Solid State Communications, Vol. 93, No. I I, pp. 915-920, 1995 Copyright (0 1995 Elsevier Science Ltd

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RAMAN SCATTERING BY CORRELATED ELECTRONS IN QUANTUM DOTS IN A MAGNETIC FIELD

Pawel Hawrylak

Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, KlA 0R6, Canada

(Received 13 October 1994 by D.J. Lockwood)

We calculate the dynamical structure factor S(Q,w) for correlated electrons in quantum dots in a magnetic field. The excitation spectrum of quantum dots, given by S(Q,w), can be measured by electronic Raman scattering. By sweeping the magnetic field one should be able to observe the incompressible (large gaps in the excitation spectrum) and compressible (soft modes associated with phase transitions) electronic structure of quantum dots.

Keywords: A. nanostructures, A. semiconductors, D. electron- electron interactions, D. fractional quantum Hall effect, E. inelastic light scattering.

QUANTUM DOTS are artificial atoms with a well controlled number of electrons N (N = 1,2,3,4) created in semiconductor quantum wells by laterally confining a two-dimensional electron gas [l-3]. The confining potentials are usually smooth and are therefore well approximated by a parabolic con- fining potential [l-4]. In contrast, electrons in normal atoms are confined by a singular, Coulomb potential. As a consequence, there are three main features which set artificial atoms apart from their natural counterparts [4]: (a) the dynamical hidden symmetries associated with the parabolic form of the confining potential; (b) the many-particle hidden symmetry associated with the separability of the Center of Mass and relative motion in a conjned system; (c) the possibility of the broken time reversal symmetry associated with very strong effective magnetic fields realizable in semiconductors. These symmetries, together with the competing effects of the electron- electron interactions, Zeeman, and kinetic energies, play an important role in determining the ground and excited states of artificial atoms [l-5]. The competi- tion between these different mechanisms, controlled by an applied magnetic field, leads to a series of incompressible ground states with “magic angular momenta” values [4, 51, familiar in the context of the

incompressible liquid of the Fractional Quantum Hall Effect [6]. This in turn controls transport [7, 81, thermodynamic [9], and optical properties [lo] of quantum dots.

A number of spectroscopic techniques have been applied experimentally and studied theoretically to unravel the electronic structure of quantum dots. It is now well known that Far Infrared Spectroscopy (FIR) measures only single particle energies [ 1, 4, 5, IO]. The information about ground state energies can be inferred indirectly from either the Photo- luminescence spectra (PL) [lo], or the addition spectra [4, 7, 81. The addition spectra can be measured via Single Electron Capacitance Spectro- scopy (SECS) [3, 41, vertical tunneling [3], or in lateral transport experiments [7, 81. The excitation spectrum is on the other hand much more difficult to measure. Recent experiments [l l] in this direction applied nonlinear tunneling spectroscopy which, like addition spectroscopy, involves tunneling of a guest electron through a quantum dot. Adding this extra electron to a highly correlated N-electron state is a large and complicated perturbation [8]. It is therefore important to employ a spectroscopic tool which does not involve an addition of electrons and allows for a direct measurement of an excitation

915

916 CORRELATED ELECTRONS IN A MAGNETIC FIELD Vol. 93, No. 11

spectrum with a fixed number of electrons. The resonant electronic Raman scattering (or any spectroscopy involving scattering of photons, phonons, or neutrons) satisfies this criterium, and is well suited for the study of the excitation spectrum of quantum dots. It has been already successfully applied to the study of excitations in a number of low dimensional semiconductor structures [12-151.

Here we study theoretically the charge density excitation spectrum of quantum dots by calculat- ing the dynamical structure factor S(Q,w), and hence the electronic Raman spectra of strongly interacting electrons in quantum dots. The exact numerical and analytical calculations of the electronic structure explicitly incorporating hidden single- and many-particle symmetries are used.

The two-dimensional artificial atom (dot) [4] contains N electrons confined by an effective parabolic potential with a characteristic energy UN. The frequency WN has a contribution from the externally imposed potential and that of a positive charge +Ne at a distance d away from the plane of the dot. The atom is placed in a magnetic field B normal to the plane of the dot. The positive charge assures charge neutrality of the atom and plays the role of the gate. The single particle Hamiltonian corresponding to a particle moving in a parabolic potential in the presence of a magnetic field is diagonalized [4] by a transformation into a pair of harmonic oscillator lowering (raising) operators (a,b). The a and b oscillators evolve into the inter- and intra-Landau level oscillators with increasing magnetic field B. The single particle energies are

E n,m = 52+(n + l/2) + sz_(m + l/2),

eigenstates

]m,n) = (b+)“(~+)“]o)/(n!m!)1’2

and the two harmonic oscillator frequencies are given

by

R,(_) = {@ZY$ + (-)w,}/2.

