- Email: [email protected]
Evidences for non-Fermi liquid behavior of quantum wires
Physica E 9 (2001) 22–30
www.elsevier.nl/locate/physe
Evidences for non-Fermi liquid behavior of quantum wires M. Sassetti, F. Napoli, B. Kramer ∗ Dipartimento di Fisica, INFM, UniversitÂa di Genova, Via Dodecaneso 33, 16146 Genova, Italy
Abstract The present understanding of the physics of correlated electrons in quantum wires is explained by using two representative examples. Non-Fermi liquid collective excitations are obtained within the Luttinger liquid model. Charge and spin density modes are considered and compared with the data from resonant Raman scattering experiments. The results indicate that interactions dominate the low-frequency collective modes. The interplay in DC-transport between interaction, spin, backscattering by impurities and inhomogeneity is discussed and compared with recent experiments. ? 2001 Elsevier Science B.V. All rights reserved. PACS: 71.10.Pm; 73.23.−b; 23.23.Hk; 78.30−j Keywords: Non-Fermi liquid behavior; Collective excitations; Resonant Raman scattering; Non-linear transport properties
1. Introduction Due to recent progresses in semiconductor preparation technology, the fabrication of AlGaAs=GaAsquantum wires of extremely high quality has become possible [1]. In these quasi-one-dimensional (1D) nanostructures, the density of electrons can be strongly reduced such that only the lowest electronic subband remains occupied. Even this can become de-populated by applying a suciently high voltage at a gate, such that eventually the region of Coulomb blockade can be reached [2]. ∗ Correspondence address: Institut fuer Theoretische Physik, Universitaet Hamburg, Jungiusstrasse 9, D-20355 Hamburg, Germany. E-mail address: [email protected] (B. Kramer).
Electronic excitations in these quantum wires are expected to behave very close to the collective excitations in 1D electron models, the so-called Tomonaga– Luttinger (TL) liquids which are paradigms of non-Fermi liquids [3–5]. Their low-frequency excitations are spin () and charge () density waves with dispersions that vanish linearly with the wave number when q → 0, but with velocities v ( = ; ) that are renormalized di erently by the interaction. This is denoted as “spin–charge separation”. With considerably improved spectroscopic techniques for measuring excitation spectra of electrons in semiconductor nanostructures, the above spectral signatures of the non-Fermi liquid should be observable. Indeed, recent results from resonant Raman spectroscopy have been found to be consistent with the predictions. In addition, it has been
1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 1 7 3 - 9
M. Sassetti et al. / Physica E 9 (2001) 22–30
possible with the TL model to solve for quantum wires a long-standing puzzle in Raman spectroscopy, namely the observation of polarization-insensitive so-called single-particle excitations (SPE), in addition to the collective spin and charge density excitations (SDE and CDE), and to explain their physical origin in the framework of collective modes [6,7]. The presence of the collective excitations leads to very peculiar transport properties at low temperatures [8]. For instance, a single, even very small, potential barrier in an otherwise ideal TL system, causes the linear conductance to vanish with decreasing temperature T according to a non-analytic power law, (T ) ˙ T 2=g−2 ;
(1)
with the interaction constant g ¡ 1 in the repulsive case. Recent transport experiments showed pronounced TL-liquid phenomena. Data for the conductance of carbon nanotubes have been found to be consistent with non-analytic temperature behavior [9]. Temperature-dependent corrections to the plateau values of the conductance of quantum wires were explained by weak impurity scattering in a TL liquid [10,11]. In the region of Coulomb blockade, evidence has been found that electron correlations are important in addition to charging e ects [2]. In this paper, we review the above results. We also provide predictions which would further support the non-Fermi liquid scenario for the low-temperature properties, if experimentally con rmed. We predict power laws analogous to Eq. (1) for the Raman cross section of intra-subband excitations, as a function of T and=or the frequency of the light [12]. We show that the behavior with T of the resonant transport through a 1D quantum dot in the region of Coulomb blockade is dominated by global CDE and SDE in the entire quantum wire while the excitations observed in non-linear transport spectroscopy are locally con ned modes. At lowest energies, they are spin excitations [13]. This is qualitatively consistent with earlier results obtained by diagonalizing few electrons con ned in a 1D island combined with a rate equation [14,15].
23
2. Tomonaga–Luttinger liquid model Let us summarize the main results needed in order to understand the evidences for TL behaviour in quantum wires. Interacting electrons in the ideal TL system of length L are described by the harmonic Hamiltonian Z ˜vF H0 = d x[2 (x) + (@x # (x))2 ] 2 Z Z 1 d x d x0 @x # (x)V (x − x0 )@x0 # (x0 ) + Z ˜vF d x[2 (x) + (@x # (x))2 ] (2) + 2 with the Fermi velocity vF and the conjugate Boson elds , and # associated with the CDE ( = ) and SDE ( = ), respectively. For simplicity, we have assumed here that the SDE propagate with the Fermi velocity vF (last term) since the spin interaction is very small. The Fourier transform Vˆ (q) of the electron–electron interaction, V (x − x0 ), determines the dispersion of the charge excitations which can be exactly obtained, together with the corresponding Boson elds, in terms of the usual Fermion creation and annihilation operators [5,16], " #1=2 2Vˆ (q) ≡ v (q)|q|: (3) ! (q) = vF |q| 1 + ˜vF The dispersion of the SDE, !2 1=2 ˆ ex V ≈ vF |q|; ! (q) = vF |q| 1 − 2˜vF
(4)
contains as the dominant part the exchange interaction Vˆ ex = Vˆ (2kF ) ≈ 0. The strength of the charge interaction is described by the parameter #−1=2 " 2Vˆ (0) : (5) g= 1+ ˜vF The densities are de ned by the standard Fermion elds cs (k) associated with spin s= ↑; ↓ and branch index = ± of the linearized dispersion of the free electrons P (6) s (q) = cs† (k + q)cs (k): k;
24
M. Sassetti et al. / Physica E 9 (2001) 22–30
The total density of charges can be written as r 2 @x # (x) (x) = ↑ (x) + ↓ (x) = 0 + √ √ +0 cos[2kF x + 2# (x)] cos[ 2# (x)]; (7) where 0 = 2kF = is the mean electron density and ↑ and ↓ correspond to the densities with spin up and spin down, respectively. The second term in Eq. (7) accounts for the slowly varying part of the density
uctuations. The third term represents the charge density wave due to the 2kF -backscattering between left and right moving electrons. It becomes e ective when backscattering is induced by a potential. It also couples the charge with the long-wave length part of the spin density r 2 @x # (x) (8) (x) = ↑ (x) − ↓ (x) ≈ which is considered with respect to a zero mean value. The contribution to the total energy of delta-function potentials P of impurities at positions x = x1 ; x2 , V (x) = i Vi (x − xi ), corresponding to the above density, is √ P Vi cos[2kF xi + 2# (xi )] HB = 0 i=1;2
√ ×cos[ 2# (xi )]:
(9)
One also obtains a contribution due to the second term in Eq. (7). As this represents forward scattering, it can be eliminated by a unitary transformation. On the other hand, for a slowly varying bias eld on the scale of kF−1 the contribution is mainly due to the long-wave length part of Eq. (7). We assume this to be the case for the bias electric eld, E(x; t) = −@x U (x; t). The corresponding term is (elementary charge e ¿ 0) r Z 2 d xU (x; t)@x # (x): (10) HU = −e The total Hamiltonian is H = H0 + HB + HU . In the following section, we consider the rst part of H and describe the experimental evidence for SDE and CDE in Raman scattering on quantum wires. We concentrate on processes within the lowest subband, though results for inter-subband scattering are also available [17–20].
