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Edge magnetoplasmon excitations in a Quantum Dot in high magnetic fields


IMSc-99/12/40

arXiv:cond-mat/9912080v1 [cond-mat.mes-hall] 6 Dec 1999

Edge magnetoplasmon excitations in a Quantum Dot in high magnetic ?elds
Subhasis Sinha
The Institute of Mathematical Sciences, Madras 600 113, India. (February 1, 2008)

Abstract

We investigate the collective magnetoplasmon excitations in a quantum dot containing ?nite number of electrons in the high magnetic ?eld limit. We consider the electrons in the lowest Landau level and neglect mixing between the higher Landau levels. The dispersion relation of these edge modes are estimated following the energy weighted sum-rule approach. In this ?nite size system the edge magnetoplasmon modes have di?erent multipolarities (or angular momemtum l). Their dependence on the magnetic ?eld and on the system size is investigated. With increasing magnetic ?eld, energy of these collective modes decreases and in the bulk limit they become gapless. We also consider the breathing mode of a dot in the presence of a strong magnetic ?eld, and the energy of this mode approaches the cyclotron frequency ? ωc . h PACS numbers: 73.20.Dx, 72.15.Rn

Typeset using REVTEX 1

Recent developments in nanofabrication technology have made it possible to manufacture electronic systems with reduced dimensionality. Quantum dot is one such nanosystem, which is very interesting both experimentally and theoretically. A quantum dot is a twodimensional electron system where the electrons are con?ned within a ?nite area by applying a gate voltage [1]. It is an example of a ?nite size quantum system where the number of electrons can be varied from a few electrons to a few thousand electrons. This nanosystem shows many interesting properties in the presence of a magnetic ?eld. One of the interesting properties in the presence of strong magnetic ?eld is the chiral edge magnetoplasmon excitations. The edge modes in the quantum Hall bar have been studied theoretically using one dimensional ?eld theory and numerical diagonalisation of the few particle system [2,3]. The edge modes of a two-dimensional electron system in the presence of positive ion background have been studied by several authors [4,5]. Giovanazzi et al [4] have calculated the dispersion relation of the edge modes of the electron gas con?ned in a jellium disk in the long wave-length limit. Recently collective excitations in the ?nite size nanostructures like quantum dot have become very interesting, because many device parameters can be controlled by adjusting the gate voltages. The number of electrons in the system can also be varied. In particular, the dipole excitation in a quantum dot with parabolic con?nement, and in the presence of a strong magnetic ?eld has been discussed in ref. [6]. Evidence has been found for a strong collective dipole mode in the presence of a magnetic ?eld in far infrared spectroscopy experiment [7,8]. Recently the collective charge and spin excitation in a dot containing about 200 electrons has been observed experimentally [9]. Non parabolic con?nement has been observed in this experimental device, but the exact nature of the con?nement is not well understood. The edge magneto-plasmon modes of a quantum dot with ?nite number of electrons have been studied by using magneto-hydrodynamics by several authors [10–13]. The collective excitations in the ?nite fermion systems like nuclei and metal clusters have been extensively studied within random phase approximation(RPA) and using sumrule approach in the last years by several authors [14,15]. This method has also been applied to analyze the multipole excitations in the two dimensional quantum dot and antidot systems [5,16,17]. Most of the theoretical works have been done numerically or by using classical magneto-hydrodynamics. The main aim of this work is to estimate the collective modes of a dot in the presence of a strong magnetic ?eld microscopically, and also to obtain analytical results for the dispersion relations of these modes. In this paper we study the low-lying multipole excitations and the breathing mode of a quantum dot in the presence of a strong magnetic ?eld by using RPA sum-rule approach. We consider the electrons in the lowest Landau level, and derive a simple semiclassical energy functional for the electrons. By minimising the energy functional we obtain the ground state density pro?le of the electrons. Then we generalise the sum-rules in the presence of a magnetic ?eld. Di?erent moments of the RPA strength distribution function are calculated by using the ground state density pro?le. Finally we obtain the analytic expressions for the dispersion relation of the low-lying magnetoplasmon modes with di?erent multi-polarities in a quantum dot. With increasing magnetic ?eld, the energy of these modes decreases and in the bulk limit these modes become gap-less. Neglecting the 1/N corrections, the dispersion relation takes a very general form, where only one parameter contains the information about the two-body interaction and the shape of the density pro?le. Also, the 1/N corrections 2