The w, is the cyclotron energy, lo = l/(m~,.)‘/~ is the magnetic length, m is the effective mass, and R = [l + (2wN/wc)2]“2. The explicit separation of the CM and the relative motion of electrons is achieved through a generalized Jacobi trans- formation [4] U/,,, = exp{i(l - l)m2n/N}/fi, trans-

forming individual particle boson creation operators (u;) into CM and relative creation operators (AL): A,f = C, U!,,,a,!$ The motion of (N - 1) relative particles is determined by electron-electron interactions and governed by the relative Hamiltonian:

H,,l = 2 R+(&A, + l/2) + R_(B;B,,, + l/2) m=2

(1) I \m=2 /I

where Q$,,, = Q*PJ,‘, - uj+m>t Q = (4x + q,$ol~ and v(q) = 2?re2/coq. The relative Hamiltonian depends only on relative coordinates A,. The CM Hamiltonian is simply given by

HCM = R+(A:A, + l/2) + Ck(B:BI + l/2),

and index 1 stands for a CM particle. The states of the relative Hamiltonian are

built from simple products of single particle states II?=‘=, JNiMi) and spin states by the use of the antisymmetric operator As = CrJ iDet(Pi)Pi, where Pi are elements of the (N - 1) dimensional representation of the permutation group SN.

As an illustration let us analyze in detail the problem of three spin polarized electrons 14-61. For N = 3 electrons there is a CM particle (1) and two “relative particles” (2), (3). The antisymmetrization of spin polarized states of the relative harmonic oscillators (coupled bosons IM2, N2), (Ms, Ns)) leads to a severe restriction on possible fermionic states of the relative Hamiltonian

IhN3, M2> Was = 5 (I&, NJ, &, MS)

where the quantum number is restricted to L=M2-M3+N2-N3=3m (m=0,1,2,3 ,... ). The Coulomb interaction conserves the total relative angular momentum R = ( M2 + M3) - ( N2 + N3), which restricts the diagonalization of the relative Hamiltonian to each value of the relative angular momentum R. This can be done almost analytically when the time-reversal symmetry is broken. This

Vol. 93, No. 11 CORRELATED ELECTRONS IN A MAGNETIC FIELD 917

“lowest Landau level” approximation is equivalent to setting N2 = Ns = 0. It is useful to label our states by quantum numbers R and L. A very simple sequence of the low angular momentum states IR, L) can be constructed in analogy with Laughlin’s approach [6]:

1{13> 3)], {I5,3)], {16,6j], {17,3)]> {I&6)], {19,9>, ]6,3)}, . . .]. The state with R = 4 is missing and degeneracies proliferate only for R > 8.

It is useful now to compare our exact states IR, L)IM1) with states one would obtain by exact diagonalization in the space of Slater determinants. Naturally, results of exact diagonalization yield states which appear to depend on the form of the electron- electron interactions. In our approach these states are simply constructed on the basis of symmetry. For example, the “magic states” IR = L, L > IO), which are an excellent approximation to ground states of the quantum dot in the strong magnetic field limit, can be expressed in terms of Slater determinants

lml,m2rm3):

hm2m3JL7 -9 = C {m III} mi+mj 1’ I

( L ){(mi::) mi + mj

+ ( )I mj i SiIl[(??Zi - mj)2n/3]

x J3”+*M!/ml!mz!m3! ’ 12)

where pairs {mi, mj} are defined on a circle

[{mi,mj) = {w,m2), {m2, m31, {m3, ml II and ml + m2 + m3 = L = R. These magic states accom- modate an integer number of flux quanta per particle and one can associate filling factors v with these states (v = 3/R, R = 3,6,9,. . .). To illustrate the charge distribution associated with these coherent states we show in Fig. 1 the occupations f(m) of single particle states

Im) [S(m) = %z,mj I(mim2msIR,L > IMi)j21 for a series of low angular momentum states. The charge distribution starts as a compact droplet at R = 3 and proceeds to a charge distribution of a “ring” at R = 6. The ring structure persists for larger angular momentum.