3. Signatures of charge and spin excitations in Raman spectra The imaginary part of the Fourier transform of the correlation function (q; t) = i(t)h[N † (q; t); N (q; 0)]i
(11)
determines the di erential cross section [21–23]. It contains the generalized density operator N (q; t). Considering only the lowest subband we have P
s c† (k + q)cs (k): (12) N (q) = D(k; q) s k; ; s The quantities s denote e ective optical transition probabilities that account for the transitions between the occupied valence and the empty conduction bands. For simplicity, we assume equal transition probabilities for parallel and perpendicular polarizations of incoming (polarization eI ) and outgoing (polarization eO ) light, independent of the spin s,
s = (eI · eO + is|eI × eO |):
(13)
The energy denominator D(k; q) = Ec (k + q) − Ev − ˜!I
(14)
contains the energy of incident photons ˜!I , a dispersionless valence band energy Ev , and a single conduction subband Ec (k) = E0 + ˜2 k 2 =2m (m e ective mass). At rst glance, this seems to be oversimpli ed in view of application to, say, AlGaAs=GaAs quantum wires. However, it is sucient to explain our main point. It is clear from the Eqs. (7) and (6) that N (q) contains all powers of the charge and spin density operators but that nevertheless the cross section can be evaluated non-perturbatively. Far from resonance, when the photon energy ˜!I is much larger than the energy gap Eg = Ec (0) − Ev , the energy denominator can be assumed constant. Here, it is easily seen that the cross section for parallel polarization of incident and scattered photons is determined by CDE. In perpendicular con guration, only SDE appear [6,7]. When resonance is approached, D(k; q) ≈ 0, the denominator cannot be assumed to be constant. It has to be taken into account for the cross section. Within the TL model, this can be done exactly and yields a triple integral which must be computed numerically. However, the essential physics can be extracted by the
M. Sassetti et al. / Physica E 9 (2001) 22–30
following approximation [12]: contributions related to SDE are related to (q; t) ˙ exp(i! (q)t). They generate peaks in the cross section near the frequency of the SDE, Im (q; !) ≈ (! − ! ) ×[(eI · eO )2 I1 + |eI × eO |2 I2 ];
(15)
where I1 (q; !I ; T ) and I2 (q; !I ; T ) are the peaks strength in parallel and perpendicular polarization, respectively. Correspondingly, when selecting (q; t) ˙ exp[i! (q)t], one gets Im (q; !) ≈ (! − ! )(eI · eO )2 I0 ;
(16)
since v is approximately constant for small q, v ≈ v (q = 0) = vF =g. While SDE gives rise to peaks in both polarizations, CDE appears as a peak only in parallel and not in perpendicular con guration, even near resonance. This can be most easily seen by considering the lowest-order term which is ˙ · in perpendicular polarization and this cannot give rise to a peak at the frequency of the CDE [6,7]. Furthermore, one can prove that the terms in a power-law expansion of N (q) that contributions near the frequency of the CDE in perpendicular polarization (i) contain at least one spin-density operator, and (ii) consist always of a product of an odd number of spin-density operators multiplied by a product of charge-density operators. Terms of this kind will not produce a peak at the frequency of the CDE. When calculating the correlator, there is always a residual pair of spin-density operators (in Heisenberg representation), (t)(0), which remains time dependent and destroys the coherence of the associated CDE terms. This annihilates any spurious CDE peak in the cross section. In the following, we consider only SDE in the regions where TL behavior can be expected to appear. The peak intensities are " 2 # dS 2 Lq 2 q2 + I1 (q; !I ; T ) = dQ 12(˜vF )2 2 ˜vF (17) with −1 = kB T (kB Boltzmann constant), and I2 (q; !I ; T ) =
Lq 2 |S(Q; T )|2 : (˜vF )2
(18)
The integral Z S(Q; T ) =
0
∞
25
dyeiQy F(y)
depends on the reduced photon wave number ˜vF q 1 : Eg + EF − ˜!I + Q= ˜vF 2
(19)
(20)
The function
−1=2 y ˜vF 1 sinh F(y) = 2 y 2 ) y ˜vF (1 + qint −2−1=2 ˜v y × (21) sinh y ˜v contains the inverse range of the interaction potential, qint , and the characteristic exponent 1 1 g+ −2 ; (22) = (g) = 8 g emphasizing that higher-order SDE in parallel con guration are dressed by CDE. Thus, the dependencies of the intensities of the SDE-peaks in resonant Raman scattering on the energy of incident photons and=or the temperature in parallel and perpendicular polarizations are governed by non-rational exponents that are characteristic for the TL liquid and contain the strength of the repulsive interaction between the electrons. In order to proceed further, we consider ¡ 0:5 (g ¿ g0 with g0 ≈ 0:2) where the integral may be approximately evaluated analytically. There are three characteristic wave numbers: the inverse range of the interaction qint , the wave number of the excitation q and the wave number corresponding to the temperature, q = 1= ˜vF . We assume qint q ¿ q since, below qint , one can expect the most important interaction-induced e ects. For Q ¿ qint , far from resonance [6,7], one can show that In ˙ (qint =Q)4=n (n = 1; 2). For qint ¿ Q we get near resonance. If Q ¿ q the temperature does not a ect the leading-order result 4(1=n−) qint : (23) In ˙ Q For q ¿ Q the characteristic temperature dependences are 4(1=n−) qint ˜vF : (24) In ˙ kB T
26
M. Sassetti et al. / Physica E 9 (2001) 22–30
For g ¡ g0 ( ¿ 0:5), the behavior is similar, but cannot be treated analytically. Presently available Raman data [24 –28] in the region of the intra-subband transitions are consistent with the above ndings. In particular, the predicted SDE-peaks near resonance in the parallel conÿguration of incident and scattered light are observed at frequencies close to those of the SDE. Formerly, these have been erroneously denoted as “single-particle excitations”. Unfortunately, systematic measurements of the dependence of these structures as a function of the temperature and the energy of the photons are only at initial stages [29]. 4. Transport in the presence of potential barriers Considerable amount of work has been done by using the TL model for the linear and non-linear transport properties. Here, we concentrate on a double potential barrier and include the electron spin. We discuss addition energies [13] and sketch the transport properties of a 1D quantum island formed between two delta-function barriers at x1 and x2 in a TL system with spin. As a model for the interaction, we use a 3D Coulomb interaction screened by the presence of a metallic plate at the distance D. Its Fourier transform for Dd (diameter d) is 2 2 dq d2 q2 =4 ˆ − 2K0 (2Dq) ; (25) E1 V (q) = V0 e 4 with V0 = e2 =40 , dielectric constant , and parabolic con nement in the x- and y-directions [13]. In this model, the interaction parameter is −1 2D 2V0 : (26)
+ 2 log g= 1+ ˜vF d The presence of the two impurities separates the charge and spin degrees of freedom at “bulk” positions x 6= x1 ; x2 from those at the barriers. Conveniently, symmetric and anti-symmetric variables for particle ( = ) and spin number densities ( = ) are introduced, r 2 ± [# (x2 ) ± # (x1 )]: (27) N = The quantity N− is associated with the uctuations of the particle number within the island as compared to
