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Collective inter-subband excitations of electron gas in quantum wells in infrared absorption, Raman scattering and nonlinear optical processes
~ )
Solid State Communications, VoL 84, Nos. 1/2, PP. 81-86, 1992. Printed in Great Britain.
0038-1098/9255.00+ .00 Pergamon Press Ltd
COLLECTIVE INTER-SUBBAND EXCITATIONSOF ELECTRON GAS IN QUANTUM WELLS IN INFRARED ABSORPTION, RAMAN SCATTERING AND NONLINEAR OPTICAL PROCESSES Ming Ya Jiang Physics Department and Laboratory for Research on the Structure of Matter University of Pennsylvania, Philadelphia, PA 19104, USA
Received May 8, 199P b~t M. Cardona We discuss the role of collective inter-subband excitations of electron gases confined in quantum wells in infrared absorption, resonant Raman scattering and nonlinear optical processes. First, exchange as well as direct Coulomb interaction are invoked in a theoretical model to calculate the shift of the collective inter-subband modes from the single particle excitation energy. We then formulate the coupling of the collective inter-subband charge density mode to the electromagnetic field. The coupling mechanisms via exchange and direct Coulomb scattering for resonant Raman processes are discussed. An experiment is designed that can differentiate between the two mechanisms, based on their dependence on wavefunction overlap. Finally, we discuss nonlinear optical processes involving inter-subband transitions and collective excitations.
1
Introduction
a large polarizability along the z direction, which leads t o various types of linear and nonlinear responses from a layered quasi-2D quantum well system. In the first part of this paper, an alternative formulation of the collective inter-subband modes is given starting from a model Hamiltonian that yields the exchange as well as the direct Coulomb screening on both the inter-subband charge density and spin density modes with a clear physical picture. In the second part, we treat the problem of coupling of the inter-subband excitations with photons that leads to a shift due to radiative interaction (Lamb shift). In the third part, we discuss the coupling mechanisms of exchange and direct Coulomb scattering for resonant Raman processes and propose an experiment that can differentiate between the two mechanisms, based on the difference in the dependence of exchange and direct Coulomb interaction on wavefunction overlap. In the last part, we discuss the role of inter-subband transitions in nonlinear optical processes and, in particular, the spectral shift upon resonant infrared pumping as a result of the optical Stark effect.
The confinement of electrons in a quantum well introduces energy gaps in single particle excitations. Collective modes, in principle, exist at every intersubband edge for an electron gas confined in a quantum well. Direct observations of collective intersubband modes were achieved experimentally with resonant Raman scattering [2] and resonant infrared absorption [3,4] in doped semiconductor heterojunctions and quantum wells. These observations, especially the Raman scattering studies, have led to various theoretical formulations of the collective inter-subband excitations. The shift of the energy of the collective charge density mode from the single particle inter-subband excitation energy has been attributed to direct Coulomb screening (i.e., the depolarization field effect) [1,6,7]. More recent experimental studies [9] indicated that the spin density excitation, which was initially thought to be unscreened, also has a sizable shift which is attributed to exchange Coulomb screening. In contrast to the collective electronic excitation (plasmon) in a 3-D electron gas which is inaccessible via optical excitation due to its longitudinal character, the intersubband plasmons in general couple to photons that have an electric field component (E,) perpendicular to the walls of the quantum well. The large oscillator strength associated with the collective mode gives rise to
2
D i r e c t and E x c h a n g e C o u l o m b Interaction
A quasi-2D electron gas is almost ideally realized with a 81
82
ELECTRON GAS IN QUANTUM WELLS
modulation-doped GaAs-(AI,Ga)As type Of direct band gap semiconductor quantum well, where electrons (or holes) move in a lattice free of impurities. Adopting the effective mass approximation, we c~n view the system as a free electron gas confined in a slab-like region. The single particle electron wave function and spectrum is determined by the confining potential. By separating the in-plane and z part of the wavefunction and energy we can write them as: ¢,,(k) = %,(z)exp(ik. p) h~k2 2rn* The Hamiltonian for such an electron gas system ignoring electron-lattice interaction is written as: e,,(k) = e,, +
Z:
i,,l II o"
+ Ei~mn
1
.