of these modes are calculated. In the strong magnetic ?eld limit the dispersion relation of these modes agrees with the dispersion relation obtained from hydrodynamics [10]. We also calculate the energy of breathing mode excitation, which is much larger than the energy of low-lying edge modes. We obtained the most general form of the dispersion relation of the edge excitations in strong magnetic ?eld limit. We also calculated the ?nite size N dependent corrections of the energy of these modes. Analyzing the dispersion relation in a strong magnetic ?eld, we have shown the stability of the self-consistent semicircular density pro?le, which shows that this semi circular density is a better ansatz for ground state density than commonly used ?at density pro?le. The hamiltonian of the two-dimensional electron system in the presence of a magnetic ?eld in the perpendicular direction is given by, H=
i

1 1 1 [(?xi + m? ωc yi )2 + (?yi ? m? ωc xi )2 ] + p p ? 2m 2 2
i<j

i

1 ? 2 2 m ω 0 ri 2 (1)

+

V (|ri ? rj |),

where m? is the e?ective mass, ω0 is the frequency of the external parabolic con?nement, and eB ωc is the cyclotron frequency m? c . Neglecting the two-body interaction, the single particle Hamiltonian can be solved exactly. The single-particle energy levels are, En,m = (2n + |m| + 1)? ?
2

m ωc , 2

(2)

2 where ? = ω0 + ωc , and m is the angular momentum quantum number, and n denotes 4 Landau level index. In the presence of a strong magnetic ?eld all electrons occupy the lowest Landau level(LLL), energy levels become almost degenarate and they form a band. In the lowest Landau level the single particle wave functions are,

ψm (r) =

1

1 z 2 2 √ ( )m e?r /2l0 , 2 πl0 m! l0

(3)

h where, l0 = m? ? , and z = x + iy is the complex coordinate. In the high magnetic ?eld ? limit the e?ect of higher Landau levels can be neglected and the many body wave function can be constructed out of these lowest Landau level basis. For electrons at the ?lling factor ν = 1 the Laughlin wave function can be obtained by constructing the Van-der Monde determinant of the above single particle states,

ψ(r1 , r2 , ....rn ) = Πi<j (zi ? zj )e?

2 z z /2l0 ? i i i

.

(4)

In general we can write the ground state many body wave function constructed out of the LLL as, ψ(r1 , r2 , ..ri ..) = f (z1 , z2 , ..zi , ..)e?
2 z z /2l0 ? i i i

.

(5)

We can calculate the energy functional in the lowest Landau level, by calculating the expectation value of the Hamiltonian with respect to the ground state wave function given above. 3

After doing some algebra and using some properties of the Bargman space [18], we arrive at the following energy functional, 1 E[ψ] = hωc |ψ|2 d2 r + m? ?(? ? ωc /2) r 2 |ψ|2 d2 r ? 2 1 d2 r1 d2 r2 V (|r1 ? r2 |)|ρ(r1 , r2 )|2 d2 r2 , + 2 where, |ρ(R1 , R2 )|2 =< r1 , r2 , ...ri , ..|
ij

(6)

δ(R1 ? ri )δ(R2 ? rj )|r1 , r2 , ...ri ... > .