the mean particle number n0 = 0 (x2 − x1 ). The corresponding excess charge is Q = −eN− . The quantity N− corresponds to the spin change N− =2. The numbers of transferred particles and spins are represented by N+ =2 ( = ; ). The DC current–voltage characteristic of the double barrier can be evaluated from the stationary limit of the charge transfer in the presence of a bias voltage e + (28) I = lim hN˙ (t)i: 2 t→∞ The brackets h· · ·i denote a thermal average over the excitations at x 6= x1 ; x2 , and a statistical one with the reduced density matrix for the variables at x = x1 ; x2 . The averages can be done with the imaginary-time path integral method [30]. The resulting action is Z ˜ d HB [N± ; N± ] Se [N± ; N± ] = 0 "Z Z ˜ ˜ P P d d 0 Nr ( ) + r=± =;
×Kr (
0
−
0
)Nr ( 0 )
0
− ;
Z
˜
0
(L → ∞):
# d
Nr ( )Lr ( )
;
(29)
The Fourier transforms, at Matsubara frequencies !n = 2n=˜ , of the dissipative kernels K± ( ) and of the e ective forces L± ( ) are Z 8vF ∞ 1 ± cos[q(x1 − x2 )] ; dq [K± (!n )]−1 = 2 ˜ 0 !n2 + !2 (q) (30) L± (!n ) =
Z ∞ 4evF ± K (! ) d x E(x; !n ) n ˜2 −∞ Z ∞ cos[q(x − x2 )] ± cos[q(x − x1 )] : dq × !n2 + !2 (q) 0 (31) K− (!n
→ 0) describe the costs in enThe quantities ergy for changing the numbers of charges and=or spins on the island between the potential barriers. The corresponding characteristic energies are E = 2K− (!n → 0);
( = ; ):
(32)
For = , this is the charging energy needed to change the charge on the island. Correspondingly, for = , the spin addition energy E is needed in order for
M. Sassetti et al. / Physica E 9 (2001) 22–30
changing the spin. The Coulomb interaction that determines the dispersion of the charge excitations increases considerably E in comparison with E . The frequency-dependent parts of the kernels describe the dynamical e ects of the leads and of the excited states in the quantum dot. Their in uence is described by spectral densities J± (!) which are related by analytic continuation to the imaginary-time kernels [31,32] J± (!) =
2 Im K± (!n → −i!): ˜
Their limits for ! → 0 are ! ! ; J± ≈ : J± ≈ 4g 4
(33)
(34)
These limits describe the dissipative in uence of the low-frequency CDE and SDE in the leads, x ¡ x1 and x ¿ x2 . In general, the driving forces L± ( ) depend in a quite complicated way on the dispersion of the collective modes and on the shape of the electric eld. We focus on the DC-limit where it is sucient to evaluate the Fourier components for !n → 0. In this case, the quantity L+ ( ) = eU=2 acts on the total transmitted charge. It depends only on the integral of the time independent electric eld over R ∞ the entire system, the source–drain voltage U ≡ −∞ d x E(x). This generalizes the result obtained previously for only one impurity [30]. On the other hand, L− ( ) acts on the excess charge on the island. It does not generate a current. It depends on the spatial shape of the electric eld and can be written in terms of the total charge QE accumulated between the points x1 and x2 as a consequence of the DC electric eld E QE L− ( ) = : (35) e For symmetry reasons, we can assume x2; 1 = ±a=2. If the e ective electric eld has inversion symmetry, L− vanishes. Without inversion symmetry, the electric eld generates a charge on the island which in uences the total current via coupling between N+ and N− due to the impurity term HB . Physically, this induced charge may be thought of as being due to a gate voltage VG which electrostatically in uences the charge on the island. Thus, Eq. (35) represents the effect of a gate voltage.
27
In order to calculate the electrical current one has to solve the equations of motion for the N± . For barriers much higher than the charging energy, Vi E , the dynamics is dominated by tunneling events connecting the minima of HB in the 4D (N+ ; N− ; N+ ; N− )-space [8]. The transitions between these minima correspond to di erent physical processes transferring electrons from one side to the other of the quantum dot. At very low temperatures, the dominant processes transfer the electron coherently through the dot. In particular, when the number of particles in the island is an odd integer the island acts as a localized magnetic impurity, similar as in the Kondo e ect [8]. If the temperature is higher than the tunneling rate through a single barrier, sequential tunneling events dominate [33,34]. The transfer of charge occurs via uncorrelated single-electron hops associated with corresponding changes in the total spin [14,15]. In the linear regime (U → 0), for T = 0, starting with the island occupied by 2n electrons, we expect that another electron can enter and leave only if the di erence between the ground state energies of 2n + 1 and 2n electrons is aligned with the chemical potential of the external semi-in nite TL systems. The ground state of even and odd numbers of electrons in the island have total spin 0 and 12 , respectively [14,15,35]. This implies U(2n + 1; ± 12 ) − U(2n; 0) = 0
(36)
with U(2n; 0) and U(2n + 1; ± 12 ) corresponding to the ground state energies with 2n or 2n + 1 particles and total spins 0 and or ± 12 , respectively. With the above charge, spin and gate terms, Eqs. (32) and (35), this is E = 0: (37) 2 The variable nG = eVG =E represents the number of induced particles due to the coupling to the gate at which the voltage VG is applied, with a proportionality factor which can be determined experimentally. One can see that the distance of the peaks of the linear conductance when changing the gate voltage is given by VG = E =e. Using independent information on from experiment, one can extract the value of the charging energy E from the experimental data. For evaluating the current as a function of temperature and=or bias voltage, one needs to consider the spectral densities given in Eq. (33). In sequential E (n − n0 − nG + 12 ) +
28
M. Sassetti et al. / Physica E 9 (2001) 22–30
tunneling, transport depends only the sum of the spectral densities [34] P P r J (!): (38) J (!) =
In the opposite limit, Da, one obtains s ˜vF 2a = : 1 − + 2 log a d
The frequency behavior of this determines the current– voltage characteristics both in the linear and in the non-linear regimes. In particular, it determines the power-law dependencies of the current as a function of temperature and=or the bias voltage. Thus, the temperature dependence of the linear conductance peaks is dominated by the interaction in the parts of the quantum wire outside of the electron island. For temperatures lower than the excitation energy of the quantum dot, one nds an intrinsic width of the peaks
For the spin excitations the spectrum is equidistant
r=± =;
(T ) ˙ T 1=ge −1 : with the e ective interaction strength 1 1 1 +1 = ge 2 g
(39)
(40)
from the low-frequency behavior of the spectral density, ! 1 +1 : (41) J (!) ≈ Jleads (!) ≈ 2 g This generalizes the results obtained previously for spinless electrons [33,34]. In non-linear transport, the current voltage characteristic provides information about the charging energy E as well as the excited quantum dot states via ne structure in the Coulomb staircase [14,15]. In the present model, the possible excitations are collective spin and charge modes. In a completely isolated island, the corresponding energy spectrum would be discrete, ! (qm ) and ! (qm ), due to the discretization of the wave number qm = m=a. The screened Coulomb interaction causes a non-linear dispersion relation for CDE in the in nite Luttinger system. This leads to non-equidistant charge excitation energies, (qm ) = ˜[! (qm+1 ) − ! (qm )]:
(42)
For aD, the rst excited charge modes are equidistant, with the charge-mode velocity v ≡ vF =g, =
˜v ˜vF = : a ag
(43)
=
˜vF : a
(44)
(45)
5. Comparison with experiment Results of the temperature dependence of the intrinsic width of the conductance peaks in the Coulomb blockade region on cleaved-edge-overgrowth quantum wires have been reported [2]. Data have been found to be consistent with power laws similar to Eq. (1) with the constant ge in Eq. (40) given by ge ≈ 0:82 and ge ≈ 0:74 for peaks closer to the onset of the conductance and the next lower one, respectively. Taking into account the spin, we nd g ≈ 0:69 and g ≈ 0:59, about 15% smaller than ge . In addition, information about energies of the excited states have been obtained via the non-linear current–voltage characteristics. A minimum of ve excited levels were observed for a given electron number. Data have been analyzed by assuming that within the quantum wire, a quantum dot has been formed between two maxima of the random potential of impurities. Experimental parameters are: L ≈ 5 m; length of the electron island a ≈ 100–200 nm; mean (non-spherical) diameter of the wire d ≈ 10–25 nm; distance to the gate D ≈ 0:5 m; charging energy, determined from the distance between the conductance peaks, is EC ≈ 2:2 meV; the Fermi energy EF ≈ 2-4 meV [36]. With these, the interaction constant can be estimated from Eqs. (30) and (32), g ≈ 0:4, clearly inconsistent with the above-mentioned values determined from the temperature dependence of the peaks. Using our results, we con rm this discrepancy. By playing with the parameters, we found that it is impossible to identify a parameter region where all of the ndings were consistent with each other. Let us explain this in more detail. First, we identify E = EC . As this is relatively insensitive with respect to changes of D=d and EF we assumed D=d = 100 with EF = 3 meV and d = 20 nm. With V0 =d = 6:02 meV for = 12
M. Sassetti et al. / Physica E 9 (2001) 22–30
(AlGaAs/GaAs), we get for the length of the island a ≈ 16d. This is roughly consistent with the earlier estimate [2]. With this, and an average of the values for g determined from the temperature behavior of the peak widths, g ≈ 0:65, one nds E = ≈ 0:8 and E = ≈ 1:2. This means that at most 1–2 excited states corresponding to a given electron number should be observed in non-linear transport! This is not consistent with the experiment. In order to observe a larger number of excited states, the interaction constant must be considerably smaller. For the above a=d = 16 and reducing g below 0.3 one can achieve at most E = = 3 and E = = 1, due to saturation of the energy ratios with D → ∞ (g → 0). This means that even for small interaction constant, one can observe at most four excited states within an interval U 6E . In order to increase the number of observed excited levels, one has to increase the ratio a=d. In order to be consistent with experiment, one has to assume at least ve excited states which would correspond to at least a=d ≈ 60(!) corresponding to a charging energy EC ≈ 1:2 meV. A more consistent t can be obtained by starting from d = 10 nm but keeping the dielectric constant xed at = 12 such that V0 =d, and thus the charging energy, is increased. Nevertheless, a small g (g ¡ 0:3) is required in any case, inconsistent with 0.65, the average value determined from the temperature behaviour of the peaks! In our opinion, this can only mean that the non-linear transport data have to be interpreted by using a di erent interaction constant than that obtained from the temperature behavior of the peaks. This is supported by the theoretical derivation: the temperature behavior is dominated by the global excitations in the whole quantum wire, while the discrete excitation spectra are related to the local interactions in the quantum dot. We are thus forced to conclude that inhomogeneity e ects are an important issue for the understanding of correlations in the electron transport in these 1D quantum wires. Another conclusion was that the TL scenario is very probably at its limits for these very low-electron densities due to the failure of linearization of the dispersion. In any case, the energetically lowest of the excited states are predicted to correspond to spin excitations, and only the energetically highest would be a charge
29
excitation! The latter could possibly be identi ed with the state denoted in Ref. [2] as “strongly coupled excited state”. Acknowledgements It is a pleasure to acknowledge very stimulating and useful discussions with Amir Yacoby which were initiated at the 225. International WE-Heraeus Seminar, October 11–15, 1999 in Bad Honnef. The work was supported by the European Union within the TMR programme, by the Deutsche Forschungsgemeinschaft within the SFB 508 of the Universitat Hamburg, and by the Italian MURST via Co nanziamento 98. References [1] A. Yacoby, H.L. Stormer, K.W. Baldwin, L.N. Pfei er, K.W. West, Solid State Commun. 101 (1997) 77. [2] O.M. Auslaender, A. Yacoby, R. de Picciotto, K.W. Baldwin, L.N. Pfei er, K.W. West, Phys. Rev. Lett. 84 (2000) 1764. [3] S. Tomonaga, Prog. Theor. Phys. 5 (1950) 544. [4] J.M. Luttinger, J. Math. Phys. 4 (1963) 1154. [5] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [6] M. Sassetti, B. Kramer, Phys. Rev. Lett. 80 (1998) 1485. [7] M. Sassetti, B. Kramer, Eur. Phys. J. B 4 (1998) 357. [8] C.L. Kane, M.P.A. Fisher, Phys. Rev. B 46 (1992) 15 233. [9] M. Bockrath, D.H. Cobden, J. Lu, A.G. Rinzler, R.E. Smalley, L. Balents, P.L. McEuen, Nature 397 (1999) 598. [10] S. Tarucha, T. Honda, T. Saku, Solid State Commun. 94 (1995) 413. [11] A. Yacoby, H.L. Stormer, N.S. Wingreen, L.N. Pfei er, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 77 (1996) 4612. [12] B. Kramer, M. Sassetti, unpublished. [13] T. Kleimann, M. Sassetti, B. Kramer, A. Yacoby (2000), unpublished. [14] D. Weinmann, W. Hausler, B. Kramer, Phys. Rev. Lett. 74 (1995) 984. [15] D. Weinmann, W. Hausler, B. Kramer, Ann. Phys. (Leipzig) 5 (1996) 652. [16] J. Voit, Rep. Prog. Phys. 57 (1995) 977. [17] M. Sassetti, F. Napoli, B. Kramer, Phys. Rev. B 59 (1999) 7297. [18] F. Napoli, B. Kramer M. Sassetti, Eur. Phys. J. B 11 (1999) 643. [19] E. Mariani, M. Sassetti, B. Kramer, Europhys. Lett. 49 (2000) 224. [20] M. Mariani, M. Sassetti, B. Kramer, Ann. Phys. (Leipzig) 8 (1999) 161. [21] F.A. Blum, Phys. Rev. B 1 (1970) 1125. [22] A. Pinczuk, L. Brillson, E. Burstein, E. Anastassakis, Phys. Rev. Lett. 27 (1971) 317.
30
M. Sassetti et al. / Physica E 9 (2001) 22–30
[23] M.V. Klein, in: M. Cardona (Ed.), Light Scattering in Solids, Springer, Berlin, 1975, p. 147. [24] A.R. Go˜ni, A. Pinczuk, J.S. Weiner, J.M. Calleja, B.S. Dennis, L.N. Pfei er, K.W. West, Phys. Rev. Lett. 67 (1991) 3298. [25] A. Schmeller, A.R. Go˜ni, A. Pinczuk, J.S. Weiner, J.M. Calleja, B.S. Dennis, L.N. Pfei er, K.W. West, Phys. Rev. B 49 (1994) 14 778. [26] R. Strenz, U. Bockelmann, F. Hirler, G. Abstreiter, G. Bohm, G. Weimann, Phys. Rev. Lett. 73 (1994) 3022. [27] C. Schuller, G. Biese, K. Keller, C. Steinebach, D. Heitmann, P. Gambow, K. Eberl, Phys. Rev. B 54 (1996) R17 304.
