standard x3 -theory [4,10]. In this limit one obtains x–m quantum beats with the binding energy of the quantum-wire m, but only for negative delay times t. For positive delay the curve is totally flat. Curve 2 is calculated for the same wire width at a slightly higher pulse strength. Surprisingly, already now one gets quantum beats for positive delay, as expected in the regime where higher-order nonlinear susceptibilities contribute [10], although the excited x density nx a20 2:2 × 1024 is still extremely small. It has been shown [2,3] that such quantum beats may also show up in the x3 -limit, if spatial disorder is introduced. Our calculations show however that disorder is not necessary to produce quantum beats for positive delay, because even for the smallest measurable x densities in quantum wires one gets contributions beyond the x3 -limit. The beats agree with the m binding energy eb1 5:52 meV, which corresponds to a period of 747 fs. Curve 3 is calculated for a broader wire with Lx 43 nm and a stronger excitation. Now two x subbands (the first and the third) are excited. There is a beating between the x, n 3 and x, n 1 transitions with the period of 511 fs and beating with the m binding energy, corresponding to a period of 1124 fs. The m beating is stronger for negative delays. The intersubband quantum beats for center-of-mass quantized xs have been observed by Kuhl et al. [18]. If more subbands (up to 11) play a role like in the top curve 4 for 100 nm wires and for a still stronger excitation, the beating gets very irregular and even seems to become chaotic, as it can be expected for the coupling of a larger number of nonlinear equations. Our results are in qualitative agreement with the observed m quantum beats in InGaAs wires [6]. For a quantitative description of the experimental results however, one would have to include a realistic treatment of the x and m states in stressed quantum wires, as well as the effects of interface disorder. In addition a microscopic treatment of the dephasing kinetics would be required.
745
Acknowledgements This work has been supported by the DFG in the framework of the Schwerpunktprojekt Quantenkoha¨renz in Halbleitern. We thank A. Forchel, M. Bayer, W. Braun and G. Bartels for stimulating discussions.
References [1] T. Saiki, M. Kuwata, T. Matsusue, H. Sakakai, Phys. Rev. B 49 (1994) 7817. [2] E.J. Mayer, G.O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, D. Bernhardt, S.W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 50 (1994) 14730. [3] T.F. Albrecht, K. Bott, T. Meier, A. Schulze, M. Koch, S.T. Cundiff, J. Feldmann, W. Stolz, P. Thomas, S.W. Koch, and E.O. Go¨bel, Phys. Rev. B 54 (1996) 4436. [4] W. Scha¨fer, D.S. Kim, J. Shah, T.C. Damen, J.E. Cunningham, K.W. Goosen, L.N. Pfeiffer, K. Ko¨hler, Phys. Rev. B 53 (1996) 16429. [5] D. Oberhauser, K.H. Pantke, W. Langbein, V.G. Lyssenko, H. Kalt, J.M. Hvam, G. Weinmann, C. Klingshirn, Phys. Stat. Sol. b 173 (1992) 53. [6] T. Baars, W. Braun, M. Bayer, A. Forchel, Phys. Rev. B 58 (1998) R1750. [7] E. Hanamura, Solid State Commun. 12 (1973) 95. [8] A.L. Ivanov, H. Haug, Phys. Rev. B 48 (1993) 1490. [9] A.L. Ivanov, M. Hasuo, N. Nagasawa, H. Haug, Phys. Rev. B 52 (1995) 11017. [10] G. Bartels, V.M. Axt, K. Victor, A. Stahl, P. Leisching, K. Ko¨hler, Phys. Rev. B 51 (1995) 11217. ¨ sterreich, K. Scho¨nhammer, L.J. Sham, Phys. Rev. Lett. [11] Th. O 74 (1995) 4698. [12] W. Braun, M. Bayer, A. Forchel, O.M. Schmitt, L. Ba´nyai, H. Haug, Phys. Rev. B 57 (1998) 12364. [13] H.P. Wagner, W. Langbein, J.M. Hvam, G. Bachner, T. Ku¨mmell, A. Forchel, Phys. Rev. B 57 (1998) 1797. [14] D.A. Kleinman, Phys. Rev. B 28 (1983) 871. [15] A.L. Ivanov, H. Haug, L.V. Keldysh, Phys. Rep. 296 (1998) 237. [16] Al.L. Efros, A.L. Efros, Sov.-Phys. Semicond. 16 (1982) 772. [17] H. Haug, A.P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer, Berlin, corr. print (1998). [18] E.J. Mayer, J.O. White, G.O. Smith, H. Lage, D. Heitmann, K. Ploog, J. Kuhl, Phys. Rev. B 49 (1994) 2993.