(7)

Within Hartree approximation we can write down a simple local density functional for lowest Landau level, E[ρ] = 1 ωc d2 r[ hωc + m? ?(? ? )r 2 ]ρ(r) ? 2 2 1 2 2 + d r1 d r2 ρ(r1 )V (|r1 ? r2 |)ρ(r2 ). 2

(8)

In the case of a ?at droplet, the Hartree energy typically goes as ? N 3/2 , and the exchange correlation energy goes as ? N, where N is the number of particles in the droplet [19]. Hence in the large N limit, the exchange term may be neglected and we obtain the above simple looking energy functional. We minimize the free energy E ? ?N with respect to ρ, keeping the total number of particles N ?xed, and obtain the following integral equation for the density, m? ?(? ? ωc /2)r 2 + 1 d2 r ′ V (|r ? r ′ |)ρ(r ′ ) = ? ? hωc . ? 2 (9)

For coulomb interaction, V (r) = e2 /r the solution of the above integral equation is [10,20], r ρ(r) = n0 1 ? ( )2 , R
2

(10)

3N where, n0 = 2πR2 , and R3 = 8m?3e πN ωc ) . From this simple energy functional we obtain the ?(?? 2 nonperturbative density of the electrons. We now consider the edge magnetoplasmon excitations in the dot. Magnetoplasmon excitation are the multipole excitations of a quantum dot in the presence of a magnetic ?eld. We denote the multipolarity of the operator by the number k. The excitation with multipolarity k is generated by the excitation operator F = z k , where z is the complex coordinate x + iy. Excitation energies of the above modes can be estimated by sum-rule method. Many useful quantities may be calculated from the strength function, that is de?ned as

S± (E) =
n

| < n|F± |0 > |2 δ(E ? En ),

(11)

where, En , and |n > are the excitation energy and excited state respectively, and F+ = F , and F? = F ? . Various energy-weighted sum rules can be derived through the moments of the strength function, and are written below, 4

m± = k

1 E k (S+ (E) ± S? (E))dE 2 1 ? ? = (< 0|F ? (H ? E0 )k F |0 > ± < 0|F (H ? E0 )k F ? |0 >). 2

(12)

Some useful moments can be written in terms of the commutators of the excitation operator F with the many body Hamiltonian H, and they are given below, 1 2 1 m+ = 1 2 1 m? = 2 2 1 m+ = 3 2 m? = 0 < 0|[F ? , F ]|0 >, < 0|[F ? , [H, F ]]|0 >, < 0|[J ? , J]|0 > < 0|[J ? , [H, J]]|0 >, (13) (14) (15) (16)

where J = [H, F ]. In the presence of magnetic ?eld +k and ?k collective modes split, and the corresponding strength distributions are sharply peaked at these collective frequencies. Near the collective excitation energy, we can approximate the strength distribution by delta function, S± = σ± δ(E ? Ec? ). Since for the multipole modes m? = 0, we obtain σ+ = σ? . 0 Within this approximation the low lying multipole modes can be estimated by using the above moments of the strength distribution function and can be written as, Ec? = m+ 3 m? 2 m? 2 3 2 , + ? ( +) ± m1 4 m1 2 (17)

where, ± sign correspond to k and ?k multipolarities. Similar expression for the excitation frequencies of the edge multipole modes can be derived by using the variational principle. Given the N electron ground state |0 >, it is possible to ?nd the collective excitation energy and the collective state |c >, if one able to ?nd an operator O ? , which satis?es the following equation of motion, ? [H, O ? ] = hωcoll O ? . ? (18)

The state O ?|0 > has the excitation energy hωcoll . This excitation energy is obtained from, ? hωcoll = ? ? < 0|[O, [H, O ?]]|0 > . < 0|[O, O ?]|0 > (19)

For k = 1 the dipole excitation operator is, O? = 1 2 (zi ? i pi+ ), ? m? ? (20)

i

where p+ = px + ipy . Similarly, we may take the variational ansatz for the higher multipole excitations in the following form, O ? = F + aJ ; 5 J = [H, F ], (21)