[28] F. Perez, B. Jusserand, B. Etienne, Physica E 7 (2000) 521. [29] A. Go˜ni, unpublished results. [30] M. Sassetti, B. Kramer, Phys. Rev. B 54 (1996) R5203. [31] A. Furusaki, N. Nagaosa, Phys. Rev. B 47 (1993) 3827. [32] M. Sassetti, F. Napoli, U. Weiss, Phys. Rev. B 52 (1995) 11 213. [33] A. Furusaki, Phys. Rev. B 57 (1998) 7141. [34] A. Braggio, M. Grifoni, M. Sassetti, F. Napoli, Europhys. Lett. 50 (2000) 136. [35] E. Lieb, D. Mattis, Phys. Rev. 125 (1962) 164. [36] A. Yacoby, private communication.
www.elsevier.nl/locate/physe
Evidences for non-Fermi liquid behavior of quantum wires M. Sassetti, F. Napoli, B. Kramer ∗ Dipartimento di Fisica, INFM, UniversitÂa di Genova, Via Dodecaneso 33, 16146 Genova, Italy
Abstract The present understanding of the physics of correlated electrons in quantum wires is explained by using two representative examples. Non-Fermi liquid collective excitations are obtained within the Luttinger liquid model. Charge and spin density modes are considered and compared with the data from resonant Raman scattering experiments. The results indicate that interactions dominate the low-frequency collective modes. The interplay in DC-transport between interaction, spin, backscattering by impurities and inhomogeneity is discussed and compared with recent experiments. ? 2001 Elsevier Science B.V. All rights reserved. PACS: 71.10.Pm; 73.23.−b; 23.23.Hk; 78.30−j Keywords: Non-Fermi liquid behavior; Collective excitations; Resonant Raman scattering; Non-linear transport properties
1. Introduction Due to recent progresses in semiconductor preparation technology, the fabrication of AlGaAs=GaAsquantum wires of extremely high quality has become possible [1]. In these quasi-one-dimensional (1D) nanostructures, the density of electrons can be strongly reduced such that only the lowest electronic subband remains occupied. Even this can become de-populated by applying a suciently high voltage at a gate, such that eventually the region of Coulomb blockade can be reached [2]. ∗ Correspondence address: Institut fuer Theoretische Physik, Universitaet Hamburg, Jungiusstrasse 9, D-20355 Hamburg, Germany. E-mail address: [email protected] (B. Kramer).
Electronic excitations in these quantum wires are expected to behave very close to the collective excitations in 1D electron models, the so-called Tomonaga– Luttinger (TL) liquids which are paradigms of non-Fermi liquids [3–5]. Their low-frequency excitations are spin () and charge () density waves with dispersions that vanish linearly with the wave number when q → 0, but with velocities v ( = ; ) that are renormalized di erently by the interaction. This is denoted as “spin–charge separation”. With considerably improved spectroscopic techniques for measuring excitation spectra of electrons in semiconductor nanostructures, the above spectral signatures of the non-Fermi liquid should be observable. Indeed, recent results from resonant Raman spectroscopy have been found to be consistent with the predictions. In addition, it has been
1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 1 7 3 - 9
M. Sassetti et al. / Physica E 9 (2001) 22–30
possible with the TL model to solve for quantum wires a long-standing puzzle in Raman spectroscopy, namely the observation of polarization-insensitive so-called single-particle excitations (SPE), in addition to the collective spin and charge density excitations (SDE and CDE), and to explain their physical origin in the framework of collective modes [6,7]. The presence of the collective excitations leads to very peculiar transport properties at low temperatures [8]. For instance, a single, even very small, potential barrier in an otherwise ideal TL system, causes the linear conductance to vanish with decreasing temperature T according to a non-analytic power law, (T ) ˙ T 2=g−2 ;
(1)
with the interaction constant g ¡ 1 in the repulsive case. Recent transport experiments showed pronounced TL-liquid phenomena. Data for the conductance of carbon nanotubes have been found to be consistent with non-analytic temperature behavior [9]. Temperature-dependent corrections to the plateau values of the conductance of quantum wires were explained by weak impurity scattering in a TL liquid [10,11]. In the region of Coulomb blockade, evidence has been found that electron correlations are important in addition to charging e ects [2]. In this paper, we review the above results. We also provide predictions which would further support the non-Fermi liquid scenario for the low-temperature properties, if experimentally con rmed. We predict power laws analogous to Eq. (1) for the Raman cross section of intra-subband excitations, as a function of T and=or the frequency of the light [12]. We show that the behavior with T of the resonant transport through a 1D quantum dot in the region of Coulomb blockade is dominated by global CDE and SDE in the entire quantum wire while the excitations observed in non-linear transport spectroscopy are locally con ned modes. At lowest energies, they are spin excitations [13]. This is qualitatively consistent with earlier results obtained by diagonalizing few electrons con ned in a 1D island combined with a rate equation [14,15].
23
2. Tomonaga–Luttinger liquid model Let us summarize the main results needed in order to understand the evidences for TL behaviour in quantum wires. Interacting electrons in the ideal TL system of length L are described by the harmonic Hamiltonian Z ˜vF H0 = d x[2 (x) + (@x # (x))2 ] 2 Z Z 1 d x d x0 @x # (x)V (x − x0 )@x0 # (x0 ) + Z ˜vF d x[2 (x) + (@x # (x))2 ] (2) + 2 with the Fermi velocity vF and the conjugate Boson elds , and # associated with the CDE ( = ) and SDE ( = ), respectively. For simplicity, we have assumed here that the SDE propagate with the Fermi velocity vF (last term) since the spin interaction is very small. The Fourier transform Vˆ (q) of the electron–electron interaction, V (x − x0 ), determines the dispersion of the charge excitations which can be exactly obtained, together with the corresponding Boson elds, in terms of the usual Fermion creation and annihilation operators [5,16], " #1=2 2Vˆ (q) ≡ v (q)|q|: (3) ! (q) = vF |q| 1 + ˜vF The dispersion of the SDE, !2 1=2 ˆ ex V ≈ vF |q|; ! (q) = vF |q| 1 − 2˜vF
(4)
contains as the dominant part the exchange interaction Vˆ ex = Vˆ (2kF ) ≈ 0. The strength of the charge interaction is described by the parameter #−1=2 " 2Vˆ (0) : (5) g= 1+ ˜vF The densities are de ned by the standard Fermion elds cs (k) associated with spin s= ↑; ↓ and branch index = ± of the linearized dispersion of the free electrons P (6) s (q) = cs† (k + q)cs (k): k;
24
M. Sassetti et al. / Physica E 9 (2001) 22–30
The total density of charges can be written as r 2 @x # (x) (x) = ↑ (x) + ↓ (x) = 0 + √ √ +0 cos[2kF x + 2# (x)] cos[ 2# (x)]; (7) where 0 = 2kF = is the mean electron density and ↑ and ↓ correspond to the densities with spin up and spin down, respectively. The second term in Eq. (7) accounts for the slowly varying part of the density
uctuations. The third term represents the charge density wave due to the 2kF -backscattering between left and right moving electrons. It becomes e ective when backscattering is induced by a potential. It also couples the charge with the long-wave length part of the spin density r 2 @x # (x) (8) (x) = ↑ (x) − ↓ (x) ≈ which is considered with respect to a zero mean value. The contribution to the total energy of delta-function potentials P of impurities at positions x = x1 ; x2 , V (x) = i Vi (x − xi ), corresponding to the above density, is √ P Vi cos[2kF xi + 2# (xi )] HB = 0 i=1;2
√ ×cos[ 2# (xi )]:
(9)
One also obtains a contribution due to the second term in Eq. (7). As this represents forward scattering, it can be eliminated by a unitary transformation. On the other hand, for a slowly varying bias eld on the scale of kF−1 the contribution is mainly due to the long-wave length part of Eq. (7). We assume this to be the case for the bias electric eld, E(x; t) = −@x U (x; t). The corresponding term is (elementary charge e ¿ 0) r Z 2 d xU (x; t)@x # (x): (10) HU = −e The total Hamiltonian is H = H0 + HB + HU . In the following section, we consider the rst part of H and describe the experimental evidence for SDE and CDE in Raman scattering on quantum wires. We concentrate on processes within the lowest subband, though results for inter-subband scattering are also available [17–20].