The basic physics of this problem is best illustrated by considering [6] the expectation value of the area S2(S = (r3 - rl) x (r2 - r,)/2) spanned by 3 electrons:

“(R, L(S2(R, L)” = {L2 + 2 + R + Nz(2 + M3)

+ Nd2 + &))/4- 13)

It is clear that for a given R (and for N2 = N3 = 0), the area spanned by electrons is strongly maximized

R=3, 13,3>(0>

R=4, (3,3+

lln R=5, 15,3>)0>

-1 0 1 2 3 4 5 6 7

m

Fig. 1. Occupation f(m) of single particle states in low exact states IR, L > (MCM).

by having the largest possible L. The largest area for a series of “magic states” with R = L = 3m minimizes any form of repulsive interaction energy, purely on the basis of many-particle symmetries. Which “magic state” is the ground state for a particular value of external parameters depends on the competition of potential energy -1/R and kinetic energy -R. An example of the extremely sparse energy spectrum of the relative Hamiltonian of a quantum dot is shown in Fig. 2(a). For these parameters (wly = 3.37 meV, B = 5 T, and other parameters applicable to GaAs) the ground state is at the value of angular momentum R = 6. For increasing value of the magnetic field higher angular momentum states (R = 9,12,15, . . .) become the ground states. We now construct a total excitation spectrum IMz, N2, MT, N3)“sMI) by attaching the Center of Mass boson (M,) with energy R-MI to relative fermions, and sorting out states according to the total angular momentum Rtot = M, + R

(Rtot remains a good quantum number). The total spectrum, shown in Fig. 2(b), is now completely equivalent to a spectrum one would obtain via direct diagonalization of the 3-electron Hamiltonian in a basis of Slater determinants.

In a confined system the quantization of kinetic energy implies a single particle gap. In the interacting system, the gap in the excitation spectrum collapses when the dot undergoes a phase transition between different angular momentum ground states, as shown in Fig. 2(c).

918 CORRELATED ELECTRONS IN A MAGNETIC FIELD Vol. 93, No. 11

CK

I’

I' ' ' (0)

I I

’ I

' I

I

I without CM motion

1 B=5T, w,=3.37meV

14.0

12.0 , II

,I 1~1,,“““‘1”“““\ I’ ’

5 10.0 ,,, III I \ I

w 8.0

I

6.0 I

,' II II

1” 1 I I” I

4.0 , I I

‘I l with CM mo!

, 1.;5 l.i5 1.45 1.'55 1.'65 I>5 Ii35

a NJ

E m

% 'C 0 .L= 2

iiT E 0.1 0.2 0.3 0.4

E/E,

Fig. 2. (a) Energy spectrum of relative spin polarized electrons; (b) total energy spectrum, including CM excitations; (c) interacting (solid line) and single particle (dashed line) energy gap as a function of the magnetic field.

Let us now turn to the calculation of the dynamical structure factor S(q,w) to demonstrate how these coherent states can be observed experimentally via electronic Raman scattering. For parallel polarization of incident and scattered photons [12-141 the incident photons lose wavevcctor q and frequency w by creating finite wavector and frequency electronic charge density fluctuations. The Raman cross section Z(q, w) is proportional to the imaginary part of the dynamical structure factor S(q, w). The detailed lineshape of the resonant Raman processes depends on a particular resonance used, and hence on the details of the valence band for a particular quantum dot. This is clearly unknown and we shall not attempt to model the resonant process here [15]. The Raman cross section is calculated using Fermi’s golden rule and exact final states If) and energies Ef coupled to the ground state IO) via the density operator p4:

Z(q,w) = c I(flPqlo)12~(Ef - Eo - 4. f

12.0

10.0

B CK 8.0

6.0

4.0

0.01

5 a 0.01 -

0.01

3 a 0.01 -

0

16>-19>

A I I’ I

I I I I

O- I I

I I

I 1 Q=0.8

n- Q=O.l

B=5T, w,=3.37meb

I I .O 0.1 0.2

Q/E0

3

Fig. 3. The low lying excitation spectrum at B = 5T and the Raman spectrum Z(Q, w) as a function of w for different values of wavevector Q. Stable ground state with large gap at R = 6.

The density operator ps = Cff= ,eiqc can be expressed in terms of the relative and CM operators in a way similar to the one used to express coulomb matrix elements in equation (1). However, p4 no longer conserves angular momentum Rtot and couples relative and CM degrees of freedom. The energy spectrum and the Raman cross section Z(Q w) (with Q = qlo/xh%) for the ground state at angular momentum R = 6 is shown in Fig. 3. This is a stable ground state with a significant gap in the excitation spectrum. The Raman spectrum at Q = 0.1 is dominated by a single peak corresponding to a CM excitation of the ground state, with energy R_. This can be easily understood by expanding the density operator for small wavevectors in terms of CM coordinates

ps M 1 + iq 2 r, z I + id%qR,. m=l

By increasing the wavevector Q the spectrum evolves from that of CM excitations to that of excitation of the internal motion. At Q = 0.5 the oscillator strength of the lowest mode is concentrated in the

Vol. 93, No. 11 CORRELATED ELECTRONS IN A MAGNETIC FIELD 919

CK

Q.* - (6~(9> I I

lO.O-

::;:p_--

I I I

I I I

II

Q=0.8

Fig. 4. The low lying excitation spectrum at B = 7T and the Raman spectrum I(Q,w) as a function of w for different values of wavevector Q. Ground state at R = 6 almost degenerate with state at R = 9, artificial atom is approaching a phase transition.

