- Email: [email protected]
Transport properties of quantum wires
Solid State Communications 131 (2004) 647–655 www.elsevier.com/locate/ssc
Transport properties of quantum wires Maura Sassettia,*, Fabio Cavalierea,b, Bernhard Kramerb a
INFM-Lamia, Dipartimento di Fisica Universita` di Genova, Via Dodecaneso 33, 16146 Genova, Italy I. Institut fu¨r Theoretische Physik, Universita¨t Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
b
Received 25 February 2004; accepted by the guest editors Available online 8 June 2004
Abstract Electron transport in Luttinger liquids connected by tunnel barriers is reviewed. The non-analytic temperature behavior of the conductance of a single tunnel barrier is derived. An overview of charge and spin transport properties through a onedimensional quantum dot formed by two impurities is given. The temperature and voltage dependences of conductance peaks reveal the non-Fermi liquid correlations. Several spin effects in the linear as well as non-linear transport are predicted. These include a parity effect in linear transport which is due to the Pauli exclusion principle, and negative differential conductances in non-linear transport. The latter are due to charge and spin selection rules and non-Fermi liquid correlations which lead to correlation-induced trapping of higher-spin states. The possibility for experimental verification of this novel effect is discussed. q 2004 Published by Elsevier Ltd. PACS: 71.10.P; 73.61 Keywords: D. Electron–electron interactions; D. Electronic transport; D. Tunneling
1. Introduction Transport properties of low dimensional systems show distinct and strong signatures of electron – electron interaction. These are closely related with the two quantities that characterize the electron: the charge and the spin. Especially in one dimension, the repulsive interaction between the electrons is of crucial importance. It destroys the Fermi liquid properties of the electron gas and gives rise to the so called Luttinger liquid behavior [1 –5]. The recent experimental realization of semiconductor-based quasi-one dimensional quantum wires [6 –8] has opened new perspectives in investigating the influence of interaction and impurities on electron transport in one dimension. Also carbon nanotubes can be controlled with such a high perfection that the systematic investigation of Luttinger liquid properties in electron transport has become possible [9 –13]. A simple example for the effect of the electron * Corresponding author. Tel.: þ39-10-35364-67; fax: þ 39-10311-066. E-mail address: [email protected] (M. Sassetti). 0038-1098/$ - see front matter q 2004 Published by Elsevier Ltd. doi:10.1016/j.ssc.2004.05.023
interaction on the transport properties is the Coulomb blockade. It has been detected for the first time in experiments done on tunnel contacts between metallic wires at milli-Kelvin temperatures [14,15]. In single electron tunneling through quantum dots, formed by applying voltages to metallic gates on electron inversion layers in AlGaAs/GaAs heterostructures, sharp resonances in the linear conductance as a function of the gate voltage have been observed [16]. Between the resonances, the electron number is fixed. This can be used to manipulate with high precision single electrons in nanostructures [17]. In order to understand the phenomenon one can use a semiclassical model in which one assumes that the almost isolated quantum dot can be represented by a small capacitor [18]. Its magnitude C can be estimated from the geometry and the size of the device. For example, a disc of radius R ¼ 0:1 mm with a dielectric constant e r ¼ 10 has a capacitance C ¼ 8e 0 e r R < 10216 F: The classical electrostatic energy stored in such a capacitor containing N electrons is EðNÞ ¼ e2 N 2 =2C: The energy needed to add one additional electron is DEðNÞ ¼ EðN þ 1Þ 2 EðNÞ ¼ e2 ð2N þ 1Þ=2C: For stationary transport, this energy has to
648
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
fulfill the condition mL , DEðNÞ , mR ; where mL and mR are the chemical potentials in the left (L) and right (R) leads that are connected to the dot by high tunnel barriers. The bias voltage is V ¼ ðmR 2 mL Þ=e (e elementary charge). In linear transport, V ! 0; mL ¼ mR ¼ DEðNÞ: Since the electron number can be controlled by a gate voltage Vg ; this implies that the conductance peaks appear at gate voltages Vg ¼ aDEðNÞ (a sample dependent constant). The distance between the peaks is thus given by e2 =C ¼ EC : This is called the charging energy. The condition for observing the effect is kB T , EC : For the above example, EC < 0:2 meV such that the typical temperature scale for observing Coulomb blockade is 1 K. The Coulomb blockade effect can be considered as a consequence of the quantization of the charge and the classical ‘charge selection rule’ DQdot ¼ ^e which is valid for transport dominated by sequential tunneling. In this transport regime, electrons tunnel one by one, and the nonlocal nature of the wave functions is unimportant. In non-linear transport, informations about the energies En of the excited states of the interacting electrons in the quantum dot can be extracted from the current– voltage characteristics. In this region, all transport processes can contribute for which the differences of En ’s are consistent with mL , DEnn0 , mR : Thus, measuring the current– voltage characteristics of a quantum dot can yield direct information about the interaction between the electrons beyond the classical Coulomb repulsion. These correlations are contained in the energies of the excited states. Also, there are selection rules related to the nature of the many-electron wave functions. The coupling between the latter and the wave functions in the leads influences the strengths of the features in the current– voltage characteristics IðVÞ: An example for the influence of quantum mechanical selection rules is the spin blockade effect. This has been predicted to occur in quantum dots containing only fewelectrons [19]. Its signature is a negative differential conductance that occurs when an excited state of the quantum dot with maximum total spin can be occupied in the tunneling processes. For describing this phenomenon ‘spin selection rules’ are included in the tunneling rates phenomenologically [20]. When exactly one electron enters or leaves a quantum dot, the total spin can change only by 1/2 and its z-component by ^ 1/2. If a state with maximum total spin is occupied dynamically with increasing bias, the number of transport channels can be reduced since the z-component of the spin can only get smaller when an electron leaves the quantum dot. This implies that states with maximum spin act as bottlenecks for the transport and the current eventually decreases. There is a variety of related effects, especially with magnetic field. The spin blockade effect has not yet been identified unequivocally in experiment. However, measurements done on quasi-one dimensional quantum dots show that the
electron spin can lead to important effects in the sequential transport. For instance, in single walled carbon nanotubes Zeeman spin splitting of peaks in the non-linear conductance spectra in the ðV; Vg Þ-plane, and parity effects in the distances between the peaks in the linear conductance have been observed [9,10]. The influence of the exchange coupling also has been studied [12]. Electron correlations influence also the shape of the structures in the differential conductance apart from their positions, and their behavior as a function of the temperature. In a one-dimensional system the collective modes lead to non-analytic power law behavior of the conductance of a tunnel barrier as a function of the temperature G / Ta
ð1Þ
where the exponent a is related to the strengths of the interaction [21,22]. In transport experiments on a quasi-one dimensional quantum dot formed by two impurities in a semiconductor quantum wire such power law has been detected in the temperature behavior of the peaks in the linear conductance [23]. Microscopically, single electron transport through a one dimensional quantum dot formed by two equal tunnel barriers in a Tomonaga – Luttinger liquid has been studied intensively both in the incoherent, tunneling region and in the coherent regime [21,22,24– 31]. The above experimentally observed signatures of the non-Fermi liquid behavior in the temperature dependence of the Coulomb blockade peaks has been analyzed within the sequential tunneling approach [32 – 34]. It has been found that the spin is necessary for understanding quantitatively the dependence of the conductance peaks on temperature. However, the range of temperatures from which the experimental data can be used is limited. For instance, an upper limit is set by the presence of excited states. Also the theoretical results are valid only in limited ranges of temperatures. For instance, dE , kB T; where dE is the level broadening due to virtual states in the quantum dot coming from coherent higher order processes. Therefore, extracting information about the correlations by analyzing the temperature behavior of conductance features is difficult. One question to be asked is therefore whether or not it is possible to detect the presence of non-Fermi liquid correlations by different means. In this article, we provide a short overview of some of the phenomena that are related to non-Fermi liquid properties. We review transport through tunnel barriers that connect Luttinger liquids including the spin. Results are summarized for correlation effects in the temperature dependences of several conductance features. We start with the power law behavior of the conductance of a tunnel barrier. The role of interactions in the distances between the linear conductance peaks of two tunnel barriers is studied. The temperature behavior of the conductance peaks is discussed in detail. Finally, some recent results obtained, for effects of
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
correlations, in the non-linear conductance are described. For two asymmetric tunnel barriers in a Luttinger liquid we show that the electron spin can lead to non-linear negative differential conductances if states with higher spins can be dynamically occupied. However, in contrast to the spin blockade phenomenon discussed earlier, this happens only if the system exhibits spin –charge separation characteristic of a Luttinger liquid. This gives rise to peculiar behaviors of the tunneling rates through the barriers which can be calculated microscopically with the bosonization technique. Experimental detection of these negative differential conductances in quasi-one dimensional quantum dots induced by the spin – charge separation provides an additional possibility for studying non-Fermi liquid behavior without relying only on the above mentioned study of the temperature dependence of the conductance.