3. Signatures of charge and spin excitations in Raman spectra The imaginary part of the Fourier transform of the correlation function (q; t) = i(t)h[N † (q; t); N (q; 0)]i
(11)
determines the di erential cross section [21–23]. It contains the generalized density operator N (q; t). Considering only the lowest subband we have P
s c† (k + q)cs (k): (12) N (q) = D(k; q) s k; ; s The quantities s denote e ective optical transition probabilities that account for the transitions between the occupied valence and the empty conduction bands. For simplicity, we assume equal transition probabilities for parallel and perpendicular polarizations of incoming (polarization eI ) and outgoing (polarization eO ) light, independent of the spin s,
s = (eI · eO + is|eI × eO |):
(13)
The energy denominator D(k; q) = Ec (k + q) − Ev − ˜!I
(14)
contains the energy of incident photons ˜!I , a dispersionless valence band energy Ev , and a single conduction subband Ec (k) = E0 + ˜2 k 2 =2m (m e ective mass). At rst glance, this seems to be oversimpli ed in view of application to, say, AlGaAs=GaAs quantum wires. However, it is sucient to explain our main point. It is clear from the Eqs. (7) and (6) that N (q) contains all powers of the charge and spin density operators but that nevertheless the cross section can be evaluated non-perturbatively. Far from resonance, when the photon energy ˜!I is much larger than the energy gap Eg = Ec (0) − Ev , the energy denominator can be assumed constant. Here, it is easily seen that the cross section for parallel polarization of incident and scattered photons is determined by CDE. In perpendicular con guration, only SDE appear [6,7]. When resonance is approached, D(k; q) ≈ 0, the denominator cannot be assumed to be constant. It has to be taken into account for the cross section. Within the TL model, this can be done exactly and yields a triple integral which must be computed numerically. However, the essential physics can be extracted by the
M. Sassetti et al. / Physica E 9 (2001) 22–30
following approximation [12]: contributions related to SDE are related to (q; t) ˙ exp(i! (q)t). They generate peaks in the cross section near the frequency of the SDE, Im (q; !) ≈ (! − ! ) ×[(eI · eO )2 I1 + |eI × eO |2 I2 ];
(15)
where I1 (q; !I ; T ) and I2 (q; !I ; T ) are the peaks strength in parallel and perpendicular polarization, respectively. Correspondingly, when selecting (q; t) ˙ exp[i! (q)t], one gets Im (q; !) ≈ (! − ! )(eI · eO )2 I0 ;
(16)
since v is approximately constant for small q, v ≈ v (q = 0) = vF =g. While SDE gives rise to peaks in both polarizations, CDE appears as a peak only in parallel and not in perpendicular con guration, even near resonance. This can be most easily seen by considering the lowest-order term which is ˙ · in perpendicular polarization and this cannot give rise to a peak at the frequency of the CDE [6,7]. Furthermore, one can prove that the terms in a power-law expansion of N (q) that contributions near the frequency of the CDE in perpendicular polarization (i) contain at least one spin-density operator, and (ii) consist always of a product of an odd number of spin-density operators multiplied by a product of charge-density operators. Terms of this kind will not produce a peak at the frequency of the CDE. When calculating the correlator, there is always a residual pair of spin-density operators (in Heisenberg representation), (t)(0), which remains time dependent and destroys the coherence of the associated CDE terms. This annihilates any spurious CDE peak in the cross section. In the following, we consider only SDE in the regions where TL behavior can be expected to appear. The peak intensities are " 2 # dS 2 Lq 2 q2 + I1 (q; !I ; T ) = dQ 12(˜vF )2 2 ˜vF (17) with −1 = kB T (kB Boltzmann constant), and I2 (q; !I ; T ) =
Lq 2 |S(Q; T )|2 : (˜vF )2
(18)
The integral Z S(Q; T ) =
0
∞
25
dyeiQy F(y)
depends on the reduced photon wave number ˜vF q 1 : Eg + EF − ˜!I + Q= ˜vF 2
(19)
(20)
The function
−1=2 y ˜vF 1 sinh F(y) = 2 y 2 ) y ˜vF (1 + qint −2−1=2 ˜v y × (21) sinh y ˜v contains the inverse range of the interaction potential, qint , and the characteristic exponent 1 1 g+ −2 ; (22) = (g) = 8 g emphasizing that higher-order SDE in parallel con guration are dressed by CDE. Thus, the dependencies of the intensities of the SDE-peaks in resonant Raman scattering on the energy of incident photons and=or the temperature in parallel and perpendicular polarizations are governed by non-rational exponents that are characteristic for the TL liquid and contain the strength of the repulsive interaction between the electrons. In order to proceed further, we consider ¡ 0:5 (g ¿ g0 with g0 ≈ 0:2) where the integral may be approximately evaluated analytically. There are three characteristic wave numbers: the inverse range of the interaction qint , the wave number of the excitation q and the wave number corresponding to the temperature, q = 1= ˜vF . We assume qint q ¿ q since, below qint , one can expect the most important interaction-induced e ects. For Q ¿ qint , far from resonance [6,7], one can show that In ˙ (qint =Q)4=n (n = 1; 2). For qint ¿ Q we get near resonance. If Q ¿ q the temperature does not a ect the leading-order result 4(1=n−) qint : (23) In ˙ Q For q ¿ Q the characteristic temperature dependences are 4(1=n−) qint ˜vF : (24) In ˙ kB T
26
M. Sassetti et al. / Physica E 9 (2001) 22–30
For g ¡ g0 ( ¿ 0:5), the behavior is similar, but cannot be treated analytically. Presently available Raman data [24 –28] in the region of the intra-subband transitions are consistent with the above ndings. In particular, the predicted SDE-peaks near resonance in the parallel conÿguration of incident and scattered light are observed at frequencies close to those of the SDE. Formerly, these have been erroneously denoted as “single-particle excitations”. Unfortunately, systematic measurements of the dependence of these structures as a function of the temperature and the energy of the photons are only at initial stages [29]. 4. Transport in the presence of potential barriers Considerable amount of work has been done by using the TL model for the linear and non-linear transport properties. Here, we concentrate on a double potential barrier and include the electron spin. We discuss addition energies [13] and sketch the transport properties of a 1D quantum island formed between two delta-function barriers at x1 and x2 in a TL system with spin. As a model for the interaction, we use a 3D Coulomb interaction screened by the presence of a metallic plate at the distance D. Its Fourier transform for Dd (diameter d) is 2 2 dq d2 q2 =4 ˆ − 2K0 (2Dq) ; (25) E1 V (q) = V0 e 4 with V0 = e2 =40 , dielectric constant , and parabolic con nement in the x- and y-directions [13]. In this model, the interaction parameter is −1 2D 2V0 : (26)
+ 2 log g= 1+ ˜vF d The presence of the two impurities separates the charge and spin degrees of freedom at “bulk” positions x 6= x1 ; x2 from those at the barriers. Conveniently, symmetric and anti-symmetric variables for particle ( = ) and spin number densities ( = ) are introduced, r 2 ± [# (x2 ) ± # (x1 )]: (27) N = The quantity N− is associated with the uctuations of the particle number within the island as compared to