~(ij IVI rtm)ai,k+q.oaj,t,, t
',o'an,k,~, (1)
kk'q¢'¢
where the wave the function normalization is taken over a unit area of the quantum well. For our purpose here we will consider only the first two subbands cl and c2 (treatments involving other subbands are similar). We can further separate the intrasubband and inter-subband interaction terms. Let us first look at the the inter-subband terms. By neglecting two electron(hole) excitation processes (where two electrons in the same subband scatter each other to the other subband) and introducing the inter-subband density fluctuation operator:
V(q,z, z') is the electron-electron interaction potential dependent on the dielectric constant of the medium as well as its geometry. For a single quantum well of width 1, dielectric constant q for the substrate, e2 for the well region, and z = 0 at the center of well, V is given [7,8] by: 2~l'e2 / i, '~ V(q, z, z') = ~ {e -~'*-~ '+ e2 - el e_q(i.+.,_q+l.+~,+tl)
e2q ~
k
= (v. - ~-v.) ~ [ ( & + +
+ &__)(e~,++ + ~21--)1
q
+ (-½v,)~[(&.,-+
- &--)(~,++
-ih~ =
+ &-+)Ce~+- + e~+-)]
q where V( and Ve are respectively the inter-subband direct and exchange Coulomb integrals defined as: (211V(q , z , z ' ) l l 2 )
= / ~o~(z)tp,(z)VcpI(z')~o2(z')dzdz' V~(q)
E ( , 2 ( k ) -- ,l(k))a~,k,oal,k,, k,a
+(~(o) - ~-~(o)) ~(al,~,<,a,,,<,<, - a[~,.a~,~,o)A,(O) (6) k,o"
By taking the thermal average value for the number operator, and assuming identical in-plane electron masses of both subbands, the energy for the charge density mode at q = 0 is given by:
(~
1
-
c,) + (~(0) - ~v.(0))(.,
-
,,~) (7)
where nl and nu represent the thermally averaged occupation number of the first and second subband per unit area. The inter-subband spin density excitations, S~(q) = i ttk+qalttka -1 - (where Si,o is the ith Pauli matrix) kota s ¢la are three-fold degenerate. Its energy at zero wavevector is given by: hfl$e(0) = (~ - ~,) - ~v.(0)(~, - ~2) ,
q
V~(q) =
= [A,, p~(0)]_
- e2,--)[
q
+ (-~v,)~[(&+_
(5)
(2)
the inter-subband part of the interaction Hamiltonian can be written as:
~,2
)
e2 + el
The scattering processes due to the direct and exchange Coulomb interactions are shown as diagrams in Fig.(la,b). Note that the singularity of V~(q) at q = 0 is cancelled by the corresponding term in the electronlattice interaction, leaving a finite value for V~ at q = O. The first term in eq.(3) represents the charge density excitation, the rest of the terms represent the three components of spin density excitations. We can find the energies of these excitations by their equations of motion. For the inter-subband charge density excitation, P~l(q) = ~ t o a2k+qaalkat at zero wavevector:
hf~(O) $~.w¢(q) = Z a~,k+q, o'amk,~
Vol. 84, Nos 1/2
= (121V(q,z,z') l l2) = f ¢p~(z)9o2(z)V~p~(z')~ol(z')dzdz'
(4)
(8)
The nature of these collective modes lies in the Coulomb coupling of the iso-energetic single particle excitations. It is straightforward to extend the above formalism to the case of cl-cs, cl-c4.., excitations. Each collective mode is shifted away from the corresponding single particle excitation energy by an amount proportional to the occupation number difference in the subbands involved. The intra-subband interaction gives rise to a finite self energy correction to the subband energy due to the exchange interaction [10]. The exchange self energy correction is generally dependent on wavevector. The main contribution is, however, a constant shift due to the qindependent part of (ii [ V(q,z, z') [ ii). It is propor-
Vol. 84, Nos I/2
ELECTRON GAS IN QUANTUM WELLS
tional to the occupation density of each subband, given
by: A~i =
- l ( i i l V ( q = O,z, z')lii)ni
(9)
The difference in self energy corrections tO each subband gives rise to another term in the shift of the intersubband modes, to be added to those given in eqs.(7) and (8). It is identical for both modes:
83
calculate the energy correction to the inter-subband collective mode via radiative interaction, which bears resemblance to the Lamb shift in atomic energy levels. Second order perturbation theory gives the following • equation for the energy of the collective mode, which is diagrammatically represented in Fig. (lc) as the virtual photon emission by one electron and subsequent absorption by another electron: *.
Aft = 1(11 ] V [ ll)nl - 1(22 [ V [ 22)n2
Photonic
(13)
(10)
We note here the difference between the RPA calculation with ours. The RPA calculation gives a manybody correction through the poles of the response function and the result is slightly different from the results of eqs.(7-10). In addition, all the approximations are identified in our formulation.