Solid State Communications 111 (1999) 741–745
Exciton and biexciton correlations for weakly confined semiconductor quantum wires O.M. Schmitt, L. Ba´nyai, H. Haug Institut fu¨r Theoretische Physik, J.W. Goethe-Universita¨t Frankfurt, Robert-Mayer-Str. 8, D-60054 Frankfurt am Main, Germany Received 22 February 1999; accepted 1 April 1999 by J. Kuhl
Abstract The contributions of excitons and biexcitons to the four-wave mixing (FWM) signal are calculated for semiconductor quantum wells with an additional weak wire confinement. In contrast to the giant oscillator model, in our description the biexciton is formed from two singlet excitons by the Coulomb interactions. The matrix elements of the repulsive and attractive interactions between two excitons are calculated explicitly using 2D quantum well wave functions and quantized center-ofmass envelop functions. By calculating the excitonic polarization in the FWM direction, we obtain coherent exciton–biexciton quantum beats as function of the delay between the two pulses and of their intensity. Quantum beat oscillations are found not only on the rising side of the FWM-signal as in spatially homogeneous x3 -theories, but also on the falling side, as it is observed experimentally. With increasing intensities the quantum beats develop into irregular signals. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Nanostructures; A. Semiconductors; A. Quantum wells; D. Optical properties; E. Time-resolved optical spectroscopies
The interaction between excitons (x) gives rise to coherent nonlinear optical effects, but influences also the dephasing and relaxation kinetics. Both effects can most clearly be studied by femtosecond four-wave mixing (FWM) spectroscopy. The most striking coherent effects are due to the formation of an excitonic molecule (m) by two singlet x with antiparallel spin alignment of the two electrons (e) and therefore also the two holes (h). In FWM experiments one observes x–m quantum beats in semiconductors with sufficiently large x and m binding energies [1–5]. Recently x–m quantum beats have also been observed in InGaAs quantum well wires [6]. The nonlinear optical properties of the bound two-x complex m has been studied mostly in terms of an effective Hamiltonian in which an m is excited from a one-x-state by absorption of a photon [7]. In this
giant oscillator model the m is formed due to the ~ interaction. In a first principle treatment, however, p~ ·A [8], an m is formed by two singlet xs which attract each other in the relative singlet state due to the Coulomb interactions between the two es and the two hs. It has been shown [9] that the latter model describes the measured m-two-photon absorption line shape better than the giant-oscillator strength model. The x–m quantum beats have been explained in the framework of multi-level Bloch equations [2,3] and in the framework of equations of motion for an e–h one and two-particle density matrices [4,10]. However, the x–x and x–m interaction matrix elements have been treated phenomenologically. Up to a thirdorder, a first-principle theory of the x–m nonlinearities [11] has been developed and numerically evaluated at least in 1D.