(k?1) ? h k ? D+ , where, D+ = where a is the variational parameter, F = i zi , and J = ( ?i? k ) i zi m? ?ω im c p+ + 2 z. The collective excitation energy in terms of the the energy weighted sum rules ? is,

hωcoll ?

m+ + 2am? + a2 m+ 2 3 . = 1 + 2 m? 2am1 + a 2

(22)

After minimising the above expression with respect to the variational parameter a, the following expression for the collective excitation frequency is obtained, hωcoll = ? m+ 3 3 m? 2 m? ? ( 2) + 2. 4 m+ 2 1 (23)

This agrees with the previous expression in eqn.(17), obtained by the approximate form the strength distribution. Now we explicitly evaluate the the sum-rules by calculating the commutators. After doing some algebra the sum-rules can be written in the following form. The ?rst energy weighted moment m+ is given by, 1 m+ (k) = 1 h2 k 2 ? < r 2(k?1) >, m? (24)

where < ... > denotes the average weighted with the ground state density. The contribution for the third moment m+ coming from the kinetic energy of the hamiltonian m+ (T ) can be 3 3 written as, m+ (T ) = ( 3 h2 k 2 2 ? )[? k(k ? 1) d2 rr 2(k?2) |?ψ|2 h m?3 +2? 2 (k ? 1)(k ? 2) < r 2(k?3) ?z > ?3? 2 k(k ? 1)m? ωc < r 2(k?2) ?z > h l2 h l 1 2 3 +? 2 m?2 ωc ( k 2 + k) < r 2(k?1) > . h 4 4

(25)

Similarly the contribution of the external potential term in m+ is, 3 m+ (V ) = ( 3 h2 k 2 ? 2 )km?2 ω0 < r 2(k?1) > . m?3 (26)

The most important contribution comes from the electron-electron interaction term, and is given by, m+ (ee) 3 h4 k 2 ? [ = 2m?2 + d2 r d2 r?2 VH (r)r 2(k?1) ρ(r) + 2(k ? 1) d2 r ′ ρ′ (r)e?ilθ d2 r ?VH 2(k?3) r ρ(r) ?r (27)

′ 1 eilθ r ′(k?1) ρ′ (r ′)], |r ? r ′ |

where VH (r) is the Hartree potential. In the presence of a magnetic ?eld there is nonvanishing second moment m? , 2 m? = 2 h2 k 2 ? [?? m? ωc k < r 2(k?1) > +2? (k ? 1) < r 2(k?2) ?z >], h h l m?2 6 (28)

where ?z is the angular momentum operator with eigenvalue m. We now consider the limit l in which the magnetic ?eld is large, so that all electrons are in the lowest landau level. Neglecting the mixing of higher Landau levels, if we consider the wave function in the following form, ψ(zi ) = f (z1 , z2 , ....zi , ...)e?
2 zi zi /2l0 ?

,

(29)

then the above sum-rules can be simpli?ed and written in terms of moments of radius r. In the lowest Landau level the sum-rules m+ and m? can be written as, 3 2 h2 k 2 2 ?2 2 ? k ωc [? m ? (3((k ? 1) ? ( ))2 + 1) < r 2(k?1) > h ?3 m 2 ? ωc 3 ? 2 2(k?2) ?3? m ?(k ? 1) (2(k ? 2) ? k( )) < r h > ? +4? 2 (k ? 1)2 (k ? 2)2 < r 2(k?3) >], h h2 k 2 ? ? h m2 = ?2 [?? m? ωc k < r 2(k?1) > m +2? (k ? 1)(m? ? < r 2(k?1) > ?? (k ? 1) < r 2(k?2) >)]. h h m+ (T ) + m+ (V ) = 3 3 We can write down the expressions for the ratios of moments in the following form, α1 ωc m? 2 ) ? ωc + 2 , + = 2(k ? 1)(? ? m1 2 R + + m3 (T ) + m3 (V ) α2 β2 k ωc = h2 ?2 [1 + 3(k ? 1 ? ( ))2 ] + 2 + 4 . ? + m1 2 ? R R (32) (33)