3
I g(q,q=) is W q - ftq
A divergence is encountered in carrying out the summation over qz which is an artifact expected in the calculation of the Lamb shift to he removed with renormalization. We will instead look at some general features based on eq.(13). The shift will depend on the fine structure constant a, and the ratio of the quantum well width to the photon wavelength. The major contribution to the shift comes from interaction with a photon of the same
Interaction
In this section we take into account the interaction between the collective mode and the electromagnetic field, in other words, the transverse (photon-like, or retarded) part of the electron-electron interaction, which differs from the longitudinal (Coulomb, or nonretarded) part of the interaction. For the q = 0 mode, the coupling between the single particle excitations can be mediated only through the longitudinal Coulomb field, but not through a transverse EM field of q = 0, which has no E, component. Therefore, there are no further shifts beyond that given in the last section. For inter-subband excitation with finite q, an additional photonic coupling arises(Fig.(lc)). This should, naturally, lead to further correction to the frequency of the collective charge density mode. This retarded interaction does not, however, affect the spin density excitations (in the absence of an external magnetic field) since the magnetic dipolar interaction is generally negligible. It is not difficult to calculate the dipolar coupling of the 1-2 collective charge density mode (cd) to the photon field (ph). The interaction Hamiltonian is written as:
=
q /lk~
X,2k'~'
(]a)
2k' - qo"
/~/ka//
lk ÷ qo
(~b)
q
+ c.c.)
%k'o"
q~'
q,qz k¢
= Eg(q'q*)(P~l(q)C~qq" @P~l(q)c~,,011) qqz
where c and c t are the photon operators and 9(q, q~) is the transition matrix element given by: g(q, qz) =
=(
(0 I ( - ~ P " A)Pll I 0)
e 2/~-~ 1
--~/ ~)(
Ip=Az [2)(m - n 2 ) ( 1 2 )
Starting from this interaction Hamiltonian, we can
(lc) i Figl. Diagrams showing inter-subband direct Coulomb (la), exchange Coulomb (lb) a~d radiative (lc) interac -: tions, l k ¢ designates electron in the 1st subhand with in plane momentum k and spin ¢(-t-, - or T, ~),etc.
84
ELECTRON GAS IN QUANTUM WELLS
energy and q as that of the collective mode. A photon of 100meV, which is the typical energy of intet;-subband transition for a quantum well of width 0.01#m, has a wavelength of 12.4pro. Therefore, the shift is expected to be very small compared with the contribution of the Coulomb interaction. The resonant splitting estimated by Dahl and Sham [7] agrees with our analysis. This by no means implies that the interaction between the EM field and the collective mode is negligible. Coupling of the inter-subband charge density mode with photons leads to a strong infrared absorption peak which directly determines the energy of the collective mode [5]. The radiative shift is expected to have an even smaller effect on the electric dipole forbidden inter-subband modes, because of the much weaker coupling strength to photons.
4
Vol. 84, Nos 1/2
D P
I
i (2~)
Resonant Raman Scattering
processes are viable for the excitation of the charge density mode; while only the exchange process is viable for the excitation of the spin density mode. An important difference between exchange and direct Coulomb coupling is: when the wavefunetions of the two scattering electrons do not overlap, the exchange Coulomb coupling vanishes while the direct Coulomb coupling still exists. This can be seen most clearly when the photoexcited probe electron is in a different spatial region than the electron gas it excites. An experiment is proposed here to elucidate the different nature of the exchange and direct Coulomb interactions. By using an asymmetric double quantum well structure, we can resonantly photoexcite an electron into one of the quantum wells and look at the excitations of the electron gas in the neighboring well. The subband wavefunctions of the two wells have a small overlap, which is tunable with the separation between the two wells or with the height of the barrier between the two wells. Specifically, for a modulation doped double quantum well (Fig. (2a)) with two wells of different widths, by tuning the incident laser to resonate with the unoccupied subband in one of the wells, one should observe the excitation of inter-subband modes in that well through polarization resolved resonant Raman spectroscopy. In
1
(2b)
In the resonant Raman scattering processes, electrons are photoexcited into one of the conduction subbands. It couples to both charge density v~nd spin density excitations primarily through nonretarded Coulomb interaction with the electron gas. The Hamiltonian describing such interaction is similar to eq.(1), except we now focus on a single externally introduced electron, which serves as a probe of the electron gas. Some insights can be gained directly from the interaction Hamiltonian. An electron couples, in general, to the inter-subband charge density mode and, depending on the spin state of the electron, one or more of the spin density modes. Two types of scattering processes are possible, namely the exchange and direct Coulomb scattering process. Both
1L I
2R !
(2c)
Fig2. In (2a) is an asymmetric double quantum well, with an interwell harrier of width D and height U, 1L designates the 1st subband of the Left well, etc. (2b,c) are the typical direct (2b) and exchange (2c) Coulomb coupling mechanisms for interwell excitations encountered in resonant Raman scattering.
addition, one should be able to observe excitation of the inter-subband charge density mode in the neighboring well through the direct Coulomb coupling across the well (Fig(2b)), while excitation of the spin-density mode in the neighboring well through the exchange Coulomb coupling across the well (Fig(2c)) should be very weak due to the small inter-well wavefunction overlap. A reduction of inter-wellspacing or the barrier height willincrease the coupling to the spin-denslty mode and demonstrate the effect of wavefunction overlap on exchange coupling.