0038-1098/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00265-3
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O.M. Schmitt et al. / Solid State Communications 111 (1999) 741–745
Here, we will use a Boson picture for x and m and study the amplitude equations of x coherently excited by a laser pulse, coupled, as discussed, via a Coulombic interaction to the amplitude equation of m. The matrix elements of the repulsive x–x interaction matrix and the attractive x–x–m interaction will be evaluated in detail from the Coulomb interactions of the es and hs. With a similar approach [12], we have been able to describe the dephasing by incoherent xs measured in a quantum well wire with weak wire confinement (see also Ref. [13]). Our theory explained the measured x-density dependence and influence of increasing confinement. Surprisingly, an increasing wire confinement is observed to result in a faster dephasing. This unexpected result finds its explanation in the increase of the resulting x–x exchange interaction matrix element with the inverse of the wire width because of the increasing wave function overlap. We will use 2D quantum well x and m wave functions together with quantized center-of-mass x and m envelope functions. For the 2D m wave function we use the variational form of Kleinman [14]. We do not use the x-polariton and the bi-polariton picture [15], because the x-coherence length in the etched quantum wires is shorter than a photon wavelength. Our theory is not limited to third order, which seems important, because most semiconductor FWM experiments are already at the lowest possible intensities beyond that order. In particular, we will consider rectangular quantum well wires, with a very strong quantum well confinement in z direction (Lz ! a0 ), where a0 is the 2D x Bohr radius, and a comparably weak wire confinement in x-direction with Lx . a0 . The remaining ydirection remains totally free, as we do not attempt to treat spatial disorder along the wire. We will make a fairly simple ansatz for the x and m states in order to have analytic expressions, with which we can calculate the repulsive x–x and the attractive x–x–m matrix elements. In the rectangular quantum well wire one can construct the x wave function from an unperturbed quasi-2D wave function for the relative motion in the x–y plane times a center-of-mass envelop function which is quantized by the extra wire confinement potential. Such a center-of-mass motion quantization is appropriate in the weak confinement regime [16], it is based on an systematic expansion in
powers of a0 =Lx . For simplicity we assume infinitely high wire barrier potentials. The x wave function has the form: ÿ ~ c2D Cn;k ~r e ; ~rh xn;k R r ; 1 1s ~ where ~ me ~re 1 mh ~rh R M and ~r ~r e 2 ~rh are 2D center-of-mass and relative coordinates of the e–h pair. The wave function for the r is the unperturbed 1s–2D– e–h relative motion c2D 1s ~ x wave function / e2r=a0 , while the center-of-mass wave function is a product of a plane wave in the wire direction y and a particle-in-a-box wave function perpendicular to the wire: ~ p1 eikRy p2 sinkn Rx ; xn;k R Lx Ly where the center-of-mass subbands are given by kn np=Lx and 0 # Rx # Lx . For the m wave function we use ÿ ÿ ~ fm ~r e1 ; ~rh1 ; ~re2 ; ~rh2 : FN;K ~r e1 ; ~rh1 ; ~re2 ; ~rh2 xN;K R 2 ~ is the m center-of-mass envelop funcHere xN;K R ~ me ~re1 1 ~re2 1 mh ~rh1 1 ~rh2 = 2me tion with R 12mh . For the wave function of the relative motion we use the variational ansatz [14] 1 2 k s 1s bk fm e 4a0 1 2 cosh t1 2 t2 C 4a0 r × k h2;h1 e2krrh2;h1 =2a0 1 le2ktrh2;h1 =2a0 3 2a0 with the five variational parameters k; b; r; l; t. s and t are determined by the particle distances ~re;h ~r e 2 ~r h si rei ;h1 1 rei ;h2 ; ti rei ;h1 2 rei ;h2 ;
4
and C is the normalization constant. The lowest m binding energy with respect to the 2x level is eb2D 0:146ER . It is obtained with the coefficients k 2:51, b 0:66, r 0:43, l 0:42, t 0:28, ER is the 2D x Rydberg. While the wire length Ly vanishes when finally the limit Ly ! ∞ is performed, the wire width has a fixed value Lx . The quantization of the center-of-mass motion results in many x and m subbands. The
O.M. Schmitt et al. / Solid State Communications 111 (1999) 741–745