(s ¼ ^; unit "=2) of the total spin. It can be decomposed into left (l) and right (r) moving parts
cls ðxÞ ¼ eikF x cls;r ðxÞ þ e2ikF x cls;l ðxÞ
cls;r ðxÞ ¼ 2cls;l ð2xÞ; cls;r ðx þ 2LÞ ¼ cls;r ðxÞ
where a21 is a short wavelength cutoff and hs is a Majorana Fermion ensuring correct anticommutation relations for electrons with different spins. The zero mode-q fields are canonically conjugated to the charge and spin numbers
We consider in this section two Luttinger liquids that are coupled by a high tunnel barrier in the presence of a bias voltage (Fig. 1). The Hamiltonian is ð" ¼ 1Þ
The bosonic fields are
Vr ðkÞ ¼
vF k; Vs ðkÞ ¼ vF k g0
with the charge interaction parameter 2U0 21=2 ,1 g0 ¼ 1 þ pvF
Fln ðxÞ ¼
ð2Þ
with vF the Fermi velocity, n†l ðkÞ; nl ðkÞ the boson operators of the charge and spin density waves, and n^ l ; s^l the zero mode operators for the excess of charge and spin with respect to their average values in the leads. We assume open boundary conditions [35,36]. The energies of the charge and spin density waves are (k ¼ pk=L; k integer) ð4Þ
ð5Þ
determined by the repulsive interaction U0 : For simplicity, we have neglected the exchange interaction such that the spin degree of freedom is not influenced by the interaction. The Fermion operator cls ðxÞ depends on the z-component
X rffiffiffiffiffiffi p 2Lk
k.0
e2
ak 2
ð9Þ "
! pffiffiffi i gn sinðkxÞ nl ðkÞ 2 n†l ðkÞ
!# 1 † 2 pffiffiffi cosðkxÞ nl ðkÞ þ nl ðkÞ gn
ð10Þ
with gr ; g0 and gs ; 1: The tunneling Hamiltonian that connects the two Luttinger leads at x ¼ 0 is X ðLÞ† ðRÞ Ht ¼ t0 ½cs;r ðx ¼ 0Þcs;r ðx ¼ 0Þ þ h:c: s¼^1
¼
X
ðHsþ þ Hs2 Þ
ð11Þ
s¼^1
with t0 the transmission amplitude of the barrier. At L ¼ ^1 the leads are assumed to be connected to reservoirs with different electrochemical potentials mR and mL : These define the external bias voltage. The corresponding operator is written in terms of the charge operators Hc ¼
eV ½^n 2 n^ L 2 R
ð12Þ
The tunneling rate for an electron with spin s to hop from the right to the left lead is calculated via Fermi’s golden rule [21]
gþ ðeVÞ ¼
Fig. 1. Two semi-infinite Luttinger liquids, L, R, at bias voltage V; connected by a barrier with tunnel amplitude t0 :
ð7Þ
The Fermion field has a very useful representation in terms of the low energy collective charge and spin bosonic fields Frl ðxÞ; Fsl ðxÞ [35] hs 2iðq^ln þsq^ls Þ i px ð^nl þs^sl Þ Frl ðxÞþsFsl ðxÞ ffie cls;r ðxÞ ¼ pffiffiffiffiffi e 2L e ð8Þ 2pa
½q^ln ; n^ l ¼ i ½q^ls ; s^l ¼ i
Here, H0 ¼ H0ðLÞ þ H0ðRÞ describes the left and right uncoupled Luttinger liquids of length Lð!1Þ [35], " # X X pvF 1 2 2 ^ ^ Vn ðkÞn†l ðkÞnl ðkÞ þ þ s H0ðlÞ ¼ n ð3Þ l 4L g20 l n¼r;s k.0
ð6Þ
Due to the boundary conditions the latter are not independent [35,36]
2. Tunneling between two Luttinger liquids
H ¼ H0 þ Ht þ Hc
649
ð1 21
dteieVt kHsþ ðtÞHs2 ð0Þlleads
ð13Þ
with Hs^ ðtÞ ¼ expð2iH0 tÞHs^ expðiH0 tÞ the Heisenberg operators. The thermal average k· · ·lleads is performed using H0 : The rate eventually turns out to be independent of the spin direction s: This process corresponds to a current
650
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
flowing from the left to the right. The total current is ðkB T ¼ b21 Þ I ¼ 2e½gþ ðeVÞ 2 gþ ð2eVÞ ¼ 2egþ ðeVÞ½1 2 e2beV ð14Þ with the factor of two accounting for the spin degeneracy. The average in Eq. (13) can be explicitly evaluated [37] t0 2 ð1 gþ ðeVÞ ¼ dteieVt e2WðtÞ ð15Þ 2pa 21 Here, the kernel ð1 J ðvÞ dv 0 2 f ðv; b; tÞ WðtÞ ¼ v 0
ð16Þ
represents the influence of the collective charge and spin modes with bv f ðv; b; tÞ ¼ coth ð17Þ ½1 2 cos vt þ isinðvtÞ 2 and J0 ðvÞ ¼
2v 2v=vc e g
ð18Þ
the corresponding spectral density. The effective interaction constant is g ¼ 2=ð1 þ g21 0 Þ: This includes the effect of the spin density excitations. The cutoff vc is the highest energy in the model connected to the shortest wave length in Eq. (8) by vc ¼ vr =a; and is of the order of the Fermi energy. By performing the integrations we get for the temperature dependent rate at the energy E ¼ eV the explicit result ðE p vc Þ [33] t2 ebE=2 bvc 12ð2=gÞ 1 bE 2 gþ ðEÞ ¼ 0 G ð19Þ 2 g vc Gð2=gÞ 2p 2pi Here, we have redefined t0 =ð2paÞ ! t0 and GðxÞ is the Euler – gamma function. In the linear limit, V ! 0; one finds from Eq. (14) the expression for the linear conductance G ¼ ›I=›VlV!0 e2 2pt0 2 G2 ð1=gÞ 2pkB T 2ðð1=gÞ21Þ GðTÞ ¼ ð20Þ p vc Gð2=gÞ vc For repulsive interaction g , 1; the conductance vanishes non-analytically according to a power law when the temperature approaches absolute zero (Eq. (1)). This is the celebrated Luttinger liquid behavior of the conductance.