5
Nonlinear Optical Processes
In this section, we discuss the role of inter-subband transitions in nonlinear optical processes. Three-wave mixing in the cases of both doped and undoped quantum
Vol. 84, Nos 1/2
ELECTRON GAS IN QUANTUM WELLS
well structures has been dealt with extensively before [11,12]. Here, I will focus on the electromagnetic radiation induced spectral shift, which is related to the ac Stark effect. For an undoped quantum well, the inter-subband transition apparently can not be the first step of a multiple step process involved in nonlinear optical phenomena. If an intense beam of p-polarized infrared light resonant with the inter-subband transition energy (say, cl - c2) is incident on the quantum well, there will be no absorption, except that due to thermally excited electrons(holes). However, if we are to put (photoexcite) a probe electron into the conduction subband(cl or c2), its energy will be altered by the presence of the EM field since it can readily exchange photons with the EM field(i.e., absorb and re--emit photons), leading to a finite self energy correction. This radiation induced self energy correction is generally known as the optical Stark effect, which has been studied in various cases, as with excitons in semiconductor quantum wells [14] and in insulators [13]. The energy shifts of the unoccupied states can be detected with linear absorption, as well as resonant sum or difference frequency generation of an input beam corresponding to the interband excitation energy mixing with the infrared beam. For a doped quantum well, an intense near-resonant p-polarized infrared radiation will cause shift of the charge density mode due to enhanced photonic coupling, as well as a change in the real population of the two subbands when resonant. This will cause the infrared absorption spectrum to shift and saturate, and may cause interesting nonlinear dynamics between the EM field and the excitations of the electron gas in the quantum well, which forms another subject in itself. As pointed out in [12], an infrared photon interacting with the collective inter-subband charge density mode forms a polariton. Therefore, three wave mixing involving a photon near the interband gap and an infrared photon can be alternatively viewed as coherent Raman scattering by a polariton, where the coupling of the virtually excited electron to the polariton has both a photonic and a Coulombic character.
85
of screening in the dielectric response function.) We also proposed an experiment to elucidate the difference between exchange and direct Coulomb coupling mechanisms through resonant Raman scattering in double quantum well structures, and discussed the optical Stark shift of inter-subband transitions induced by an intense EM field. D e d i c a t i o n - - I t is my pleasure to dedicate this paper to Professor Elias Burstein on the occasion of his 75th birthday.
Acknowledgements---I thank Prof. Burstein who has stimulated much of this work through many discussions, and Dr. Aron Pinczuk for the communication of his paper [9] before publication.
References [1] W.P. Chen, Y.J. Chen and E. Burstein, Surf. Sci. 58, 263(1976). [2] G. Abstreiter, M. Cardona and A. Pinczuk, Light Scattering in Solids IV, eds., M. Cardona and G. GSntherodt, p5., Springer (1984) [3] L.C. West and S.J. Eglash, Appl. Phys. Left. 46, 1156(1985). [4] P. yon Allmen, M. Berz, F.K. Reinhart and G. Harbeke, Superlattices and Micmstructures, 5, 259(1989). [5] G, Harbeke, Progress in Electron Properties of Solids eds., E. Doni et al, p373, Kluwer Academic Publishers (1989). [6] S.J. Allen Jr., D.C. Tsui and B. Vinter, Solid State Commun. 20, 425(1976). [7] D.A. DaM and L.J. Sham, Phys. Rev. B16, 651(1977). [8] F. Stern, Phys. Rev. Left. 18, 546 (1967).
6
Summary
In summary, we have formulated explicitly the intersubband collective electronic excitations in a semiconductor quantum well and derived the shift of the frequency of the inter-subband collective modes from the bare single particle inter-subband spacing. The shift in frequency is a many-body effect, it is induced by the direct and exchange Coulomb interactions including self energy correction, as well as interaction mediated by electromagnetic field. (We have left out the interaction mediated by the lattice optical phonons. The electronLO phonon interaction induces a further correction to the charge density excitation, but not to the spin density excitation. It has been treated adequately in terms
[9] A. Pinczuk, S. Schmitt-Rink, G. Danan, J.P. Valladares, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 63, 1633(1989). [10] G. D. Mahan, Many Particle Physics Plenum Press (1981). [11] M.Y. Jiang and E. Burstein, Progress in Electron Properties of Solids eds., E. Doni et al, p395, Kluwer Academic Publishers (1989) [12] E.Burstein and M.Y. Jiang, Light Scattering in Semiconductor Structures and Superlattices eds., D.J. Lockwood and J.F. Young, p441, Plenum Press (1991).
86
F.J.2~CTRON GAS IN QUANTUM WELLS
[13] D. Fr6hlieh, R. Wille, W. Schlapp and G. Weiman, Phys. Rev. Left. 59, 1748 (1987).
Vol. 84, Nos 1/2
[14] A. yon Lehman, D. Chemla; J.E. Zueker and J.P. Heritage, Optics Lett. 11,609 (1986).
