collective excitations in a quasi-3-dimensional electron system

Physica B 184 (1993) 100-105 North-Holland PHYSICA

Collective excitations in a quasi-3-dimensional electron system

H . D . D r e w , X. Ying and K. Karrai '

Joint Program for Advanced Electronic Materials, Laboratory for Physical Sciences, College Park, MD, USA and Physics Department, University of Maryland, College Park, MD, USA

M. S h a y e g a n and M. Santos Department of Electrical Engineering, Princeton University, N J, USA

The far-infrared magneto-optical properties of remotely doped wide-parabolically graded AIGaAs quantum wells is reviewed. The optical response of the system is the same as that of a single-component three-dimensional plasma. Extensive studies were made of the plasma-shifted cyclotron resonance in the Voigt geometry. It is shown that the resonance frequency is independent of the areal density which demonstrates the generalized Kohn theorem. The internal modes of the system are excited by using wells with periodic and aperiodic perturbations superimposed on the parabolic potential. Dimensional modes and grating modes are identified which permit the wave-vector assignment of the internal magneto-plasma oscillations. From results on different samples and magnetic fields the dispersion relation for magneto- plasmons is measured. A roton excitation is observed at q ~2/ l o which is good agreement with the predictions of calculations based on the single-mode approximation.

Strong interactions in many-body systems gen- erally lead to highly correlated ground states. These correlations are manifest by modulations in the pair-correlation function which can be determined, in principle, from elastic diffraction measurements of the static structure factor. The interactions also lead to collective excited states of these systems that deviate significantly from the single-particle excitations of noninteracting systems. These collective excitations can be described in terms of density waves and their wave- vector dispersion is expected to reflect the ground-state correlations through the spectral features in the static structure factor.

The classic example is superfluid 4He in which the collective excitations are phonons at low wave vector q and rotons at high q [1]. The dispersion relation of these excitations has been measured by inelastic neutron scattering. The

Correspondence to: H.D. Drew, Department of Physics and Astronomy, University of Maryland, College Park, MD 20724, USA.

theoretical relationship between the roton spectrum and the peaks in the static structure factor was developed by Feynman in his single-mode approximation (SMA) [1]. This theoretical ap- proach was also used by Girvin et al. [2] to describe the collective excitations of fractional quantum-Hall-effect ground states in 2-dimensional electron systems. In this case the collective excitations are magneto-plasma waves. For fractional Landau level filling these systems have two branches of magneto-plasmons corresponding to intra- and inter-Landau-level excitations. Both branches have a magneto-roton feature at q = 1 / l 0 (l 0 is the magnetic length) [2,3]. The softening of the roton minimum in the intra Landau level branch is expected to be the precursor to the transition into the Wigner solid phase. Measure- ments of the magneto-plasma dispersion relation for these systems in the interesting wave-vector range (q = 1/10) has not been reported because of the difficulty of coupling to large q. Measure- ments of the density of modes for the upper branch has been reported by Pinczuk et al. [4] by Raman scattering. However, the mechanism that

0921-4526/93/$06.00 ~ 1993 - Elsevier Science Publishers B.V. All rights reserved

H.D. Drew et al. / Collective excitations in a quasi-3-dimensional electron system 101

allows coupling to the high-q modes is not yet understood [4].

Exotic ground states have also been predicted for three-dimensional electron systems in high magnetic fields, including a variety of charge- density-wave states. At sufficiently high magnetic field and T = 0 the ground state of these systems is also a Wigner crystal [5,6]. So far there has been no convincing evidence for electronic phase transitions in three-dimensional systems except for the case of graphite [7]. Doped bulk semi- conductors have been studied extensively; however, the Coulomb interaction with the impurity ions dominates the behavior of these systems leading instead to a Mot t -Anderson metal-insu- lator transition [8]. In the case of metals the electron density is too high. Collective excitations are also expected in interacting three- dimensional electron systems in high magnetic fields. For wave vectors perpendicular to the magnetic field B the excitation spectrum is sim- ple and has a structure similar to the 2D case with two branches of magneto-plasmons corresponding to the inter- and intra-Landau-level transitions.