(30)

(31)

For large number of electrons, if we neglect the terms O(1/R2) and the higher order terms, then the excitation energies can be written in the most general form, given below, Ecoll (k) = (k ? 1)? (? ? h ωc )+ 2 h2 ?2 + ?(e) ? ? ωc . 2 (34)

The coe?cients α1 , α2 , β2 depend on the density of the electrons, and the function ?(e) depends on the nature of the two body interaction and also on the shape of the density pro?le. Now we can estimate the low-lying multipole excitations by using the semi-circular density pro?le. The parameters in the above form of the collective frequency can be evaluated by using the semicircular density pro?le ρ = n0 1 ? (r/R)2 , ?(e) = 2? 2 ?(? ? h ωc Γ(k + 1 ) 2 )[ ? k], 2 Γ(k)Γ(3/2) α1 = h?(k ? 1)(2k + 1), ? 3 ωc α2 = ? h2 ?2 (k ? 1)(2k + 1)(2(k ? 2) ? k( )), ? 2 ? β2 = h2 ?2 (k ? 1)(k ? 2)(2k + 1)(2k ? 1). ? (35) (36) (37) (38)

Since the semi-circular density pro?le is the exact solution of the Hartree equation, the dispersion relation of the edge modes obtained by using this density is a non-perturbative 7

result. Further the asymptotic results do not contain the coupling constant e2 /l0 , only the ?nite size corrections depend on the coupling constant. This density pro?le goes to zero at the turning point R and the difussive tail is absent in this case. But the difussive tail in the exact density distribution can only contribute 1/N corrections to the spectrum of low-lying excitations [16]. From the energy levels of the non-interacting electrons, we can estimate the magnetic ?eld above which all N particles go into the lowest Landau level. This gives the √ condition 2? ? > (N ? 1)(? ? ωc /2)(or ω0 /ωc < 1/ N). If k << N, which is true for the h low-lying modes, then in the strong ?eld limit we can expand the expression for collective modes in terms of ?(e)/? 2 ?2 , and obtain the following result in strong magnetic ?eld limit h Ec (k) = ω0 ω0 ω0 Γ(k + 1/2) + O(( )4 ). ωc Γ(k)Γ(3/2) ωc (39)

This result now be compared with the dispersion relation obtained by, Shikin et al [10] using classical hydrodynamics,
? ωk = 2 k?2 + ωc /4 ? ωc /2, kk 2 Γ(k + 1/2) , = ω0 Γ(k)Γ(3/2)

(40) (41)

k?2 kk

where hωk is the same as Ec (k) in our notation. If we expand this expression in terms of ? ? ω0 , the leading term agrees with the dispersion relation obtained from sum-rule approach. ωc For comparison, we may also estimate the edge excitations in a quantum dot by using the density pro?le of the non-interacting electrons at the ?lling factor ν = 1. In the large N 1 limit, the density pro?le of the system can be approximated by, ρ0 θ(R ? r), where, ρ0 = πl2 . 0 This density pro?le is very sharp near the edge, and therefore singularities arise in evaluating ?, but after doin the integrals and then taking the limit(r → R) at the edge, the singularities cancel out and we obtain ?nite value of the parameter ?. The parameter ? in this case can be derived as, ?(e) = 2? ? h
k 1 e2 k √ [1 ? ]. l0 N m=1 2m ? 1

(42)

This result agrees with the value obtained by Giovanazzi et al [4]. In this case the excitation energies of various multipole modes vanishes at di?erent values of magnetic ?eld. The approximate value of the magnetic ?eld, where excitation energy of the k th mode becomes negative is, (
k 1 ω0 3/2 e2 ) <√ [ ? 1], ωc 2N ?0 hω0 m=1 2m ? 1 l ?