Solid State Communications, VoL 84, Nos. 1/2, PP. 81-86, 1992. Printed in Great Britain.
0038-1098/9255.00+ .00 Pergamon Press Ltd
COLLECTIVE INTER-SUBBAND EXCITATIONSOF ELECTRON GAS IN QUANTUM WELLS IN INFRARED ABSORPTION, RAMAN SCATTERING AND NONLINEAR OPTICAL PROCESSES Ming Ya Jiang Physics Department and Laboratory for Research on the Structure of Matter University of Pennsylvania, Philadelphia, PA 19104, USA
Received May 8, 199P b~t M. Cardona We discuss the role of collective inter-subband excitations of electron gases confined in quantum wells in infrared absorption, resonant Raman scattering and nonlinear optical processes. First, exchange as well as direct Coulomb interaction are invoked in a theoretical model to calculate the shift of the collective inter-subband modes from the single particle excitation energy. We then formulate the coupling of the collective inter-subband charge density mode to the electromagnetic field. The coupling mechanisms via exchange and direct Coulomb scattering for resonant Raman processes are discussed. An experiment is designed that can differentiate between the two mechanisms, based on their dependence on wavefunction overlap. Finally, we discuss nonlinear optical processes involving inter-subband transitions and collective excitations.
1
Introduction
a large polarizability along the z direction, which leads t o various types of linear and nonlinear responses from a layered quasi-2D quantum well system. In the first part of this paper, an alternative formulation of the collective inter-subband modes is given starting from a model Hamiltonian that yields the exchange as well as the direct Coulomb screening on both the inter-subband charge density and spin density modes with a clear physical picture. In the second part, we treat the problem of coupling of the inter-subband excitations with photons that leads to a shift due to radiative interaction (Lamb shift). In the third part, we discuss the coupling mechanisms of exchange and direct Coulomb scattering for resonant Raman processes and propose an experiment that can differentiate between the two mechanisms, based on the difference in the dependence of exchange and direct Coulomb interaction on wavefunction overlap. In the last part, we discuss the role of inter-subband transitions in nonlinear optical processes and, in particular, the spectral shift upon resonant infrared pumping as a result of the optical Stark effect.
The confinement of electrons in a quantum well introduces energy gaps in single particle excitations. Collective modes, in principle, exist at every intersubband edge for an electron gas confined in a quantum well. Direct observations of collective intersubband modes were achieved experimentally with resonant Raman scattering [2] and resonant infrared absorption [3,4] in doped semiconductor heterojunctions and quantum wells. These observations, especially the Raman scattering studies, have led to various theoretical formulations of the collective inter-subband excitations. The shift of the energy of the collective charge density mode from the single particle inter-subband excitation energy has been attributed to direct Coulomb screening (i.e., the depolarization field effect) [1,6,7]. More recent experimental studies [9] indicated that the spin density excitation, which was initially thought to be unscreened, also has a sizable shift which is attributed to exchange Coulomb screening. In contrast to the collective electronic excitation (plasmon) in a 3-D electron gas which is inaccessible via optical excitation due to its longitudinal character, the intersubband plasmons in general couple to photons that have an electric field component (E,) perpendicular to the walls of the quantum well. The large oscillator strength associated with the collective mode gives rise to
2
D i r e c t and E x c h a n g e C o u l o m b Interaction
A quasi-2D electron gas is almost ideally realized with a 81
82
ELECTRON GAS IN QUANTUM WELLS
modulation-doped GaAs-(AI,Ga)As type Of direct band gap semiconductor quantum well, where electrons (or holes) move in a lattice free of impurities. Adopting the effective mass approximation, we c~n view the system as a free electron gas confined in a slab-like region. The single particle electron wave function and spectrum is determined by the confining potential. By separating the in-plane and z part of the wavefunction and energy we can write them as: ¢,,(k) = %,(z)exp(ik. p) h~k2 2rn* The Hamiltonian for such an electron gas system ignoring electron-lattice interaction is written as: e,,(k) = e,, +
Z:
i,,l II o"
+ Ei~mn
1
.