Recently have quasi-three-dimensional systems based on wide quantum wells with a parabolic potential profile been proposed as an interesting class of interacting electron systems [6]. Using MBE technology remotely doped AIAS/GaAs wide parabolic quantum wells (WPQW) have been grown that provide a high- mobility slab of electrons - 1 0 0 0 / ~ wide with a nearly constant electron density n o that is controlled by the curvature of the parabolic potential profile [9]. At B = 0 typically 3-4 electric subbands are occupied in these systems [10]. For large B parallel to the plane, however, the electric subbands are depopulated and the extreme magnetic limit conditions can be achieved [11]. It is in this geometry that these systems are most nearly three-dimensional. Since n o is nearly constant the width W of the slab is given by W = N] no, which is precise for W ~> l 0.

The optical response of WPQWs are found to be consistent with that of the three-dimensional electron plasma. Two resonances are observed whose frequencies are given by

122 sine0 122 cos20 1 = : 2 + 2 ( 1 )

¢.O - - ¢ . O c O.)

where 0 is the angle that the magnetic field makes with the normal to the plane of the electron slab, and 12 is the plasma frequency

2 oop = noeZ/m*eoe, where e is the static dielectric constant [12]. In the limit of N s ~ 0 the response can be described in terms of the hybridized harmonic oscillator and inter Landau level resonances and then 12 is the harmonic oscillator frequency to 2 = k/m*, where k is the curvature of the parabolic potential [13]. Since n o is determined from the well potential by the condition that the Coulomb potential of the electron distribution cancels the well potential V 0 = ½kz 2 it follows that % : w 0.

In fig. 1 we show the far-infrared magnetotransmission spectrum of a WPQW in the Fara- day (0 = 0 ) and Voigt (0 = 90 °, E_I_B) geomet- ries. A narrow cyclotron resonance is observed in the Faraday geometry whose line width variations and (small) position variations with Ns can be understood in terms of the effects of the AI concentration experienced by the electron system [14]. In the Voigt geometry the resonance is depolarization-shifted from the cyclotron resonance. This Voigt resonance corresponds to the plasma-shifted cyclotron resonance (PSCR) for a

I I i I

VOIGT EIIB

r ~

m~ 0.9

VOIGT E.LB

0.8 T=4.2K

FARADAY B = 3 T

0.7 t r i I t 30 40 50 60 70 80

FREQUENCY cm -1

Fig. 1. Faraday CR and Voigt PSCR mcasurcd at 4.2 K for a wide parabolic quantum well at B = 3 T. The frequency shift is due to depolarization fields induced by the magnetic field in the E L B geometry. No resonance is observed in the whole frequency range in the E[IB geometry.

102 H.D. Drew et al. / Collective excitations in a quasi-3-dimensional electron system

t h r e e - d i m e n s i o n a l sys tem [15]. I t is not o b s e r v e d in h e t e r o j u n c t i o n s o r square wells. The resonance f r equency is g iven by w 2 = w~ + ~o~. The m a g n e t i c field d e p e n d e n c e of the P S C R has been r e p o r t e d e l s ewhere [15]. The o t h e r m o d e in this g e o m e t r y is a D r u d e m o d e at ze ro f r equency which c o r r e s p o n d s to the in t ra L a n d a u level t rans i t ions . In fig. 2 we show the r e sonance f r equenc ies d e t e r m i n e d f rom b r o a d - b a n d far- i n f r a red t ransmiss ion m e a s u r e m e n t s at fixed fields for severa l d i f fe ren t va lues of 0 c o m p a r e d wi th the p r e d i c t e d curves using eq. (1) [15]. T h e s e resul ts d e m o n s t r a t e tha t W P Q W s behave op t i ca l ly l ike a t h r e e - d i m e n s i o n a l e l ec t ron p la sma .