(43)

h ? where ?0 = m? ω0 . Energy of the higher multipole modes vanish at lower magnetic ?eld. For l example the mode with multipolarity k = 10 becomes gapless at a magnetic ?eld ? 9.65T , in a dot with N = 40 and hω0 = 5.4meV . Softening of these edge modes indicate the edge ? reconstruction of the dot and the formation of new ground state. This phenomena indicates the instability of the ?at density. The critical magnetic ?eld where the instability sets in is obtained from [19,21],

8

(

.5139e2 ω0 3/2 √ ]. ) ≈[ ?0 hω0 N ωc l ?

(44)

Although ? is negative for both the cases, the excitation energy for low-lying modes for the semi-circular density pro?le are positive and asymptotically go to zero, which indicates the stability of the ground state. For comparisn, we have shown in Fig.1 ? for di?erent multipolarities, evaluated using two di?erent ground state densities. In Fig.2, the variation of few low-lying modes with magnetic ?eld is shown, using the dispersion law given in eqn.(34) and eqn.(35). These modes vanish as ? 1/B with the increasing magnetic ?eld. These low-lying modes are important for the thermodynamics of the system at very low temperatures. Finally we consider the breathing mode of the dot within the same formalism. The breathing mode is excitated by the excitation operater, F = r 2 . The operator J is, J = [H, F ] = ( The ?rst moment m+ is given by, 1 m+ 1 2? 2 h = ? < r2 > . m (46) ?i? h )[?x x + x?x + py y + y py ]. p p ? ? m? (45)

The third moment can be written in the following way, m+ = 3 1 ?2 ? < η|H|η > |η=0 , 2 ?η 2 (47)

where |η >= eηJ |0 >. Now the moment m+ can be evaluated by using the scaling properties 3 of the wave function [22],
h eηJ |ψ >= e?2η?
2

/m?

h |ψ(xe?2η?

2

/m?

h , ye?2η?

2

/m?

)>.

(48)

From the above scaling property of the wave function, we obtain, m+ = 8( 3 h2 2 ? 1 h2 ? ? ) [< T > + m? ?2 < r 2 >] + 2( ? )2 Eint , ? m 2 m 2 2 h ? h ? = 8( ? )2 [m? ?2 < r 2 >] + 2( ? )2 Eint , m m (49) (50)

? where T is the kinetic energy operator and Eint is the interaction energy of the system. In deriving the second step, we used the properties of the lowest-Landau level wave functions. The general expression of the excitation energy of the breathing mode is derived as, Eb = 4? 2 ?2 + h h2 Eint ? . m? < r 2 > (51)

Using the semi-circular density for the ground state of the electrons, we obtain the nonperturbative result for the breathing mode, 9

ωc 1 Eb = 2? ? 1 + (1 ? h ). 4 2?

(52)

In the strong ?eld limit this mode approaches the bulk collective mode hωc . ? To summerise, we have considered the quantum dot in the lowest Landau level. We have derived a simple local density functional for the electrons in the LLL within the Hartree approximation. For the two-body coulomb interaction a semicircular density pro?le of the electrons has been derived. We have estimated the low-lyimg edge multipole modes within the sum-rule approach. Since the semi-circular density is exact within Hartree approximation, we obtain the non-perturbative results for the dispersion realtions of the collective modes by using this density pro?le. The energy of these collective modes decreases with increasing magnetic ?eld. In the strong ?eld limit the most general expression for the dispersion relation of the edge modes has been derived and is given by, E(k) = ?2 + ?(e) ? ωc + (k ? 1)ω0 ω0 . 2 ωc The parameter ?(e) depends on the exact nature of two-body interaction and on the number of electrons in the system. We evaluate the parameter ?(e) by using exact density, as well as by using the density pro?le of the non interacting electrons at ?lling factor ν = 1. In both cases ?(e) is negative. In the case of ?at density the energy of the low-lying edge modes become zero at some magnetic ?eld and the softening of the edge modes indicate the instability of the ground state. But for the semi-circular ground state density the low-lying modes are positive and asymptotically go to zero, which shows the stability of the self-consistent density pro?le. We have also calculated the breathing mode of a quantum dot, and in the strong magnetic ?eld the energy of the breathing mode approaches the cyclotron frequency ωc . The main results of this paper are, the derivation of the most general dispersion relation of the edge modes in a quantum dot in a strong magnetic ?eld. The 1/N corrections of the energy of these modes are also obtained. Stability of the self consistent density pro?le has been shown by analysing the dispersion relation of the low-lying edge modes. This result shows that the semicircular density is a better ansatz for the ground state density than commonly used ?at density pro?le, in the strong magnetic ?eld limit. These edge modes are important for the edge excitation and edge reconstruction of quantum dot. They also determine the low-temperature thermodynamic properties of the quantum dot. I would like to thank M. V. N Murthy for his helpful comments. I would also like to thank Tapash Chakraborty and R. Shankar for critical reading of the manuscript.