3. The conductance of a Luttinger liquid quantum dot In this section, we summarize several results for the conductance through two tunnel barriers that connect three Luttinger liquids in the incoherent tunneling region (Fig. 2). The Hamiltonian has the same form as in Eq. (2). However, H0 contains now the contribution of the ‘quantum dot’, the Luttinger system between tunnel barriers at 2a=2
Fig. 2. Three Luttinger liquids connected by tunnel barrier at x ¼ ^a=2: Bias voltages ^V=2 are applied to left (L) and right (R) leads. Gate voltage Vg is applied to the quantum dot (d) between 2a=2 and þa=2:
and a=2; in addition to the parts representing the left and right leads, " # X X pvF 1 2 ðdÞ † 2 H0 ¼ vn ðqÞn ðqÞnðqÞ þ n^ þ s^ ð21Þ 4a g2r n¼r;s q.0 Here, n† ðqÞ; nðqÞ are the boson operators of the collective charge and spin density waves in the quantum dot at wave numbers q ¼ ðpm=aÞ ðm $ 1Þ: The energy spectra are v vn ðqÞ ¼ vn q; vn ¼ F ð22Þ gn with the interaction parameters 2V0 21=2 gs ¼ 1 gr ¼ 1 þ pvF
ð23Þ
where V0 is the forward ðq ! 0Þ scattering contribution of the repulsive interaction. In general, gr – g0 : The zero mode operators n^ and s^ represent the excess of number of charges and spins with respect to their average values on the ground state. The eigenvalues of the zero mode operators are integer n and s with the constraint n þ s ¼ even: The zero mode energy contributions in Eq. (21) is the energy needed for changing the total charge and spin. The tunnel barriers at x ¼ ^a=2 are described by i X X h ðlÞ† Ht ¼ tl cs;r ðxl ÞcðdÞ ð24Þ s;r ðxl Þ þ h:c: s¼^1 l¼L;R
with cðdÞ s;r the right moving fermions operators and tL;R the transmission amplitudes of the barriers [35,36]. As before, we assume high tunnel barriers such that Ht can be treated as a perturbation. The operator of the external bias and gate voltages for electrically controlling current and charge density is obtained by including in Hc the term Cg dC Hgate ¼ 2e Vþ Vg n^ ð25Þ 2CS CS Here, Vg is the gate voltage, CS ¼ CL þ CR þ Cg the total capacitance with CL ; CR and Cg the capacitances of the leads and the gate, and dC ¼ CL 2 CR : For high barriers and not too low temperatures, tunneling through the two barriers can be described as sequential transfers of electrons with spin up or down through the
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
quantum dot region ð2a=2; a=2Þ: In this case, Ht is treated as a perturbation in the lowest order. Higher order coherent processes can be safely neglected as long as kB T q dE; with dE the level broadening due to virtual states, proportional to the tunneling rates [28]. During these incoherent processes the state of the dot is changed between initial lil and final lf l configurations. These are characterized by the excess of number of electrons ðnÞ and spin ðsÞ; and by the occupation numbers of charge ðrÞ and spin ðsÞ density excitations at different wave numbers q: We denote them as ln; s; {lqr }; {lqs }l: The energy of the above configuration will be Er E Uðn; s; lr ; ls Þ ¼ ðn 2 ng Þ2 þ s s2 þ lr 1r þ ls 1s 2 2
i i
The plasmon energy 1r is affects by the Coulomb interaction, with in general 1r . 1s : The energetic difference between the charge and spin density wave is a direct manifestation of the spin– charge separation occurring in a Luttinger liquid. From the above discussion we can identify a ‘hierarchy’ of energy scales 2Es ¼ 10 ¼ 1s , 1r , Er ; so that we choose Es as the natural energy unit. Due to the sequential nature of the tunneling the initial and final configurations have to fulfill the selection rules ð29Þ
We assume that the states containing collective excitations,
f
f
21
Here, the factor expðiDU tÞ ensures energy conservation,
ð26Þ
where 10 ¼ pvF =a plays the role of constant level spacing in the non-interacting case. These terms are different from zero even in the non-interacting limit, a fact which reflects the discrete nature of the energy levels inside the quantum dot and the Pauli exclusion principle. Despite the microscopic model provides an estimate for the charge addition energy, many physical effects have been here oversimplified. For instance, the coupling with gates, the long range interaction effects and the coupling with the nearby 2DEG [23] can strongly affect Er leading to significant deviations from the simple expression Eq. (27) [34]. Therefore, in the following we will treat Er as a free adjustable parameter, and we assume Er q Es : On the contrary, for the spin energy we use the microscopic expression Es ¼ 10 =2: The second two terms in Eq. (26) represent the charge and spin density waves contributions. Since the excitation spectra are linear in the wave number, the energies of the collective modes P depend on the total number of excitation quanta ln ¼ q qlqn only via the discrete excitation energies pvn 1 ¼ 0 ð28Þ 1n ¼ a gn
Dn ¼ nf 2 ni ¼ ^1; Ds ¼ sf 2 si ¼ ^1
although they can be occupied when an electron enters the quantum dot region, decay very rapidly into the corresponding ground state with lr ¼ ls ¼ 0: This implies that when calculating the tunnel rates the thermal average includes averaging over the initial collective excitations of the quantum dot in addition to those of the leads. Then, the tunneling rates for barriers l ¼ L; R is ð1 GðlnlÞ;s l!ln ;s l ¼ tl2 dteiDU t e2Wl ðtÞ e2Wd ðtÞ ð30Þ
DU ¼
Here, the first two terms represent the charge and spin addition energy contributions. In the charge sector we have included the term ðenCg Vg =CS Þ of Eq. (25) with the gate voltage induced number of electrons ng ¼ Cg Vg =e: From the microscopic Hamiltonian one has ðn ¼ r; sÞ pvn 1 ¼ 02 ð27Þ En ¼ 2agn 2gn
651
Er E ½1 þ 2ðni 2 ng ÞDn þ s ½1 þ 2si Ds 2 2 2 dC eV 71 Dn CS 2
ð31Þ
The signs 7 refers to the right (2) and to the left (þ ) barrier. The correlation function ðlÞ† e2Wl ðtÞ ¼ kcs;r ðxl ; tÞcðs;rlÞ ðxl ; 0Þlleads
ð32Þ
results from the trace over the excitations in the leads as in Section 2 [28]. The correlation function ðdÞ e2Wd ðtÞ ¼ kcðdÞ† s;r ðxl ; tÞcs;r ðxl ; 0Þldot
ð33Þ
contains the thermal average over the collective modes in the dot, performed with respect to H0ðdÞ : The dissipative functions Wl;d ðtÞ have the same form as in Section 2. However, for Wl ; in Eq. (16) the spectral density is now J0 ðvÞ=2: For Wd ; J0 ðvÞ must be replaced by the spectral function of the quantum dot, Jd ðvÞ ¼ v
1 X en X dðv 2 me n Þe2v=vc 2g n n¼r;s m¼1
ð34Þ
The structure of this spectral density of the dot indicates that, although they are infinitely quickly relaxing, charge and spin density waves still contribute to the tunneling dynamics. Even if the relaxation prevents the collective excitations to be initial states for the tunneling, it is still possible to reach an excited state as a ‘final’ state for a given energy when an electron enters the dot. Using the bosonization method one can calculate exactly all of the above averages as in Section 2 [35]. For simplicity, we do not specify in the following the initial and final states. Exploiting the discrete nature of the dot spectral density one rewrite Eq. (30) as a function of the energy difference E ¼ DU X GðlÞ ðEÞ ¼ Gð0lÞ alr als gðE 2 lr 1r 2 ls 1s Þ ð35Þ lr ;ls
where
Gð0lÞ ¼
1r vc
1=2gr
1s vc
1=2gs
2vc Gl e2 Gð1=gÞ
ð36Þ
2 2 2 with Gl ¼ R21 l ¼ pe tl =vc the intrinsic conductances of the barriers. As before, the function gðxÞ is determined by
652
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
peaks is affected by the spin that leads to a parity effect in the distances between peaks
the leads [38] 1 bx=2 1 bx 2 2p ð1=gÞ21 e G 2 gðxÞ ¼ 2p 2g 2pi bvc At T ¼ 0 we have ð1=gÞ21 x g0 ðxÞ ¼ uðxÞ vc
ð37Þ
ð38Þ
The weights aln are due to the quantum dot. At T ¼ 0; they can be exactly evaluated a0ln ¼
Gð1=2gn þ ln Þ uðln Þ Gð1=2gn Þln !