In fig. 3(a) we show the fa r - in f ra red magne to - t r ansmiss ion o f a W P Q W as a funct ion of N s in the Voigt g e o m e t r y . The p lasma-sh i f t ed cyclo- t ron r e s o n a n c e f r equency is seen to be indepen- den t o f N S. This resul t is an i l lus t ra t ion of the g e n e r a l i z e d K o h n t h e o r e m which s ta tes tha t the r e s p o n s e of an e l ec t ron sys tem, e m b e d d e d in a

.,-i

0

- 5 0 0 50 - 5 0 0 50 - 5 0 0 50

.._ 1.0 ~ _ ~

~-.~ 0.9

0.8 ' i ,(a~l" 5 6

Z ( n m

(b I

5 6

B(T)

I , ( e t

5 6

Fig. 3. (a) The SIMS data (top panel) for a pure parabolic well and the corresponding transmission spectrum shown for different N~. (b) Potential profile (top panel) of a WPQW with a periodic array of AlAs spikes and the corresponding transmission spectrum (bottom panel). (c) Potential profile (top panel) and transmission spectrum (bottom spectrum) of a WPQW sample containing a single AlAs spike in its center.

)..- n-" I.~ Z

120 crn -~

I00

80

60

40

20

I I I I I f I I

SAMPLE M77 y T=4.2 K /

_.'°I ~o , - , " .,.%+d

_ / ° °

O V ~ , J J J ~ I . ¢ . 4 1 ÷ ' . . $ " 0 10 2 0 3 0 ,<,v4" ~ " ~ + f

o°O +.s ANGLE (degrees} / +~'L ~ / "

~ ! " " ~ ' I - - A /

~' .~ ' . . - . " """" '~ o TILT ANGLE=O* + °

,, ~,,~ "" • = 50* ¢~t ~'. -- THEORY

I I I I I I I I I 2 :5 4 5 6 7 8 9

MAGNETIC FIELD Fig. 2. Frequency dependence of the transmission minima vs magnetic field applied at 0 °, 13 ° and 30 ° with respect to the sample normal. The normal modes predicted by eq. (1) are the solid lines. The anticrossing field is 3.5 T. Inset: energy separation between the two modes at the anticrossing field. The full line is the prediction of eq. (1).


harmonic confining potential, to a uniform elec- t rodynamic excitation is independent of interactions [16,17]. The essence of the theorem is that uniform incident light excites only the center of mass modes of the electron system.

In order to probe the effects of the interactions it is necessary to excite the internal modes of the electron system. These modes can be excited only by violating the conditions of the Kohn theorem. This is accomplished either by probing the system with a nonuniform electro- magnetic wave or by studying wells whose potential profiles deviate f rom quadratic. Through M B E technology it is possible to produce wells with controlled AI concentration (and therefore potential) profiles in the growth direction. We have studied two kinds of wells with potential per turbat ions superimposed on the parabolic profile [18]. In one type we have placed a single narrow A1 concentrat ion spike at the center of the well. The other type has a periodic array or grating of such A! spikes with period a. The transmission spectra of examples of these wells are shown in fig. 3(b) and (c). In these per turbed wells satellite resonances are seen to develop on the high-field side of the PSCR. The magnetic field dependence of the spectra for a grating- per turbed well is shown in fig. 4. The oscillator strength of the satellite resonance is seen to grow more rapidly with magnetic field than the main resonance. This is understood in terms of the dependence of the transition matrix elements on q l o [19]. In order to see more clearly the effects

I i I

~'~ ~ IG3 .03

a=2Onm

T=I.4K Voigt geometry

I [ I

0 2

I I [ I

1 1 8 . 8 3 1 0 3 . 1 2

I i I i

4 6 B(T)

Fig. 4. Magnetotransmission spectra for different wave- lengths of a WPQW perturbed with periodic potential spikes. The numbers above the spectra indicate the infrared wave- length in I~m. The spectra are displaced vertically for clarity.

of the perturbat ions we have used a front gate to sweep the electrons across the AI spike in the centered single-spike well. The position of the electron slab in the well can be deduced f rom an analysis of the cyclotron mass in Faraday geometry measurements . At zero gate voltage the front surface of the electron slab is found to be not yet quite at the well center. As the gate voltage is increased the front surface of the electron slab advances across the A1 spike reaching the symmetric condition at approximately 0.5V. The dependence of the transmission spectrum of this sample on gate voltage is shown in fig. 5. The satellite structure is seen to develop in strength and complexity as the electrons move into the region of the perturbat ion [19].