~(ij IVI rtm)ai,k+q.oaj,t,, t
',o'an,k,~, (1)
kk'q¢'¢
where the wave the function normalization is taken over a unit area of the quantum well. For our purpose here we will consider only the first two subbands cl and c2 (treatments involving other subbands are similar). We can further separate the intrasubband and inter-subband interaction terms. Let us first look at the the inter-subband terms. By neglecting two electron(hole) excitation processes (where two electrons in the same subband scatter each other to the other subband) and introducing the inter-subband density fluctuation operator:
V(q,z, z') is the electron-electron interaction potential dependent on the dielectric constant of the medium as well as its geometry. For a single quantum well of width 1, dielectric constant q for the substrate, e2 for the well region, and z = 0 at the center of well, V is given [7,8] by: 2~l'e2 / i, '~ V(q, z, z') = ~ {e -~'*-~ '+ e2 - el e_q(i.+.,_q+l.+~,+tl)
e2q ~
k
= (v. - ~-v.) ~ [ ( & + +
+ &__)(e~,++ + ~21--)1
q
+ (-½v,)~[(&.,-+
- &--)(~,++
-ih~ =
+ &-+)Ce~+- + e~+-)]
q where V( and Ve are respectively the inter-subband direct and exchange Coulomb integrals defined as: (211V(q , z , z ' ) l l 2 )
= / ~o~(z)tp,(z)VcpI(z')~o2(z')dzdz' V~(q)
E ( , 2 ( k ) -- ,l(k))a~,k,oal,k,, k,a
+(~(o) - ~-~(o)) ~(al,~,<,a,,,<,<, - a[~,.a~,~,o)A,(O) (6) k,o"
By taking the thermal average value for the number operator, and assuming identical in-plane electron masses of both subbands, the energy for the charge density mode at q = 0 is given by:
(~
1
-
c,) + (~(0) - ~v.(0))(.,
-
,,~) (7)
where nl and nu represent the thermally averaged occupation number of the first and second subband per unit area. The inter-subband spin density excitations, S~(q) = i ttk+qalttka -1 - (where Si,o is the ith Pauli matrix) kota s ¢la are three-fold degenerate. Its energy at zero wavevector is given by: hfl$e(0) = (~ - ~,) - ~v.(0)(~, - ~2) ,
q
V~(q) =
= [A,, p~(0)]_
- e2,--)[
q
+ (-~v,)~[(&+_
(5)
(2)
the inter-subband part of the interaction Hamiltonian can be written as:
~,2
)
e2 + el
The scattering processes due to the direct and exchange Coulomb interactions are shown as diagrams in Fig.(la,b). Note that the singularity of V~(q) at q = 0 is cancelled by the corresponding term in the electronlattice interaction, leaving a finite value for V~ at q = O. The first term in eq.(3) represents the charge density excitation, the rest of the terms represent the three components of spin density excitations. We can find the energies of these excitations by their equations of motion. For the inter-subband charge density excitation, P~l(q) = ~ t o a2k+qaalkat at zero wavevector:
hf~(O) $~.w¢(q) = Z a~,k+q, o'amk,~
Vol. 84, Nos 1/2
= (121V(q,z,z') l l2) = f ¢p~(z)9o2(z)V~p~(z')~ol(z')dzdz'
(4)
(8)
The nature of these collective modes lies in the Coulomb coupling of the iso-energetic single particle excitations. It is straightforward to extend the above formalism to the case of cl-cs, cl-c4.., excitations. Each collective mode is shifted away from the corresponding single particle excitation energy by an amount proportional to the occupation number difference in the subbands involved. The intra-subband interaction gives rise to a finite self energy correction to the subband energy due to the exchange interaction [10]. The exchange self energy correction is generally dependent on wavevector. The main contribution is, however, a constant shift due to the qindependent part of (ii [ V(q,z, z') [ ii). It is propor-
Vol. 84, Nos I/2
ELECTRON GAS IN QUANTUM WELLS
tional to the occupation density of each subband, given
by: A~i =
- l ( i i l V ( q = O,z, z')lii)ni
(9)
The difference in self energy corrections tO each subband gives rise to another term in the shift of the intersubband modes, to be added to those given in eqs.(7) and (8). It is identical for both modes:
83
calculate the energy correction to the inter-subband collective mode via radiative interaction, which bears resemblance to the Lamb shift in atomic energy levels. Second order perturbation theory gives the following • equation for the energy of the collective mode, which is diagrammatically represented in Fig. (lc) as the virtual photon emission by one electron and subsequent absorption by another electron: *.
Aft = 1(11 ] V [ ll)nl - 1(22 [ V [ 22)n2
Photonic
(13)
(10)
We note here the difference between the RPA calculation with ours. The RPA calculation gives a manybody correction through the poles of the response function and the result is slightly different from the results of eqs.(7-10). In addition, all the approximations are identified in our formulation.