ð39Þ
For finite temperature, they have to be numerically determined. In the presence of spin charge separation the above weights modify crucially the rate Eq. (35) which shows non-integer quantized steps at different energies 1r – 1s : For the calculation of the current voltage characteristic one has to determine the occupation probabilities Pn;s for the charge and spin states ln; sl of the quantum dot from the master equation X X ›t Pn;s ðtÞ ¼ n0 ¼n^1 s0 ¼s^1
h i Pn0 ;s0 ðtÞGln0 ;s0 l!ln;sl 2 Pn;s ðtÞGln;sl!ln0 ;s0 l ð40Þ P ðlÞ where G ; l G : In the stationary limit, ›t Pn;s ¼ 0; one has to solve a homogeneous system P of linear equations. The solution must be normalized, n;s Pn;s ¼ 1: The stationary current is h i X X ðRÞ I¼e Pn;s GðRÞ 2 G ð41Þ ln;sl!lnþ1;sþDsl ln;sl!ln21;sþDsl n;s Ds¼^1
If both eV; kB T , Er ; at most two charge states can enter the dynamics, and the current is given by tunneling events corresponding to transitions of the type ln; sl ! ln þ 1; s0 l ! ln; s00 l: As an example, we consider the linear conductance G at low temperatures, kB T p Es for symmetric barriers ðtR ¼ tL Þ: One can solve the master equation in the subspace of the ground states with n and n þ 1 electrons. The position of the conductance peak at T ¼ 0 is determined by the electrochemical potential of the quantum dot, 1 E ð42Þ md ¼ Er n 2 ng þ þ ð21Þn s ¼ 0 2 2
dn;nþ1 ¼ 1 þ ð21Þnþ1
Es Er
ð45Þ
For T – 0 the conductance peak positions shift linearly with temperature res nþ1 nres fðgÞ g ðTÞ ¼ ng þ ð21Þ
kB T Er
ð46Þ
Without interaction one recover the well-known result fð1Þ ¼ log 2=2 [38]. With interaction, fðgÞ increases monotonically increasing interactions, reaching the maximum value fð0Þ ¼ log 2: The temperature dependence of the conductance maximum, which is around j ¼ 0; is determined only by the excitations in the leads present in GðRÞ ð0Þ since one cannot excite charge and spin modes in the dot ðlr ¼ ls ¼ 0Þ Gmax ðTÞ / T ð1=gÞ22
ð47Þ
A sequence of conductance peaks obtained for two temperatures which show the characteristic low temperature features is shown in Fig. 3.
4. Non-linear transport In this section, we discuss some of the non-linear transport features. The master equation for obtaining the stationary probability distribution Pn;s is solved numerically. From this, the current – voltage characteristic and the differential conductance are calculated. We consider asymmetric tunnel barriers. The asymmetry is described by the ratio of the conductances of the left and right tunnel barrier, A ¼ GL =GR : Note that for A . 1; V . 0 and kB T , Es ; the electrons traverse the higher right barrier into the dot, and the lower left barrier out of the dot. Thus, states ðn; sÞ with n ¼ even will have a higher occupation probability as compared to the states ðn þ 1; s þ 1Þ: This will eventually create a ‘trapping’ phenomenon at sufficiently large
The conductance for kB T , Es is
be2 GðRÞ ðjÞe2bj=2 G ¼ pffiffi 8 cosh{½bj þ ð21Þn log 2=2}
ð43Þ
where j ¼ Er ðng 2 nres g Þ corresponds to the deviation from the resonance peak position at T ¼ 0; nres g ¼nþ
1 E þ ð21Þn s 2 2Er
ð44Þ
The rate GðRÞ ðjÞ is given in Eq. (35). The position of the
Fig. 3. Linear conductance (unit e2 G0ðRÞ =Es ) of a quantum dot as a function of the number gate voltage ng for symmetric tunnel barriers at temperatures kB T equal 0:02Es (full curve) and 0:5Es (dotted), with Er ¼ 5Es ; g ¼ 0:9; gr ¼ 0:6:
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
asymmetries, inducing the rich features that will be described below. Fig. 4 shows the conductance G in the ðV; ng Þ-plane including several regions of Coulomb blockade for A ¼ 100: The parity effect mentioned above is clearly observed. The rich structure for V – 0 provides in principle detailed information about the spin addition features and the collective spin and charge excitations of the quantum dot. It is obvious that in order to extract this information a detailed analysis of the possible transitions including the behavior of the corresponding tunnel rates is necessary. Of particular importance are the conductance features indicated by the white lines. These correspond to negative differential conductance. The non-linear transport features are rather stable against changes in the temperature as can be seen by comparing the left and right panels. As an example we discuss in some detail the behavior of the negative differential conductance indicated by the white transition lines in Fig. 4. They are more clearly shown in the magnification of the region near the Coulomb peak between charge numbers n, n þ 1 (n even) shown in Fig. 5. The physical origin of the negative differential conductance is here related to the dynamical occupation of excited states with higher spin (states with spin s and 2s are degenerate, Eq. (26)). As a consequence of the spin selection rule they can act as traps in certain regions of parameters. To understand the phenomenon let us identify the different lines in Fig. 5. The two fundamental black transition lines starting at V ¼ 0 with slope Er ðng 2 nres g Þ ¼ ^eV=2 are the boundaries of the stable Coulomb blockade regions with n and n þ 1 charges and correspond to transitions from the lowest spins ðn þ 1; 1Þ ! ðn; 0Þ; and ðn; 0Þ ! ðn þ 1; 1Þ; respectively. Here, the resonant value of the gate voltage corresponds to nres g ¼ 0:52: Once the state ðn þ 1; 1Þ is occupied, transition to higher spin as ðn þ 1; 1Þ ! ðn; 2Þ becomes
Fig. 4. The differential conductance G (arbitrary unit), as a function of bias voltage V (unit Es =e) and number of charges ng : System parameters: n even, Er ¼ 5Es ; g0 ¼ 1; gr ¼ 0:63; asymmetry A ¼ kB T ¼ 7:5 £ 1023 Es ; right kB T ¼ 100; CL ¼ CR ; left: 22 7:5 £ 10 Es : Right: gray scale.
653
Fig. 5. The differential conductance G (arbitrary unit) as a function of bias voltage V (unit Es =e) and ng ; for kB T ¼ 7:5 £ 1023 Es : Other parameters as in Fig. 4, apart Er ¼ 25Es : Right: gray scale.
available by increasing the voltage. The activation threshold in this case is given by Er ðng 2 nres g Þ ¼ eV=2 2 2Es starting, in the above figure, at ðV; ng Þ ¼ ð2Es ; 0:48Þ: Along this line negative differential conductance regions occur. By iterating the procedure with increasing voltage even higher spin states become occupied. Because of the asymmetry A . 1 the lines with negative conductances always correspond to transitions from odd s to s þ 1 even. So far, we only have considered the excited states of the dot with higher spins. In order to complete the picture, we must include all of the transitions involving collective charge and spin excitations. This enhances considerably the complexity of the spectrum since at sufficiently high bias voltage each transition of the type ðn; sÞ ! ðn0 ; s0 Þ can also occur via the channels ðn; sÞ ! ðn0 ; s0 ; lr ; ls Þ ! ðn0 ; s0 Þ: For example, consider charge and spin density waves for the transition ðn; 0Þ ! ðn þ 1; 1Þ: These correspond to the lines Er ðng 2 nres g Þ ¼ ð2eV=2Þ þ lr 1r þ ls 1s : They are parallel to the fundamental line ðn; 0Þ ! ðn þ 1; 1Þ: Analogously, including charge and spin collective modes in the transition ðn þ 1; 1Þ ! ðn; 0Þ give rise to Er ðng 2 nres g Þ ¼ ðeV=2Þ þ lr 1r þ ls 1s with transition parallel to the fundamental line ðn þ 1; 1Þ ! ðn; 0Þ: Now, in order to better understand the first transitions ðs ¼ 0; ^1; ^2Þ and the corresponding occupation probabilities, we consider Fig. 5 at fixed gate voltage ng ¼ 0:49 as a function of V: Starting in the Coulomb blockade region with charge n and V ¼ 0; by increasing the bias one eventually populates the ðn þ 1; 1Þ-state via a tunneling process through the right barrier near V ¼ 1:5Es : At this voltage a current starts to flow since this state can easily be emptied again via tunneling through the left barrier. When the bias is further increased, the transition ðn þ 1; 1Þ ! ðn; 2Þ becomes also possible at V ¼ 2:5Es : The state ðn; 2Þ can decay into the ground state ðn; 0Þ only via the ðn þ 1; 1Þ-state, i.e. if first another electron enters the quantum dot. This could increase the current further. However, if the occupation of the
654
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
where
Fig. 6. Occupation probabilities Pn;0 (dashed), Pn;2 (dashed – dotted), Pnþ1;1 (dotted), and current I (unit eG0ðRÞ ; full line) as a function of V (unit Es =e); system parameters as in Fig. 5.