In the case of a single AI spike the satellites correspond to the excitation of dimensional modes in which the thickness of the electron slab corresponds to an integer number n of half wave- lengths of the magneto-plasma wave [18]. There- fore, the wave vector of the excited magneto- plasmon depends on the well width as q = 2"trn/

W. This leads to a shifting of the frequency of these satellite modes with filling as can be seen in fig. 5. For the case of symmetric filling of the well only even n leads to a finite dipole moment and optical absorption [18]. For the grating samples magneto-plasmons of wave vector q = 2"rr/a

I ; ~ i I i I i

I I M 2 0 8 F 1 I ~ V o i g t

~ ..~.~.._.~... . .~_ o.~.

I i I i I iVg~O'104V

4.5 5.0 5.5 6.0 6.5 B(Tesla)

Fig. 5. Transmission spectra at different gate voltages for a WPQW with a single AIAs spike in its center. The numbers on the right are the gate voltages.

104 H.D. Drew et al. / Collective excitations in a quasi-3-dimensional electron system

are excited in addition to the dimensional modes.

By combining the data from different samples at different magnetic fields and different N s, we can construct the magneto plasma dispersion relation as shown in fig. 6. The data points represent the results from both single and grating AI spike samples at different magnetic fields and N~. We have plotted the data in normalized units in order to display the results for different B and to o . Although the theory does not scale in these units except at q = 0, the deviations grow slowly with q as is seen for two fields in fig. 6. The solid lines connecting the two theory curves correspond to the theoretically predicted resonances for the grating samples over the magnetic field range of the experimental data. The theoretical curves are produced from SMA calculations in the high-field limit [18]. The ground-state static structure factor, used in the SMA, is calculated in the Har t r ee -Fock approximation. In fig. 7 we show the magnetic field dependence of resonances for the 30 nm-period grating sample to- gether with the predictions of the SMA.

Although the theory is in good agreement with the shape of the magneto-plasmon dispersion,

1.1 ' I ' f ' I ' I ' [ '

n =2.5 10ta/crn a V o

1 . 0 -., ........ " a * ~ ~ ' ' ~ ' " " " ~ ........................ p " -

a, , 7T ~" ' ,""

X

0 . 7 -- • a--3Onm

T=1.45 K"'" • q=4~'/w 0.5 i I , I J I , I , I ,

0 1 2 3 4 5 6

ql o

Fig. 6. Normalized dispersion of the magnetoplasma modes of WPQWs. The dashed lines are the dispersion relations calculated for two magnetic fields in the SMA. The solid symbols correspond to the satellite modes and open symbols correspond to the Kohn mode plotted vertically above the satellite resonances.

' T = 1 . 4 K "~-'""-

2 2 2 .~" ..'"'"

>5/,...

'~rS ...."" nv=2.SxlOl°/cm a 0 " " " I I I I I I

0 2 0 4 0 6 0 ]3 ~ (T 2)

Fig. 7. to 2 vs B 2 dependence of the plasma-shifted cyclotron resonance (circles) and of the satellite resonance (triangles) for the 30 nm-period sample (wave vectors near the roton minimum). The dashed line is the calculated position of the magneto-plasma mode. The dotted line is the cyclotron resonance position.

there is a quantitative discrepancy with the mode frequencies. This raises the important question concerning the degree to which the three-dimensional theory applies to the Q3D experiments. The fact that optical data on WPQWs obey eq. (1) is a necessary but not sufficient condition for three-dimensionality. Three-dimensional behavior is clearly expected in the large-N s limit. Experimental evidence that these systems behave three-dimensionally for N~ in the range of the experiments reported here is found in studies of the inhomogeneous electron plasma in asym- metric wide parabolic quantum wells [20]. Also, indications that the 3D calculations may provide an adequate description of these quasi-three- dimensional systems can be found in theoretical calculations for a square well in zero magnetic field [21]. Therefore, we believe that the experimental results presented here represent, at least qualitatively, the behavior of three-dimensional systems. Another open question is how the perturbations affect the mode frequencies and how they permit the coupling to the radiation.