3
I g(q,q=) is W q - ftq
A divergence is encountered in carrying out the summation over qz which is an artifact expected in the calculation of the Lamb shift to he removed with renormalization. We will instead look at some general features based on eq.(13). The shift will depend on the fine structure constant a, and the ratio of the quantum well width to the photon wavelength. The major contribution to the shift comes from interaction with a photon of the same
Interaction
In this section we take into account the interaction between the collective mode and the electromagnetic field, in other words, the transverse (photon-like, or retarded) part of the electron-electron interaction, which differs from the longitudinal (Coulomb, or nonretarded) part of the interaction. For the q = 0 mode, the coupling between the single particle excitations can be mediated only through the longitudinal Coulomb field, but not through a transverse EM field of q = 0, which has no E, component. Therefore, there are no further shifts beyond that given in the last section. For inter-subband excitation with finite q, an additional photonic coupling arises(Fig.(lc)). This should, naturally, lead to further correction to the frequency of the collective charge density mode. This retarded interaction does not, however, affect the spin density excitations (in the absence of an external magnetic field) since the magnetic dipolar interaction is generally negligible. It is not difficult to calculate the dipolar coupling of the 1-2 collective charge density mode (cd) to the photon field (ph). The interaction Hamiltonian is written as:
=
q /lk~
X,2k'~'
(]a)
2k' - qo"
/~/ka//
lk ÷ qo
(~b)
q
+ c.c.)
%k'o"
q~'
q,qz k¢
= Eg(q'q*)(P~l(q)C~qq" @P~l(q)c~,,011) qqz
where c and c t are the photon operators and 9(q, q~) is the transition matrix element given by: g(q, qz) =
=(
(0 I ( - ~ P " A)Pll I 0)
e 2/~-~ 1
--~/ ~)(
Ip=Az [2)(m - n 2 ) ( 1 2 )
Starting from this interaction Hamiltonian, we can
(lc) i Figl. Diagrams showing inter-subband direct Coulomb (la), exchange Coulomb (lb) a~d radiative (lc) interac -: tions, l k ¢ designates electron in the 1st subhand with in plane momentum k and spin ¢(-t-, - or T, ~),etc.
84
ELECTRON GAS IN QUANTUM WELLS
energy and q as that of the collective mode. A photon of 100meV, which is the typical energy of intet;-subband transition for a quantum well of width 0.01#m, has a wavelength of 12.4pro. Therefore, the shift is expected to be very small compared with the contribution of the Coulomb interaction. The resonant splitting estimated by Dahl and Sham [7] agrees with our analysis. This by no means implies that the interaction between the EM field and the collective mode is negligible. Coupling of the inter-subband charge density mode with photons leads to a strong infrared absorption peak which directly determines the energy of the collective mode [5]. The radiative shift is expected to have an even smaller effect on the electric dipole forbidden inter-subband modes, because of the much weaker coupling strength to photons.
4
Vol. 84, Nos 1/2
D P
I
i (2~)
Resonant Raman Scattering
processes are viable for the excitation of the charge density mode; while only the exchange process is viable for the excitation of the spin density mode. An important difference between exchange and direct Coulomb coupling is: when the wavefunetions of the two scattering electrons do not overlap, the exchange Coulomb coupling vanishes while the direct Coulomb coupling still exists. This can be seen most clearly when the photoexcited probe electron is in a different spatial region than the electron gas it excites. An experiment is proposed here to elucidate the different nature of the exchange and direct Coulomb interactions. By using an asymmetric double quantum well structure, we can resonantly photoexcite an electron into one of the quantum wells and look at the excitations of the electron gas in the neighboring well. The subband wavefunctions of the two wells have a small overlap, which is tunable with the separation between the two wells or with the height of the barrier between the two wells. Specifically, for a modulation doped double quantum well (Fig. (2a)) with two wells of different widths, by tuning the incident laser to resonate with the unoccupied subband in one of the wells, one should observe the excitation of inter-subband modes in that well through polarization resolved resonant Raman spectroscopy. In
1
(2b)
In the resonant Raman scattering processes, electrons are photoexcited into one of the conduction subbands. It couples to both charge density v~nd spin density excitations primarily through nonretarded Coulomb interaction with the electron gas. The Hamiltonian describing such interaction is similar to eq.(1), except we now focus on a single externally introduced electron, which serves as a probe of the electron gas. Some insights can be gained directly from the interaction Hamiltonian. An electron couples, in general, to the inter-subband charge density mode and, depending on the spin state of the electron, one or more of the spin density modes. Two types of scattering processes are possible, namely the exchange and direct Coulomb scattering process. Both
1L I
2R !
(2c)
Fig2. In (2a) is an asymmetric double quantum well, with an interwell harrier of width D and height U, 1L designates the 1st subband of the Left well, etc. (2b,c) are the typical direct (2b) and exchange (2c) Coulomb coupling mechanisms for interwell excitations encountered in resonant Raman scattering.
addition, one should be able to observe excitation of the inter-subband charge density mode in the neighboring well through the direct Coulomb coupling across the well (Fig(2b)), while excitation of the spin-density mode in the neighboring well through the exchange Coulomb coupling across the well (Fig(2c)) should be very weak due to the small inter-well wavefunction overlap. A reduction of inter-wellspacing or the barrier height willincrease the coupling to the spin-denslty mode and demonstrate the effect of wavefunction overlap on exchange coupling.