higher-spin state ðn; 2Þ is favored at the expense of the ground state ðn; 0Þ; it will accumulate occupation probability (Fig. 6). This will eventually lead to a reduction of the current, and to a negative differential conductance. Asymmetric tunnel barriers provide only one possible mechanism for preferring the higher-spin state. In the parameter region considered here, the right tunnel barrier is higher that the left which implies that the n-electron states will be more frequently occupied on the average. There are also other mechanisms that could lead to such asymmetry in the occupation, for instance due to spin dependent tunneling [39]. However, it is important to realize that in the present case trapping alone is not sufficient to induce negative differential conductance. The detailed analysis of the master equation in the limit of large asymmetry, which is the most favorable case to induce negative conductance, shows that the crucial ingredient for the phenomenon discussed here is the spin charge separation in the quantum dot. Without spin– charge separation ðgr ¼ gs ¼ 1Þ; no negative differential conductance can occur. This is intimately related to the behavior of the rate Eq. (35) which, in the absence of spin – charge separation, has equal steps centered at equidistant energy. This behavior will never strongly promote higher spin at the expense of the lower ones, giving always an increasing current and a positive conductance. Only non-equal steps, induced by gr – 1; provide this possibility. In order to make this crucial ingredient more transparent, we have reconsidered the parameter region taking into account only the five dominant spin states. In this case the master equation is solved analytically. We quote here the result for the transition line ðn þ 1; 1Þ ! ðn; 2Þ for voltages low enough not too excite charge modes (first white line in Fig. 5). We assume non-interacting leads, kB T p Es and spin charge separation ðgr , 1Þ: One gets
G ¼ eL
ðLÞ 2GR0!1 GR2!1 ›G1!2 ðVÞ 2 ›V D
ð48Þ
ðLÞ ðRÞ L ¼ 2GR0!1 GR2!1 þ G1!0 ðG2!1 2 2GðRÞ 0!1 Þ
ð49Þ
ðLÞ ðRÞ ðRÞ ðLÞ D ¼ G1!0 G2!1 þ 2GðRÞ 0!1 ðG2!1 þ G1!2 Þ
ð50Þ
with the rates evaluated at the transition energies U0!1 ¼ eV 2 2Es ; U2!1 ¼ eV; U1!0 ¼ 2Es ; U1!2 ¼ 0: The condition for negative differential conductance is determined solely by L , 0: The latter, when evaluated along the first part of the line ðn þ 1; 1Þ ! ðn; 2Þ has rates values GðLÞ 1!0 ¼ ðRÞ ðRÞ ð3=2ÞG0ðLÞ ; G0!1 ¼ G0ðRÞ ; while GðRÞ 2!1 ¼ ð3=2ÞG0 : We then have negative differential conductance if A $ 8: If gr ¼ 1 ðLÞ ðRÞ the values of the rates change as G1!0 ¼ 2GðLÞ 0 and G2!1 ¼ ðRÞ 2G0 yielding L . 0 for any asymmetry. Similarly, one can understand other regions of negative differential conductances in Fig. 5. To conclude we show in Fig. 7 the behavior of the differential conductance with temperature. The negative peak corresponds to the above discussed transition line ðn þ 1; 1Þ ! ðn; 2Þ: The positive peak is contributed by the first collective charge excitation above the ðn þ 1; 1Þ-ground state. With increasing temperature, the conductance peaks are broadened due to the Fermi distributions in the left and right leads. The negative conductance features look quite robust against increasing temperature, despite the position of the peak is shifted linearly towards the positive one.
5. Conclusion We have reviewed results of extensive studies of the electron transport through tunnel barriers connecting
Fig. 7. Middle: two differential conductance peaks, one negative (white) and one positive (black), as a function of voltage (unit Es =e) and temperature (unit 1023 Es ), for ng ¼ 0:5: Right: gray scale. Top: the conductance as a function of V (unit Es =e) for kB T ¼ 2 £ 1022 Es : Bottom: as in the top but with kB T ¼ 2 £ 1023 Es : System parameters: Er ¼ 25Es ; g0 ¼ 1; gr ¼ 0:63; A ¼ 100; CL ¼ CR ; conductance unit: e2 GðRÞ 0 =Es :
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
Luttinger liquids with spin. We have considered the temperature region of sequential tunneling. Special emphasis has been devoted to the signatures of non-Fermi liquid correlations. For a single barrier connecting two semiinfinite leads the well-known non-analytic temperature behavior has been recovered. For two barriers connecting a Luttinger liquid of finite length with leads the temperature behavior of the linear conductance peaks has been briefly mentioned. In the non-linear transport region, the energy scales of the excited states have been detected in the differential conductance. Negative differential conductance features have been found that are related to the occupation of states with higher spins in the quantum dot. The physical origin of these features is the presence of the non-Fermi liquid correlations which provide not only different energy scales for charge and spin excitations but also influence the magnitude and temperature behavior of the tunnel rates. A strong asymmetry enhances the chance to create a negative differential conductance but it would be not sufficient alone. In the figures described above we have used high asymmetries in order to display the effect more clearly. However, as shown by analytic analyses, negative differential conductances can be found also at smaller asymmetries. Most importantly, the phenomenon is rather stable against increasing the temperature (Figs. 4 and 7). In order to see whether or not such phenomenon could be observed experimentally we consider a one dimensional quantum dot of the length 0.2 mm. The corresponding energy scale is Es ¼ ðpvF =aÞ < 1 meV if we assume a Fermi velocity of order 105 m/s. This corresponds to a temperature of about 10 K and the temperatures used in obtaining the above results kB T < 1022 Es is roughly 0.1 K. It might be more difficult to obtain sufficient asymmetry in the tunnel resistances still fulfilling the conditions for sequential tunneling. However, it seems to us that asymmetric barriers are the genuine situation in experiment, rather than the exception. So one can hope that in experiment on semiconductor quantum dots as well as in Carbon nanotubes the effect predicted above might be observable.
References [1] [2] [3] [4] [5] [6]
S. Tomonaga, Prog. Theor. Phys. 5 (1950) 544. J.M. Luttinger, J. Math. Phys. 4 (1963) 1154. F.D.M. Haldane, J. Phys. C 14 (1981) 2585. H.J. Schulz, Int. J. Mod. Phys. B 5 (1991) 57. J. Voit, Rep. Prog. Phys. 58 (1995) 977. S. Tarucha, T. Honda, T. Saku, Solid State Commun. 94 (1995) 413. [7] A. Yacoby, H.L. Stormer, N.S. Wingreen, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 77 (1996) 4612. [8] A. Yacoby, H.L. Stormer, K.W. Baldwin, L.N. Pfeiffer, K.W. West, Solid State Commun. 101 (1997) 77. [9] D.H. Cobden, M. Bockrath, P.L. McEuen, A.G. Rinzler, R.E. Smalley, Phys. Rev. Lett. 81 (1998) 681.