resulting x and m energy dispersions are (with mx me 1 mh ) "2 k2 " 2 np 2 exn;k ex1s 1 1 5 2mx 2mx Lx
emN;K
"2 K 2 "2 e 1 1 4mx 4mx m
Np Lx
2
:
6
In this situation the usual m binding energy ebm 2Ex 2 Em is replaced by a whole spectrum of binding energies ebn1 ;n2 ;N exn1 1 exn2 2 em N . For the lowest subband x and m one gets, e.g. a growing m binding energy with decreasing wire width 3 "2 p 2 eb1;1;1 eb2D 1 : 4 mx L x We consider in the following two types of singlet x with the polarization s^ , for the e spin orientations se "; # and correspondingly sh #; ". Note, that as a simplification we do not use the fully antisymmetric spin singlet state. We assume that the x and m operators b and c, respectively, obey Bose commutation relations, which holds if x densities are not too high (nx a20 p 1) and the x and m binding energies are sufficiently large. We further assume linearly polarized coherent light pulses, so that both xs are excited equally. We split the Hamiltonian in the following form H Hx 1 Hm 1 Hx;x 1 Hx;m 1 Hx;E :
7
With the quantum number n n; k, the free x and m Hamiltonians are X x † X m † e n bn ; s bn ; s ; Hm en c n c n : 8 Hx n
n;s^
The repulsive x–x interaction Hamiltonian between two singlet xs with the same e spin alignment is 1 X w b† b† b b : 9 Hx;x 4 {n };s n1 ;n2 ;n3 ;n4 n1 ;s n2 ;s n3 ;s n4 ;s i
We couple the m to the x system similarly by the attractive Coulombic interaction between two xs in the relative singlet state X vn;n1;n2 c†n bn1;1 bn2;2 1 h:c: ; 10 Hx;m n;n1;n2
where h.c. denotes the Hermitian conjugate of the
743
preceding term. The coupling of the xs with k 0 to the linearly polarized coherent light field E t is given by 1X p p Hx;E 2 g E tbn;s 1 h:c:; 11 2 n;s n;s with the x optical matrix element gn;s gn . The matrix elements g, w and v are calculated within the underlying e–h picture. For the x-dipole matrix element one obtains with the interband dipole matrix element dcv Z gn d2 re dvc cn;k ~re ; ~re dk;0 q dvc Lx Ly
2
p3=2 a
0n
1 1 21 n11 dk;0 :
12
In the considered weak-confinement limit only the exchange contributes to the x–x interaction [12] Z Z Z Z wn1 ;n2 ;n3 ;n4 2 d2 re1 d 2 re2 d2 rh1 d2 rh2 × Cn1 ~re1 ; ~rh1 Cn2 ~r e2 ; ~rh2 Cn3 ~re2 ; ~rh1 Cn4 ~re1 ; ~rh2 0 1 X X V ~rj1 2 ~rj2 2 V ~r j1 2 ~ri±j2 A; 13 ×@ je;h
i;je;h
which yields [12] a wn1 ;n2 ;n3 ;n4 Cx 0 ER dk1 1k2 ;k3 1k4 an1 ;n2 ;n3 ;n4 : Lx
14
where a describes the inter-subband selection rules: 1X an1 ;n2 ;n3 ;n4 d 1 dn1 1n2 ;^n3 1^n4 8 ^ n1 2n2 ;^n3 7n4 2 dn1 2n2 ;^n3 ^n4 2 dn1 1n2 ;^n3 7n4 :
15
The remaining six dimensional integral denoted by Cx has to be calculated numerically by Monte Carlo integration [12]. An identical procedure for the x–m coupling yields in leading order of a0 =Lx Y Z vN;K;n1 ;k1 ;n2 ;k2 2 exn1 1 exn2 2 em d2 rjl N je;h;l1;2
× Fpm ~re1 ; ~rh1 ; ~r e2 ; ~rh2 C re1;h1 C re2;h2 :
16
The resulting integrals have the same structure as
744
O.M. Schmitt et al. / Solid State Communications 111 (1999) 741–745
amplitudes, the Heisenberg equations yield d i X x 2 iDn kbn;^ tl 2 v 0 kc lkb† 0 l dt " N;n 0 N;n ;n N n ;7
100 4 3 2 1
10
1
-4
-2
0
τ (ps)
2
4
6
8
Fig. 1. Calculated time-integrated FWM signals versus delay time for the following parameters: Curve 1: Pulse strength 0.004 p, wire width Lx 29 nm, x density nx a20 3:3 × 1025 ; curve 2: 0.01 p, 29 nm, 2:2 × 1024 ; curve 3: 0.01 p, 43 nm, 2:3 × 1024 ; curve 4:0.02 p, 100 nm, 1:02 × 1023 .
those of the repulsive x–x exchange interaction, however with different selection rules b and a different coupling constant Cmx : vN;K;n1 ;k1 ;n2 ;k2 Cmx dK2k1 2k2 bN;n1 ;n2
17
with bN;n1 ;n2
" 1X 1 1 21N7n1 7n2 11 1 2 dN;^n1 ^n2 2 4 ^ N 7 n1 7 n2
1 1 212 N7n1 ^n2 21 2 1 2 dN;^n1 7n2 N 7 n1 ^ n2
# :
18
Again the interaction strength Cmx is calculated by Monte Carlo integration. In Ref. [12] we calculated the dephasing of the coherent x amplitude due to scattering with thermal xs excited by a prepulse. Here, we concentrate on the coherent effects of the x–x and x–m couplings, which will give rise to x–m quantum beats as observed in the time-integrated FWM. The coherent light pulses excite a coherent x amplitude kbn;k0;s l kbn;s l for both polarizations which in turn will drive a coherent m amplitude Considering only coherent kcN;K0 l kcN l.