10

REFERENCES
[1] T. Chakraborty, Quantum Dot, A Survey of the Properties of Arti?cial Atoms, (Elsevier, North-Holand, 1999) [2] X. G. Wen , Phys. Rev. B 44, 5708 (1991). [3] M. Stone , Phys. Rev. B 42 , 212 (1990) [4] S. Giovanazzi, L. Pitaevskii and S. Stringari, Phys. Rev. Lett. 72, 3230 (1994). [5] E. Lipparini, N. Barberan, M. Barranco, M. Pi, and L. Serra et al, Phys. Rev. B, 56, 12375 (1997); A. Emperador, M. Barranco, E. Lipparini, M. Pi, and L. Serra condmat/9902357. [6] P. A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65, 108 (1990). [7] Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989). [8] T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. Lett. 64, 788 (1990). [9] C. Schuller, K. Keller, G. Biese, E. Ulrichs, L. Rolf, C.Steinebach, and D. Heitmann, Phys. Rev. Lett. 80, 2673 (1998). [10] V. Shikin, S. Nazin, D. Heitmann, and T. Demel, Phys. Rev. B, 43, 11903(1991). [11] D. B. Mast, A. J. Dahm, and A. L. Fetter, Phys. Rev. Lett. 54, 1706 (1985). [12] B. Partoeus, A. Matulis and F. M. Peeters, cond-mat 9712066 [13] Z. L. Ye and E. Zaremba , Phys. Rev. B 50 , 17217 (1994). [14] E. Lipparini and S. Stringari, phys. Rep. 175, 103 (1989). [15] M. Brack, Rev. Mod. Phys,65, 677 (1993). W. de Heer, Rev. Mod. Phys,65, 611 (1993). [16] S. Sinha, Preprint [17] L. Serra, M. Barranco, A. Emperador, M. Pi, and E. Lipparini, e-print cond-mat 9806104 [18] S. M. Girvin and T. Jach , Phys. Rev. B 29, 5617 (1984). [19] A. H. MacDonald, S. R. Eric Yang and M. D. Johnson , Aust. J. Phys 46, 345 (1993). [20] E. Lieb, J. P. Solovej and J. Yngvason , Phys. Rev. B 51, 10646 (1995). [21] M. Ferconi and G. Vignale, Phys. Rev. B 56, 12108 (1997). [22] O. Bohigas, A. M. Lane, and J. Martorell, Phys. Rep. 51, 267 (1979).

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FIGURES
FIG. 1. The parameter ?(e) in units of h2 ω0 , for di?erent multipolarities k, for a dot with ? 2 e2 N = 50, ? ? ω = 0.8 and ωc /ω0 = 10. The open triangles represent values calculated with l0 h 0 semi-circular density and the solid triangles denote the same parameter for ?at density pro?le. FIG. 2. The variation of four lowest multipole modes with magnetic ?eld, in a quantum dot with semi-circular density pro?le. Di?erent values of k denote di?erent multipolarities.

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