655
[10] D.H. Cobden, J. Nyga˚rd, Phys. Rev. Lett. 89 (2002) 046803. [11] M.R. Buitelaar, A. Bachtold, T. Nussbaumer, M. Iqbal, C. Scho¨nenberger, Phys. Rev. Lett. 88 (2002) 156801. [12] W. Liang, M. Bockrath, H. Park, Phys. Rev. Lett. 88 (2002) 126801. [13] H.W.Ch. Postma, M. de Jonge, Z. Yao, C. Dekker, Phys. Rev. B 62 (2000) R10653. [14] T.A. Fulton, P.L. Gammel, D.J. Bishop, L.N. Dunkleberger, Phys. Rev. Lett. 63 (1989) 1307. [15] L.J. Geerligs, V.F. Anderegg, P.A.M. Holweg, J.E. Mooij, Phys. Rev. Lett. 64 (1989) 2691. [16] U. Meirav, M.A. Kastner, S. Wind, Phys. Rev. Lett. 65 (1990) 771. [17] For timely overviews of the field, see, e.g. (a), in: B. Kramer (Ed.), Quantum Coherence in Mesoscopic Systems, NATO ASI Series B 254, Plenum, New York, 1992. (b) in: M. Devoret, H. Grabert (Eds.), Single Charge Tunneling, NATO ASI Series B 294, 1992, p. 21. (c) in: B. Kramer (Ed.), Quantum Transport in Semiconductor Submicron Structures, NATO ASI Series E 326, Kluwer, Dordrecht, 1996. (d) in: L.L. Sohn, L.P. Kouwenhoven, G. Scho¨n (Eds.), Mesoscopic Electron Transport, NATO ASI Series E 345, Kluwer, Dordrecht, 1997. [18] G.-L. Ingold, Y. Nazarov, Single Charge Tunneling, NATO ASI Series B 294, 1992, p. 21. [19] D. Weinmann, W. Ha¨usler, B. Kramer, Phys. Rev. Lett. 74 (1995) 984. [20] K. Jauregui, W. Ha¨usler, B. Kramer, Phys. Rev. B 53 (1997) R1713. [21] (a) C.L. Kane, M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 1220. (b) C.L. Kane, M.P.A. Fisher, Phys. Rev. B 46 (1992) 15233. [22] A. Furusaki, N. Nagaosa, Phys. Rev. B 47 (1993) 3827. [23] O.M. Auslaender, A. Yacoby, R. de Picciotto, K.W. Baldwin, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 84 (2000) 1764. [24] M. Sassetti, F. Napoli, U. Weiss, Phys. Rev B 52 (1995) 11213. [25] H. Maurey, T. Giamarchi, Europhys. Lett. 38 (1997) 681. [26] R. Egger, H. Grabert, Phys. Rev. B 55 (1997) 9929. [27] M.P.A. Fisher, L.I. Glazman, Mesoscopic Electron Transport, NATO ASI Series B 345, Kluwer, Dordrecht, 1997. [28] A. Furusaki, Phys. Rev. B 57 (1998) 7141. [29] A. Braggio, M. Grifoni, M. Sassetti, F. Napoli, Europhys. Lett. 50 (2000) 236. [30] Yu.V. Nazarov, L.I. Glazman, Phys. Rev. Lett. 91 (2003) 126804. [31] D.G. Polyakov, I.V. Gornyi, Phys. Rev. B 68 (2003) 035421. [32] T. Kleimann, M. Sassetti, B. Kramer, A. Yacoby, Phys. Rev. B 62 (2000) 8144. [33] A. Braggio, M. Sassetti, B. Kramer, Phys. Rev. Lett. 87 (2001) 146802. [34] T. Kleimann, F. Cavaliere, M. Sassetti, B. Kramer, Phys. Rev. B 66 (2002) 165311. [35] M. Fabrizio, A.O. Gogolin, Phys. Rev. B 51 (1995) 17827. [36] (a) S. Eggert, H. Johannesson, A. Mattsson, Phys. Rev. Lett. 76 (1996) 1505. (b) A.E. Mattsson, S. Eggert, H. Johannesson, Phys. Rev. B 56 (1997) 15615. [37] U. Weiss, Quantum Dissipative Systems, Second ed., World Scientific, Singapore, 1999. [38] G. Usaj, H.U. Baranger, Phys. Rev. B 64 (2001) 201319. [39] M. Ciorga, A. Wensauer, M. Pioro-Ladriere, M. Korkusinski, J. Kyriakidis, A.S. Sachrajda, P. Hawrylak, Phys. Rev. Lett. 88 (2002) 256804.
Transport properties of quantum wires Maura Sassettia,*, Fabio Cavalierea,b, Bernhard Kramerb a
INFM-Lamia, Dipartimento di Fisica Universita` di Genova, Via Dodecaneso 33, 16146 Genova, Italy I. Institut fu¨r Theoretische Physik, Universita¨t Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
b
Received 25 February 2004; accepted by the guest editors Available online 8 June 2004
Abstract Electron transport in Luttinger liquids connected by tunnel barriers is reviewed. The non-analytic temperature behavior of the conductance of a single tunnel barrier is derived. An overview of charge and spin transport properties through a onedimensional quantum dot formed by two impurities is given. The temperature and voltage dependences of conductance peaks reveal the non-Fermi liquid correlations. Several spin effects in the linear as well as non-linear transport are predicted. These include a parity effect in linear transport which is due to the Pauli exclusion principle, and negative differential conductances in non-linear transport. The latter are due to charge and spin selection rules and non-Fermi liquid correlations which lead to correlation-induced trapping of higher-spin states. The possibility for experimental verification of this novel effect is discussed. q 2004 Published by Elsevier Ltd. PACS: 71.10.P; 73.61 Keywords: D. Electron–electron interactions; D. Electronic transport; D. Tunneling
1. Introduction Transport properties of low dimensional systems show distinct and strong signatures of electron – electron interaction. These are closely related with the two quantities that characterize the electron: the charge and the spin. Especially in one dimension, the repulsive interaction between the electrons is of crucial importance. It destroys the Fermi liquid properties of the electron gas and gives rise to the so called Luttinger liquid behavior [1 –5]. The recent experimental realization of semiconductor-based quasi-one dimensional quantum wires [6 –8] has opened new perspectives in investigating the influence of interaction and impurities on electron transport in one dimension. Also carbon nanotubes can be controlled with such a high perfection that the systematic investigation of Luttinger liquid properties in electron transport has become possible [9 –13]. A simple example for the effect of the electron * Corresponding author. Tel.: þ39-10-35364-67; fax: þ 39-10311-066. E-mail address: [email protected] (M. Sassetti). 0038-1098/$ - see front matter q 2004 Published by Elsevier Ltd. doi:10.1016/j.ssc.2004.05.023
interaction on the transport properties is the Coulomb blockade. It has been detected for the first time in experiments done on tunnel contacts between metallic wires at milli-Kelvin temperatures [14,15]. In single electron tunneling through quantum dots, formed by applying voltages to metallic gates on electron inversion layers in AlGaAs/GaAs heterostructures, sharp resonances in the linear conductance as a function of the gate voltage have been observed [16]. Between the resonances, the electron number is fixed. This can be used to manipulate with high precision single electrons in nanostructures [17]. In order to understand the phenomenon one can use a semiclassical model in which one assumes that the almost isolated quantum dot can be represented by a small capacitor [18]. Its magnitude C can be estimated from the geometry and the size of the device. For example, a disc of radius R ¼ 0:1 mm with a dielectric constant e r ¼ 10 has a capacitance C ¼ 8e 0 e r R < 10216 F: The classical electrostatic energy stored in such a capacitor containing N electrons is EðNÞ ¼ e2 N 2 =2C: The energy needed to add one additional electron is DEðNÞ ¼ EðN þ 1Þ 2 EðNÞ ¼ e2 ð2N þ 1Þ=2C: For stationary transport, this energy has to
648
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
fulfill the condition mL , DEðNÞ , mR ; where mL and mR are the chemical potentials in the left (L) and right (R) leads that are connected to the dot by high tunnel barriers. The bias voltage is V ¼ ðmR 2 mL Þ=e (e elementary charge). In linear transport, V ! 0; mL ¼ mR ¼ DEðNÞ: Since the electron number can be controlled by a gate voltage Vg ; this implies that the conductance peaks appear at gate voltages Vg ¼ aDEðNÞ (a sample dependent constant). The distance between the peaks is thus given by e2 =C ¼ EC : This is called the charging energy. The condition for observing the effect is kB T , EC : For the above example, EC < 0:2 meV such that the typical temperature scale for observing Coulomb blockade is 1 K. The Coulomb blockade effect can be considered as a consequence of the quantization of the charge and the classical ‘charge selection rule’ DQdot ¼ ^e which is valid for transport dominated by sequential tunneling. In this transport regime, electrons tunnel one by one, and the nonlocal nature of the wave functions is unimportant. In non-linear transport, informations about the energies En of the excited states of the interacting electrons in the quantum dot can be extracted from the current– voltage characteristics. In this region, all transport processes can contribute for which the differences of En ’s are consistent with mL , DEnn0 , mR : Thus, measuring the current– voltage characteristics of a quantum dot can yield direct information about the interaction between the electrons beyond the classical Coulomb repulsion. These correlations are contained in the energies of the excited states. Also, there are selection rules related to the nature of the many-electron wave functions. The coupling between the latter and the wave functions in the leads influences the strengths of the features in the current– voltage characteristics IðVÞ: An example for the influence of quantum mechanical selection rules is the spin blockade effect. This has been predicted to occur in quantum dots containing only fewelectrons [19]. Its signature is a negative differential conductance that occurs when an excited state of the quantum dot with maximum total spin can be occupied in the tunneling processes. For describing this phenomenon ‘spin selection rules’ are included in the tunneling rates phenomenologically [20]. When exactly one electron enters or leaves a quantum dot, the total spin can change only by 1/2 and its z-component by ^ 1/2. If a state with maximum total spin is occupied dynamically with increasing bias, the number of transport channels can be reduced since the z-component of the spin can only get smaller when an electron leaves the quantum dot. This implies that states with maximum spin act as bottlenecks for the transport and the current eventually decreases. There is a variety of related effects, especially with magnetic field. The spin blockade effect has not yet been identified unequivocally in experiment. However, measurements done on quasi-one dimensional quantum dots show that the