5
Nonlinear Optical Processes
In this section, we discuss the role of inter-subband transitions in nonlinear optical processes. Three-wave mixing in the cases of both doped and undoped quantum
Vol. 84, Nos 1/2
ELECTRON GAS IN QUANTUM WELLS
well structures has been dealt with extensively before [11,12]. Here, I will focus on the electromagnetic radiation induced spectral shift, which is related to the ac Stark effect. For an undoped quantum well, the inter-subband transition apparently can not be the first step of a multiple step process involved in nonlinear optical phenomena. If an intense beam of p-polarized infrared light resonant with the inter-subband transition energy (say, cl - c2) is incident on the quantum well, there will be no absorption, except that due to thermally excited electrons(holes). However, if we are to put (photoexcite) a probe electron into the conduction subband(cl or c2), its energy will be altered by the presence of the EM field since it can readily exchange photons with the EM field(i.e., absorb and re--emit photons), leading to a finite self energy correction. This radiation induced self energy correction is generally known as the optical Stark effect, which has been studied in various cases, as with excitons in semiconductor quantum wells [14] and in insulators [13]. The energy shifts of the unoccupied states can be detected with linear absorption, as well as resonant sum or difference frequency generation of an input beam corresponding to the interband excitation energy mixing with the infrared beam. For a doped quantum well, an intense near-resonant p-polarized infrared radiation will cause shift of the charge density mode due to enhanced photonic coupling, as well as a change in the real population of the two subbands when resonant. This will cause the infrared absorption spectrum to shift and saturate, and may cause interesting nonlinear dynamics between the EM field and the excitations of the electron gas in the quantum well, which forms another subject in itself. As pointed out in [12], an infrared photon interacting with the collective inter-subband charge density mode forms a polariton. Therefore, three wave mixing involving a photon near the interband gap and an infrared photon can be alternatively viewed as coherent Raman scattering by a polariton, where the coupling of the virtually excited electron to the polariton has both a photonic and a Coulombic character.
85
of screening in the dielectric response function.) We also proposed an experiment to elucidate the difference between exchange and direct Coulomb coupling mechanisms through resonant Raman scattering in double quantum well structures, and discussed the optical Stark shift of inter-subband transitions induced by an intense EM field. D e d i c a t i o n - - I t is my pleasure to dedicate this paper to Professor Elias Burstein on the occasion of his 75th birthday.
Acknowledgements---I thank Prof. Burstein who has stimulated much of this work through many discussions, and Dr. Aron Pinczuk for the communication of his paper [9] before publication.
References [1] W.P. Chen, Y.J. Chen and E. Burstein, Surf. Sci. 58, 263(1976). [2] G. Abstreiter, M. Cardona and A. Pinczuk, Light Scattering in Solids IV, eds., M. Cardona and G. GSntherodt, p5., Springer (1984) [3] L.C. West and S.J. Eglash, Appl. Phys. Left. 46, 1156(1985). [4] P. yon Allmen, M. Berz, F.K. Reinhart and G. Harbeke, Superlattices and Micmstructures, 5, 259(1989). [5] G, Harbeke, Progress in Electron Properties of Solids eds., E. Doni et al, p373, Kluwer Academic Publishers (1989). [6] S.J. Allen Jr., D.C. Tsui and B. Vinter, Solid State Commun. 20, 425(1976). [7] D.A. DaM and L.J. Sham, Phys. Rev. B16, 651(1977). [8] F. Stern, Phys. Rev. Left. 18, 546 (1967).
6
Summary
In summary, we have formulated explicitly the intersubband collective electronic excitations in a semiconductor quantum well and derived the shift of the frequency of the inter-subband collective modes from the bare single particle inter-subband spacing. The shift in frequency is a many-body effect, it is induced by the direct and exchange Coulomb interactions including self energy correction, as well as interaction mediated by electromagnetic field. (We have left out the interaction mediated by the lattice optical phonons. The electronLO phonon interaction induces a further correction to the charge density excitation, but not to the spin density excitation. It has been treated adequately in terms
[9] A. Pinczuk, S. Schmitt-Rink, G. Danan, J.P. Valladares, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 63, 1633(1989). [10] G. D. Mahan, Many Particle Physics Plenum Press (1981). [11] M.Y. Jiang and E. Burstein, Progress in Electron Properties of Solids eds., E. Doni et al, p395, Kluwer Academic Publishers (1989) [12] E.Burstein and M.Y. Jiang, Light Scattering in Semiconductor Structures and Superlattices eds., D.J. Lockwood and J.F. Young, p441, Plenum Press (1991).
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F.J.2~CTRON GAS IN QUANTUM WELLS
[13] D. Fr6hlieh, R. Wille, W. Schlapp and G. Weiman, Phys. Rev. Left. 59, 1748 (1987).
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[14] A. yon Lehman, D. Chemla; J.E. Zueker and J.P. Heritage, Optics Lett. 11,609 (1986).