2
i X w kb† lkb lkb l 2" n1 ;n2 ;n3 n;n1 ;n2 ;n3 n1 ;^ n2 ;^ n3 ;^
1
i kb l gn E0 t 2 n;^ ; 2" T2;x
19
and d i X 0 kb 2 iDm v lkb 0 l N kcN tl 2 dt " n;n 0 N;n;n n;1 n ;2 2
kcN l ; T2;m
20
where the detunings are given for x by Dxn "v 2 exn and for m DmN 2"v 2 emN . Similar equations in other geometries and approximation schemes have been treated also in Refs. [4,8,10]. A special feature of these equations is that the m amplitude is not driven directly by the light field, but is only excited via the x–x–m interaction. In order to get realistic FWM signals we added to these coherent amplitude equations phenomenological dephasing terms with T2;x and T2;m 12 T2;x . For the numerical evaluation we took the material parameters of the etched quantum well wires investigated in Ref. [12]. We assume Gaussian pulses with a FWHM pulse width of 150 fs, tuned to the lowest 2Dx level. For a 29 nm wire, e.g. the central frequency of the pulses is detuned by Dx1 22:22 meV below the lowest x resonance. The two colinearly polarized pulses of equal strength are traveling in the directions k1 and k2 with a delay t. We calculate the x-polarization, i.e. the x amplitude in the FWM direction 2k2 2 k1 by a projection technique described, e.g. in Ref. [17]. In Fig. 1 the calculated time-integrated FWM signals are shown versus the delay time t for various wire widths Rand various excitation intensities ∞ measured by 1 2 ∞ dtdcv E0 t xp in fractions of a p-pulse. The x dephasing time has chosen as T2;x 8ps. Curve 1 for Lx 29 nm corresponds to an extremely low excitation. With the width of only 29 nm the level splitting is so large that only the lowest subband of x and of m are excited. The system
O.M. Schmitt et al. / Solid State Communications 111 (1999) 741–745
should behave similarly as a 2D quantum well. For such an extremely low excitation one reproduces indeed the result of the standard x3 -theory [4,10]. In this limit one obtains x–m quantum beats with the binding energy of the quantum-wire m, but only for negative delay times t. For positive delay the curve is totally flat. Curve 2 is calculated for the same wire width at a slightly higher pulse strength. Surprisingly, already now one gets quantum beats for positive delay, as expected in the regime where higher-order nonlinear susceptibilities contribute [10], although the excited x density nx a20 2:2 × 1024 is still extremely small. It has been shown [2,3] that such quantum beats may also show up in the x3 -limit, if spatial disorder is introduced. Our calculations show however that disorder is not necessary to produce quantum beats for positive delay, because even for the smallest measurable x densities in quantum wires one gets contributions beyond the x3 -limit. The beats agree with the m binding energy eb1 5:52 meV, which corresponds to a period of 747 fs. Curve 3 is calculated for a broader wire with Lx 43 nm and a stronger excitation. Now two x subbands (the first and the third) are excited. There is a beating between the x, n 3 and x, n 1 transitions with the period of 511 fs and beating with the m binding energy, corresponding to a period of 1124 fs. The m beating is stronger for negative delays. The intersubband quantum beats for center-of-mass quantized xs have been observed by Kuhl et al. [18]. If more subbands (up to 11) play a role like in the top curve 4 for 100 nm wires and for a still stronger excitation, the beating gets very irregular and even seems to become chaotic, as it can be expected for the coupling of a larger number of nonlinear equations. Our results are in qualitative agreement with the observed m quantum beats in InGaAs wires [6]. For a quantitative description of the experimental results however, one would have to include a realistic treatment of the x and m states in stressed quantum wires, as well as the effects of interface disorder. In addition a microscopic treatment of the dephasing kinetics would be required.
745
Acknowledgements This work has been supported by the DFG in the framework of the Schwerpunktprojekt Quantenkoha¨renz in Halbleitern. We thank A. Forchel, M. Bayer, W. Braun and G. Bartels for stimulating discussions.
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