the mean particle number n0 = 0 (x2 − x1 ). The corresponding excess charge is Q = −eN− . The quantity N− corresponds to the spin change N− =2. The numbers of transferred particles and spins are represented by N+ =2 ( = ; ). The DC current–voltage characteristic of the double barrier can be evaluated from the stationary limit of the charge transfer in the presence of a bias voltage e + (28) I = lim hN˙ (t)i: 2 t→∞ The brackets h· · ·i denote a thermal average over the excitations at x 6= x1 ; x2 , and a statistical one with the reduced density matrix for the variables at x = x1 ; x2 . The averages can be done with the imaginary-time path integral method [30]. The resulting action is Z ˜ d HB [N± ; N± ] Se [N± ; N± ] = 0 "Z Z ˜ ˜ P P d d 0 Nr ( ) + r=± =;
×Kr (
0
−
0
)Nr ( 0 )
0
− ;
Z
˜
0
(L → ∞):
# d
Nr ( )Lr ( )
;
(29)
The Fourier transforms, at Matsubara frequencies !n = 2n=˜ , of the dissipative kernels K± ( ) and of the e ective forces L± ( ) are Z 8vF ∞ 1 ± cos[q(x1 − x2 )] ; dq [K± (!n )]−1 = 2 ˜ 0 !n2 + !2 (q) (30) L± (!n ) =
Z ∞ 4evF ± K (! ) d x E(x; !n ) n ˜2 −∞ Z ∞ cos[q(x − x2 )] ± cos[q(x − x1 )] : dq × !n2 + !2 (q) 0 (31) K− (!n
→ 0) describe the costs in enThe quantities ergy for changing the numbers of charges and=or spins on the island between the potential barriers. The corresponding characteristic energies are E = 2K− (!n → 0);
( = ; ):
(32)
For = , this is the charging energy needed to change the charge on the island. Correspondingly, for = , the spin addition energy E is needed in order for
M. Sassetti et al. / Physica E 9 (2001) 22–30
changing the spin. The Coulomb interaction that determines the dispersion of the charge excitations increases considerably E in comparison with E . The frequency-dependent parts of the kernels describe the dynamical e ects of the leads and of the excited states in the quantum dot. Their in uence is described by spectral densities J± (!) which are related by analytic continuation to the imaginary-time kernels [31,32] J± (!) =
2 Im K± (!n → −i!): ˜
Their limits for ! → 0 are ! ! ; J± ≈ : J± ≈ 4g 4
(33)
(34)
These limits describe the dissipative in uence of the low-frequency CDE and SDE in the leads, x ¡ x1 and x ¿ x2 . In general, the driving forces L± ( ) depend in a quite complicated way on the dispersion of the collective modes and on the shape of the electric eld. We focus on the DC-limit where it is sucient to evaluate the Fourier components for !n → 0. In this case, the quantity L+ ( ) = eU=2 acts on the total transmitted charge. It depends only on the integral of the time independent electric eld over R ∞ the entire system, the source–drain voltage U ≡ −∞ d x E(x). This generalizes the result obtained previously for only one impurity [30]. On the other hand, L− ( ) acts on the excess charge on the island. It does not generate a current. It depends on the spatial shape of the electric eld and can be written in terms of the total charge QE accumulated between the points x1 and x2 as a consequence of the DC electric eld E QE L− ( ) = : (35) e For symmetry reasons, we can assume x2; 1 = ±a=2. If the e ective electric eld has inversion symmetry, L− vanishes. Without inversion symmetry, the electric eld generates a charge on the island which in uences the total current via coupling between N+ and N− due to the impurity term HB . Physically, this induced charge may be thought of as being due to a gate voltage VG which electrostatically in uences the charge on the island. Thus, Eq. (35) represents the effect of a gate voltage.
27
In order to calculate the electrical current one has to solve the equations of motion for the N± . For barriers much higher than the charging energy, Vi E , the dynamics is dominated by tunneling events connecting the minima of HB in the 4D (N+ ; N− ; N+ ; N− )-space [8]. The transitions between these minima correspond to di erent physical processes transferring electrons from one side to the other of the quantum dot. At very low temperatures, the dominant processes transfer the electron coherently through the dot. In particular, when the number of particles in the island is an odd integer the island acts as a localized magnetic impurity, similar as in the Kondo e ect [8]. If the temperature is higher than the tunneling rate through a single barrier, sequential tunneling events dominate [33,34]. The transfer of charge occurs via uncorrelated single-electron hops associated with corresponding changes in the total spin [14,15]. In the linear regime (U → 0), for T = 0, starting with the island occupied by 2n electrons, we expect that another electron can enter and leave only if the di erence between the ground state energies of 2n + 1 and 2n electrons is aligned with the chemical potential of the external semi-in nite TL systems. The ground state of even and odd numbers of electrons in the island have total spin 0 and 12 , respectively [14,15,35]. This implies U(2n + 1; ± 12 ) − U(2n; 0) = 0
(36)
with U(2n; 0) and U(2n + 1; ± 12 ) corresponding to the ground state energies with 2n or 2n + 1 particles and total spins 0 and or ± 12 , respectively. With the above charge, spin and gate terms, Eqs. (32) and (35), this is E = 0: (37) 2 The variable nG = eVG =E represents the number of induced particles due to the coupling to the gate at which the voltage VG is applied, with a proportionality factor which can be determined experimentally. One can see that the distance of the peaks of the linear conductance when changing the gate voltage is given by VG = E =e. Using independent information on from experiment, one can extract the value of the charging energy E from the experimental data. For evaluating the current as a function of temperature and=or bias voltage, one needs to consider the spectral densities given in Eq. (33). In sequential E (n − n0 − nG + 12 ) +
28
M. Sassetti et al. / Physica E 9 (2001) 22–30
tunneling, transport depends only the sum of the spectral densities [34] P P r J (!): (38) J (!) =
In the opposite limit, Da, one obtains s ˜vF 2a = : 1 − + 2 log a d
The frequency behavior of this determines the current– voltage characteristics both in the linear and in the non-linear regimes. In particular, it determines the power-law dependencies of the current as a function of temperature and=or the bias voltage. Thus, the temperature dependence of the linear conductance peaks is dominated by the interaction in the parts of the quantum wire outside of the electron island. For temperatures lower than the excitation energy of the quantum dot, one nds an intrinsic width of the peaks
For the spin excitations the spectrum is equidistant
r=± =;
(T ) ˙ T 1=ge −1 : with the e ective interaction strength 1 1 1 +1 = ge 2 g
(39)
(40)
from the low-frequency behavior of the spectral density, ! 1 +1 : (41) J (!) ≈ Jleads (!) ≈ 2 g This generalizes the results obtained previously for spinless electrons [33,34]. In non-linear transport, the current voltage characteristic provides information about the charging energy E as well as the excited quantum dot states via ne structure in the Coulomb staircase [14,15]. In the present model, the possible excitations are collective spin and charge modes. In a completely isolated island, the corresponding energy spectrum would be discrete, ! (qm ) and ! (qm ), due to the discretization of the wave number qm = m=a. The screened Coulomb interaction causes a non-linear dispersion relation for CDE in the in nite Luttinger system. This leads to non-equidistant charge excitation energies, (qm ) = ˜[! (qm+1 ) − ! (qm )]:
(42)
For aD, the rst excited charge modes are equidistant, with the charge-mode velocity v ≡ vF =g, =
˜v ˜vF = : a ag
(43)
=
˜vF : a
(44)
(45)