The results presented in this paper represent the first measurement of roton dispersion in an electron system. They demonstrate the collective nature of the low-lying excitations in quasi-three- dimensional systems. It should be possible to extend these measurements to study the low-


frequency branch which, because of its relation to the Wigner crystal condensation, is of great fundamental interest. When it becomes possible to grow large-area potential perturbations in the plane of heterostructures these techniques could also be used to study the roton spectrum in the two-dimensional electron systems.

We acknowledge useful discussions with S. Das Sarma, N. Johnson, A. MacDonald and E. Yang. This work was supported in part by NSF under grant nos. DMR 8705002 and DMR 8704670.

References

[1] D.R. Tilley and J. Tilley, Superfluidity and Supercon- ductivity (Adam Hilger, New York, 1990); R.P. Feyn- man, Statistical Mechanics (Addison-Wesley, New York, 1972).

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[3] C. Kallin and B.I. Halperin, Phys. Rev. B 30 (1984) 5655; 31 (1985) 3635.

[4] A. Pinczuk et al., Phys. Rev. Lett. 61 (1988) 2701; 68 (1992) 3623.

[5] A.H, MacDonald and G.W. Bryant, Phys. Rev. Lett 58 (1987) 515.

[6] B.I. Halperin, Jpn. J. Appl. Phys. 26, Suppl. 26-3 (1987) 1913.

[7] Y. Iye et al., The Physics and Chemistry of Organic Superconductors, eds. G. Saito and S. Kagoshima (Springer-Verlag, Berlin, 1990) p. 324.

[8] M. Shayegan, V.J. Goldman and H.D. Drew, Phys. Rev. B 38 (1988) 5585.

[9] M. Shayegan, T. Sajoto, M. Santos and C. Silvestre, Appl. Phys. Lett. 53 (1988) 791; M. Shayegan, T. Sajoto, M. Santos and L. Engel, Surf. Sci. 229 (1990) 83; M. Sundaram, A.C. Gossard, J.H. English and R.M. Westervelt, Superlatt. Microstruct. 4 (1988) 683.

[10] T. Sajoto, J. Jo, M. Santos and M. Shayagan, Appl. Phys. Lett. 55 (1989) 1430.

[11] T. Sajoto, J. Jo, H.P. Wei, M. Santos and M. Shayegan, J. Vac. Sci. Technol. B 7 (1989) 311.

[12] I.B. Bernstein, Phys. Rev. 104 (1958) 10. [13] R. Merlin, Solid State Commun. 64 (1987) 99. [14] K. Karrai, M. Stopa, X. Ying, H.D. Drew, S. Das

Sarma and M. Shayegan, Phys. Rev. B 42 (1990) 9732. [15] K. Karrai', H.D. Drew, M.W. Lee and M. Shayagan,

Phys. Rev. B 39 (1989) 1426. [16] L. Brey, N.F. Johnson and B.I. Halperin, Phys. Rev. B

40 (1989) 10647. [17] Q.P. Li, K. Karrai', S.K. Yip, S. Das Sarma and H.D.

Drew, Phys. Rev. B 43 (1991) 5151. [18] K. Karrai, X. Ying, H.D. Drew, M. Santos, M.

Shayegan, S.-R.E. Yang and A.H. MacDonald, Phys. Rev. Lett. 67 (1991) 3428.

[19] X. Ying, PhD thesis, University of Maryland, to be published.

[20] X. Ying, K. Karral, H.D. Drew, M. Santos and M. Shayegan, Phys. Rev. B 46 (1992) 1823.

[21] W.G. Teich and G. Mahler, Phys. Stat. Sol. B 138 (1986) 607.

collective excitations in a quasi-3-dimensional electron system

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