transition between two “magic states” 16,6) and 19,9). The energy of this transition is nevertheless almost degenerate with the energy of the CM excitation. These energies can be untangled by changing the value of the magnetic field. In Fig. 4 we show the energy and Raman spectrum for the value of B = 7T, corresponding to the situation where the ground state of the dot at R = 6 is almost degenerate with the state at R = 9. For values of B > 7T, R = 9 becomes a ground state [4, lo]. Despite this complex behavior, the spectrum at small Q is still dominated by the CM excitations with energy a_. However, with increasing Q a well resolved peak at low energies appears. It signals a phase transition between two almost degenerate states R = 6 and R = 9. By sweeping the magnetic field while maintaining the significant wavevector transfer one should be able to observe the incom- pressible (large gaps in the excitation spectrum) and compressible (soft modes associated with phase transitions) electronic structure of artificial atoms.

In summary, we have calculated the charge

density excitation spectrum, including spectral weights, of strongly interacting electrons in quantum dots in a strong magnetic field. The wavevector and frequency dependence of the dynamical structure factor S(Q, w), measured in electronic Raman, phonon, and neutron scattering, is dominated by collective excitations of the Center of Mass at low scattering wavevectors, but shows a significant oscillator strength due to transition between incom- pressible states of the dot at larger wavevectors. By sweeping the magnetic field at sufficiently large wavevector transfers, the excitation spectrum, and in particular, the opening and closing of energy gaps tuned by the magnetic field can be measured directly.

The choice of the scattering technique is dictated by the fact that the required scattering wavevectors Q should be simultaneously of the order of the size of the quantum dot and the wavelength of the scattering particle.

Acknowledgement - We thank J.J. Palacios, D.J. Lockwood, G.C. Aers, J.J. Quinn and C. Dharma- Wardana for discussions.

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REFERENCES

For recent reviews and references see M. Kastner, Physics Today, 24, Janauary (1993); T. Chakraborty, Comments in Cond. Matter Phys. 16, 35 (1992). Ch. Sikorski & U. Merkt, Phys. Rev. Lett. 62, 2164 (1989); B. Meurer, D. Heitman & K. Ploog, Phys. Rev. Lett. 68, 1371 (1992). R.C. Ashoori, H.L. Stormer, J.S. Weiner, L.N. Pfeiffer, K.W. Baldwin & K.W. West, Phys. Rev. Lett. 71, 613 (1993); Bo Su, V.J. Goldman & J.E. Cunningham, Science, 255, 313 (1992). P. Hawrylak, Phys. Rev. Lett. 71, 3347 (1993). P. Maksym, Physica B184, 385 (1993); P.A. Maksym & Tapash Chakraborty, Phys. Rev. Lett. 65, 108 (1990). R.B. Laughlin, Phys. Rev. B27, 3383 (1983); S.M. Girvin & Terrence Jach, Phys. Rev. B28, 4506 (1983). P.L. McEuen, E.B. Foxman, Jari Kinaret, U. Meirav, M.A. Kastner, N.S. Wingreen & S.J. Wind, Phys. Rev. B45, 1141 (1992); A.S. Sachrajda, R.P. Taylor, C. Dharma-Wardana, P. Zawadzki, J.A. Adams & P.T. Coleridge, Phys. Rev. B47, 6811 (1993). J.J. Palacios et al., Eur. Phys. Lett. 23, 495 (1993); S.-E. Eric Yang, A.H. MacDonald & M.D. Johnson, Phys. Rev. Lett. 71,3194 (1993). P.A. Maksym & Tapash Chakraborty, Phys. Rev. B45, 1947 (1992). P. Hawrylak & D. Pfannkuche, Phys. Rev. Lett. 70,485 (1993); D. Pfannkuche et al., SoIid State Electron. 37, 1221 (1994).

920 CORRELATED ELECTRONS IN A MAGNETIC FIELD Vol. 93, No. 11

11. J. Weis et al., Phys. Rev. Lett. 71, 4019 (1993); P. Hawrylak, J.-W. Wu & J.J. Quinn, Phys. A.T. Johnson et al., Phys. Rev. Lett. 69, 1592 Rev. B32, 5169 (1985). (1992). 13. A. Pinczuk et al., Phys. Rev. Lett. 70,3983 (1993).

12. A. Pinczuk & G. Abstreiter, Light Scattering in 14. A.R.Gonietal., Phys. Rev. Lett.70,1151 (1993). Solids V (Edited by M. Cardona & G. 15. A. Govorov & L.I. Magarill, Phys. Low-Dim. Guntherodt), p. 153. Springer, Berlin (1989); Struct. 2, 61 (1994).