5. Comparison with experiment Results of the temperature dependence of the intrinsic width of the conductance peaks in the Coulomb blockade region on cleaved-edge-overgrowth quantum wires have been reported [2]. Data have been found to be consistent with power laws similar to Eq. (1) with the constant ge in Eq. (40) given by ge ≈ 0:82 and ge ≈ 0:74 for peaks closer to the onset of the conductance and the next lower one, respectively. Taking into account the spin, we nd g ≈ 0:69 and g ≈ 0:59, about 15% smaller than ge . In addition, information about energies of the excited states have been obtained via the non-linear current–voltage characteristics. A minimum of ve excited levels were observed for a given electron number. Data have been analyzed by assuming that within the quantum wire, a quantum dot has been formed between two maxima of the random potential of impurities. Experimental parameters are: L ≈ 5 m; length of the electron island a ≈ 100–200 nm; mean (non-spherical) diameter of the wire d ≈ 10–25 nm; distance to the gate D ≈ 0:5 m; charging energy, determined from the distance between the conductance peaks, is EC ≈ 2:2 meV; the Fermi energy EF ≈ 2-4 meV [36]. With these, the interaction constant can be estimated from Eqs. (30) and (32), g ≈ 0:4, clearly inconsistent with the above-mentioned values determined from the temperature dependence of the peaks. Using our results, we con rm this discrepancy. By playing with the parameters, we found that it is impossible to identify a parameter region where all of the ndings were consistent with each other. Let us explain this in more detail. First, we identify E = EC . As this is relatively insensitive with respect to changes of D=d and EF we assumed D=d = 100 with EF = 3 meV and d = 20 nm. With V0 =d = 6:02 meV for = 12
M. Sassetti et al. / Physica E 9 (2001) 22–30
(AlGaAs/GaAs), we get for the length of the island a ≈ 16d. This is roughly consistent with the earlier estimate [2]. With this, and an average of the values for g determined from the temperature behavior of the peak widths, g ≈ 0:65, one nds E = ≈ 0:8 and E = ≈ 1:2. This means that at most 1–2 excited states corresponding to a given electron number should be observed in non-linear transport! This is not consistent with the experiment. In order to observe a larger number of excited states, the interaction constant must be considerably smaller. For the above a=d = 16 and reducing g below 0.3 one can achieve at most E = = 3 and E = = 1, due to saturation of the energy ratios with D → ∞ (g → 0). This means that even for small interaction constant, one can observe at most four excited states within an interval U 6E . In order to increase the number of observed excited levels, one has to increase the ratio a=d. In order to be consistent with experiment, one has to assume at least ve excited states which would correspond to at least a=d ≈ 60(!) corresponding to a charging energy EC ≈ 1:2 meV. A more consistent t can be obtained by starting from d = 10 nm but keeping the dielectric constant xed at = 12 such that V0 =d, and thus the charging energy, is increased. Nevertheless, a small g (g ¡ 0:3) is required in any case, inconsistent with 0.65, the average value determined from the temperature behaviour of the peaks! In our opinion, this can only mean that the non-linear transport data have to be interpreted by using a di erent interaction constant than that obtained from the temperature behavior of the peaks. This is supported by the theoretical derivation: the temperature behavior is dominated by the global excitations in the whole quantum wire, while the discrete excitation spectra are related to the local interactions in the quantum dot. We are thus forced to conclude that inhomogeneity e ects are an important issue for the understanding of correlations in the electron transport in these 1D quantum wires. Another conclusion was that the TL scenario is very probably at its limits for these very low-electron densities due to the failure of linearization of the dispersion. In any case, the energetically lowest of the excited states are predicted to correspond to spin excitations, and only the energetically highest would be a charge
29
excitation! The latter could possibly be identi ed with the state denoted in Ref. [2] as “strongly coupled excited state”. Acknowledgements It is a pleasure to acknowledge very stimulating and useful discussions with Amir Yacoby which were initiated at the 225. International WE-Heraeus Seminar, October 11–15, 1999 in Bad Honnef. The work was supported by the European Union within the TMR programme, by the Deutsche Forschungsgemeinschaft within the SFB 508 of the Universitat Hamburg, and by the Italian MURST via Co nanziamento 98. References [1] A. Yacoby, H.L. Stormer, K.W. Baldwin, L.N. Pfei er, K.W. West, Solid State Commun. 101 (1997) 77. [2] O.M. Auslaender, A. Yacoby, R. de Picciotto, K.W. Baldwin, L.N. Pfei er, K.W. West, Phys. Rev. Lett. 84 (2000) 1764. [3] S. Tomonaga, Prog. Theor. Phys. 5 (1950) 544. [4] J.M. Luttinger, J. Math. Phys. 4 (1963) 1154. [5] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [6] M. Sassetti, B. Kramer, Phys. Rev. Lett. 80 (1998) 1485. [7] M. Sassetti, B. Kramer, Eur. Phys. J. B 4 (1998) 357. [8] C.L. Kane, M.P.A. Fisher, Phys. Rev. B 46 (1992) 15 233. [9] M. Bockrath, D.H. Cobden, J. Lu, A.G. Rinzler, R.E. Smalley, L. Balents, P.L. McEuen, Nature 397 (1999) 598. [10] S. Tarucha, T. Honda, T. Saku, Solid State Commun. 94 (1995) 413. [11] A. Yacoby, H.L. Stormer, N.S. Wingreen, L.N. Pfei er, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 77 (1996) 4612. [12] B. Kramer, M. Sassetti, unpublished. [13] T. Kleimann, M. Sassetti, B. Kramer, A. Yacoby (2000), unpublished. [14] D. Weinmann, W. Hausler, B. Kramer, Phys. Rev. Lett. 74 (1995) 984. [15] D. Weinmann, W. Hausler, B. Kramer, Ann. Phys. (Leipzig) 5 (1996) 652. [16] J. Voit, Rep. Prog. Phys. 57 (1995) 977. [17] M. Sassetti, F. Napoli, B. Kramer, Phys. Rev. B 59 (1999) 7297. [18] F. Napoli, B. Kramer M. Sassetti, Eur. Phys. J. B 11 (1999) 643. [19] E. Mariani, M. Sassetti, B. Kramer, Europhys. Lett. 49 (2000) 224. [20] M. Mariani, M. Sassetti, B. Kramer, Ann. Phys. (Leipzig) 8 (1999) 161. [21] F.A. Blum, Phys. Rev. B 1 (1970) 1125. [22] A. Pinczuk, L. Brillson, E. Burstein, E. Anastassakis, Phys. Rev. Lett. 27 (1971) 317.
30
M. Sassetti et al. / Physica E 9 (2001) 22–30
[23] M.V. Klein, in: M. Cardona (Ed.), Light Scattering in Solids, Springer, Berlin, 1975, p. 147. [24] A.R. Go˜ni, A. Pinczuk, J.S. Weiner, J.M. Calleja, B.S. Dennis, L.N. Pfei er, K.W. West, Phys. Rev. Lett. 67 (1991) 3298. [25] A. Schmeller, A.R. Go˜ni, A. Pinczuk, J.S. Weiner, J.M. Calleja, B.S. Dennis, L.N. Pfei er, K.W. West, Phys. Rev. B 49 (1994) 14 778. [26] R. Strenz, U. Bockelmann, F. Hirler, G. Abstreiter, G. Bohm, G. Weimann, Phys. Rev. Lett. 73 (1994) 3022. [27] C. Schuller, G. Biese, K. Keller, C. Steinebach, D. Heitmann, P. Gambow, K. Eberl, Phys. Rev. B 54 (1996) R17 304.
[28] F. Perez, B. Jusserand, B. Etienne, Physica E 7 (2000) 521. [29] A. Go˜ni, unpublished results. [30] M. Sassetti, B. Kramer, Phys. Rev. B 54 (1996) R5203. [31] A. Furusaki, N. Nagaosa, Phys. Rev. B 47 (1993) 3827. [32] M. Sassetti, F. Napoli, U. Weiss, Phys. Rev. B 52 (1995) 11 213. [33] A. Furusaki, Phys. Rev. B 57 (1998) 7141. [34] A. Braggio, M. Grifoni, M. Sassetti, F. Napoli, Europhys. Lett. 50 (2000) 136. [35] E. Lieb, D. Mattis, Phys. Rev. 125 (1962) 164. [36] A. Yacoby, private communication.