electron spin can lead to important effects in the sequential transport. For instance, in single walled carbon nanotubes Zeeman spin splitting of peaks in the non-linear conductance spectra in the ðV; Vg Þ-plane, and parity effects in the distances between the peaks in the linear conductance have been observed [9,10]. The influence of the exchange coupling also has been studied [12]. Electron correlations influence also the shape of the structures in the differential conductance apart from their positions, and their behavior as a function of the temperature. In a one-dimensional system the collective modes lead to non-analytic power law behavior of the conductance of a tunnel barrier as a function of the temperature G / Ta
ð1Þ
where the exponent a is related to the strengths of the interaction [21,22]. In transport experiments on a quasi-one dimensional quantum dot formed by two impurities in a semiconductor quantum wire such power law has been detected in the temperature behavior of the peaks in the linear conductance [23]. Microscopically, single electron transport through a one dimensional quantum dot formed by two equal tunnel barriers in a Tomonaga – Luttinger liquid has been studied intensively both in the incoherent, tunneling region and in the coherent regime [21,22,24– 31]. The above experimentally observed signatures of the non-Fermi liquid behavior in the temperature dependence of the Coulomb blockade peaks has been analyzed within the sequential tunneling approach [32 – 34]. It has been found that the spin is necessary for understanding quantitatively the dependence of the conductance peaks on temperature. However, the range of temperatures from which the experimental data can be used is limited. For instance, an upper limit is set by the presence of excited states. Also the theoretical results are valid only in limited ranges of temperatures. For instance, dE , kB T; where dE is the level broadening due to virtual states in the quantum dot coming from coherent higher order processes. Therefore, extracting information about the correlations by analyzing the temperature behavior of conductance features is difficult. One question to be asked is therefore whether or not it is possible to detect the presence of non-Fermi liquid correlations by different means. In this article, we provide a short overview of some of the phenomena that are related to non-Fermi liquid properties. We review transport through tunnel barriers that connect Luttinger liquids including the spin. Results are summarized for correlation effects in the temperature dependences of several conductance features. We start with the power law behavior of the conductance of a tunnel barrier. The role of interactions in the distances between the linear conductance peaks of two tunnel barriers is studied. The temperature behavior of the conductance peaks is discussed in detail. Finally, some recent results obtained, for effects of
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
correlations, in the non-linear conductance are described. For two asymmetric tunnel barriers in a Luttinger liquid we show that the electron spin can lead to non-linear negative differential conductances if states with higher spins can be dynamically occupied. However, in contrast to the spin blockade phenomenon discussed earlier, this happens only if the system exhibits spin –charge separation characteristic of a Luttinger liquid. This gives rise to peculiar behaviors of the tunneling rates through the barriers which can be calculated microscopically with the bosonization technique. Experimental detection of these negative differential conductances in quasi-one dimensional quantum dots induced by the spin – charge separation provides an additional possibility for studying non-Fermi liquid behavior without relying only on the above mentioned study of the temperature dependence of the conductance.
(s ¼ ^; unit "=2) of the total spin. It can be decomposed into left (l) and right (r) moving parts
cls ðxÞ ¼ eikF x cls;r ðxÞ þ e2ikF x cls;l ðxÞ
cls;r ðxÞ ¼ 2cls;l ð2xÞ; cls;r ðx þ 2LÞ ¼ cls;r ðxÞ
where a21 is a short wavelength cutoff and hs is a Majorana Fermion ensuring correct anticommutation relations for electrons with different spins. The zero mode-q fields are canonically conjugated to the charge and spin numbers
We consider in this section two Luttinger liquids that are coupled by a high tunnel barrier in the presence of a bias voltage (Fig. 1). The Hamiltonian is ð" ¼ 1Þ
The bosonic fields are
Vr ðkÞ ¼
vF k; Vs ðkÞ ¼ vF k g0
with the charge interaction parameter 2U0 21=2 ,1 g0 ¼ 1 þ pvF
Fln ðxÞ ¼
ð2Þ
with vF the Fermi velocity, n†l ðkÞ; nl ðkÞ the boson operators of the charge and spin density waves, and n^ l ; s^l the zero mode operators for the excess of charge and spin with respect to their average values in the leads. We assume open boundary conditions [35,36]. The energies of the charge and spin density waves are (k ¼ pk=L; k integer) ð4Þ
ð5Þ
determined by the repulsive interaction U0 : For simplicity, we have neglected the exchange interaction such that the spin degree of freedom is not influenced by the interaction. The Fermion operator cls ðxÞ depends on the z-component
X rffiffiffiffiffiffi p 2Lk
k.0
e2
ak 2
ð9Þ "
! pffiffiffi i gn sinðkxÞ nl ðkÞ 2 n†l ðkÞ
!# 1 † 2 pffiffiffi cosðkxÞ nl ðkÞ þ nl ðkÞ gn
ð10Þ
with gr ; g0 and gs ; 1: The tunneling Hamiltonian that connects the two Luttinger leads at x ¼ 0 is X ðLÞ† ðRÞ Ht ¼ t0 ½cs;r ðx ¼ 0Þcs;r ðx ¼ 0Þ þ h:c: s¼^1
¼
X
ðHsþ þ Hs2 Þ
ð11Þ
s¼^1
with t0 the transmission amplitude of the barrier. At L ¼ ^1 the leads are assumed to be connected to reservoirs with different electrochemical potentials mR and mL : These define the external bias voltage. The corresponding operator is written in terms of the charge operators Hc ¼
eV ½^n 2 n^ L 2 R
ð12Þ
The tunneling rate for an electron with spin s to hop from the right to the left lead is calculated via Fermi’s golden rule [21]
gþ ðeVÞ ¼
Fig. 1. Two semi-infinite Luttinger liquids, L, R, at bias voltage V; connected by a barrier with tunnel amplitude t0 :
ð7Þ
The Fermion field has a very useful representation in terms of the low energy collective charge and spin bosonic fields Frl ðxÞ; Fsl ðxÞ [35] hs 2iðq^ln þsq^ls Þ i px ð^nl þs^sl Þ Frl ðxÞþsFsl ðxÞ ffie cls;r ðxÞ ¼ pffiffiffiffiffi e 2L e ð8Þ 2pa
½q^ln ; n^ l ¼ i ½q^ls ; s^l ¼ i
Here, H0 ¼ H0ðLÞ þ H0ðRÞ describes the left and right uncoupled Luttinger liquids of length Lð!1Þ [35], " # X X pvF 1 2 2 ^ ^ Vn ðkÞn†l ðkÞnl ðkÞ þ þ s H0ðlÞ ¼ n ð3Þ l 4L g20 l n¼r;s k.0
ð6Þ
Due to the boundary conditions the latter are not independent [35,36]
2. Tunneling between two Luttinger liquids
H ¼ H0 þ Ht þ Hc
649
ð1 21
dteieVt kHsþ ðtÞHs2 ð0Þlleads
ð13Þ
with Hs^ ðtÞ ¼ expð2iH0 tÞHs^ expðiH0 tÞ the Heisenberg operators. The thermal average k· · ·lleads is performed using H0 : The rate eventually turns out to be independent of the spin direction s: This process corresponds to a current
650
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
flowing from the left to the right. The total current is ðkB T ¼ b21 Þ I ¼ 2e½gþ ðeVÞ 2 gþ ð2eVÞ ¼ 2egþ ðeVÞ½1 2 e2beV ð14Þ with the factor of two accounting for the spin degeneracy. The average in Eq. (13) can be explicitly evaluated [37] t0 2 ð1 gþ ðeVÞ ¼ dteieVt e2WðtÞ ð15Þ 2pa 21 Here, the kernel ð1 J ðvÞ dv 0 2 f ðv; b; tÞ WðtÞ ¼ v 0
ð16Þ
represents the influence of the collective charge and spin modes with bv f ðv; b; tÞ ¼ coth ð17Þ ½1 2 cos vt þ isinðvtÞ 2 and J0 ðvÞ ¼
2v 2v=vc e g
ð18Þ
the corresponding spectral density. The effective interaction constant is g ¼ 2=ð1 þ g21 0 Þ: This includes the effect of the spin density excitations. The cutoff vc is the highest energy in the model connected to the shortest wave length in Eq. (8) by vc ¼ vr =a; and is of the order of the Fermi energy. By performing the integrations we get for the temperature dependent rate at the energy E ¼ eV the explicit result ðE p vc Þ [33] t2 ebE=2 bvc 12ð2=gÞ 1 bE 2 gþ ðEÞ ¼ 0 G ð19Þ 2 g vc Gð2=gÞ 2p 2pi Here, we have redefined t0 =ð2paÞ ! t0 and GðxÞ is the Euler – gamma function. In the linear limit, V ! 0; one finds from Eq. (14) the expression for the linear conductance G ¼ ›I=›VlV!0 e2 2pt0 2 G2 ð1=gÞ 2pkB T 2ðð1=gÞ21Þ GðTÞ ¼ ð20Þ p vc Gð2=gÞ vc For repulsive interaction g , 1; the conductance vanishes non-analytically according to a power law when the temperature approaches absolute zero (Eq. (1)). This is the celebrated Luttinger liquid behavior of the conductance.
3. The conductance of a Luttinger liquid quantum dot In this section, we summarize several results for the conductance through two tunnel barriers that connect three Luttinger liquids in the incoherent tunneling region (Fig. 2). The Hamiltonian has the same form as in Eq. (2). However, H0 contains now the contribution of the ‘quantum dot’, the Luttinger system between tunnel barriers at 2a=2
Fig. 2. Three Luttinger liquids connected by tunnel barrier at x ¼ ^a=2: Bias voltages ^V=2 are applied to left (L) and right (R) leads. Gate voltage Vg is applied to the quantum dot (d) between 2a=2 and þa=2:
and a=2; in addition to the parts representing the left and right leads, " # X X pvF 1 2 ðdÞ † 2 H0 ¼ vn ðqÞn ðqÞnðqÞ þ n^ þ s^ ð21Þ 4a g2r n¼r;s q.0 Here, n† ðqÞ; nðqÞ are the boson operators of the collective charge and spin density waves in the quantum dot at wave numbers q ¼ ðpm=aÞ ðm $ 1Þ: The energy spectra are v vn ðqÞ ¼ vn q; vn ¼ F ð22Þ gn with the interaction parameters 2V0 21=2 gs ¼ 1 gr ¼ 1 þ pvF
ð23Þ
where V0 is the forward ðq ! 0Þ scattering contribution of the repulsive interaction. In general, gr – g0 : The zero mode operators n^ and s^ represent the excess of number of charges and spins with respect to their average values on the ground state. The eigenvalues of the zero mode operators are integer n and s with the constraint n þ s ¼ even: The zero mode energy contributions in Eq. (21) is the energy needed for changing the total charge and spin. The tunnel barriers at x ¼ ^a=2 are described by i X X h ðlÞ† Ht ¼ tl cs;r ðxl ÞcðdÞ ð24Þ s;r ðxl Þ þ h:c: s¼^1 l¼L;R
with cðdÞ s;r the right moving fermions operators and tL;R the transmission amplitudes of the barriers [35,36]. As before, we assume high tunnel barriers such that Ht can be treated as a perturbation. The operator of the external bias and gate voltages for electrically controlling current and charge density is obtained by including in Hc the term Cg dC Hgate ¼ 2e Vþ Vg n^ ð25Þ 2CS CS Here, Vg is the gate voltage, CS ¼ CL þ CR þ Cg the total capacitance with CL ; CR and Cg the capacitances of the leads and the gate, and dC ¼ CL 2 CR : For high barriers and not too low temperatures, tunneling through the two barriers can be described as sequential transfers of electrons with spin up or down through the
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
quantum dot region ð2a=2; a=2Þ: In this case, Ht is treated as a perturbation in the lowest order. Higher order coherent processes can be safely neglected as long as kB T q dE; with dE the level broadening due to virtual states, proportional to the tunneling rates [28]. During these incoherent processes the state of the dot is changed between initial lil and final lf l configurations. These are characterized by the excess of number of electrons ðnÞ and spin ðsÞ; and by the occupation numbers of charge ðrÞ and spin ðsÞ density excitations at different wave numbers q: We denote them as ln; s; {lqr }; {lqs }l: The energy of the above configuration will be Er E Uðn; s; lr ; ls Þ ¼ ðn 2 ng Þ2 þ s s2 þ lr 1r þ ls 1s 2 2
i i
The plasmon energy 1r is affects by the Coulomb interaction, with in general 1r . 1s : The energetic difference between the charge and spin density wave is a direct manifestation of the spin– charge separation occurring in a Luttinger liquid. From the above discussion we can identify a ‘hierarchy’ of energy scales 2Es ¼ 10 ¼ 1s , 1r , Er ; so that we choose Es as the natural energy unit. Due to the sequential nature of the tunneling the initial and final configurations have to fulfill the selection rules ð29Þ
We assume that the states containing collective excitations,
f
f
21
Here, the factor expðiDU tÞ ensures energy conservation,
ð26Þ
where 10 ¼ pvF =a plays the role of constant level spacing in the non-interacting case. These terms are different from zero even in the non-interacting limit, a fact which reflects the discrete nature of the energy levels inside the quantum dot and the Pauli exclusion principle. Despite the microscopic model provides an estimate for the charge addition energy, many physical effects have been here oversimplified. For instance, the coupling with gates, the long range interaction effects and the coupling with the nearby 2DEG [23] can strongly affect Er leading to significant deviations from the simple expression Eq. (27) [34]. Therefore, in the following we will treat Er as a free adjustable parameter, and we assume Er q Es : On the contrary, for the spin energy we use the microscopic expression Es ¼ 10 =2: The second two terms in Eq. (26) represent the charge and spin density waves contributions. Since the excitation spectra are linear in the wave number, the energies of the collective modes P depend on the total number of excitation quanta ln ¼ q qlqn only via the discrete excitation energies pvn 1 ¼ 0 ð28Þ 1n ¼ a gn
Dn ¼ nf 2 ni ¼ ^1; Ds ¼ sf 2 si ¼ ^1
although they can be occupied when an electron enters the quantum dot region, decay very rapidly into the corresponding ground state with lr ¼ ls ¼ 0: This implies that when calculating the tunnel rates the thermal average includes averaging over the initial collective excitations of the quantum dot in addition to those of the leads. Then, the tunneling rates for barriers l ¼ L; R is ð1 GðlnlÞ;s l!ln ;s l ¼ tl2 dteiDU t e2Wl ðtÞ e2Wd ðtÞ ð30Þ
DU ¼
Here, the first two terms represent the charge and spin addition energy contributions. In the charge sector we have included the term ðenCg Vg =CS Þ of Eq. (25) with the gate voltage induced number of electrons ng ¼ Cg Vg =e: From the microscopic Hamiltonian one has ðn ¼ r; sÞ pvn 1 ¼ 02 ð27Þ En ¼ 2agn 2gn
651
Er E ½1 þ 2ðni 2 ng ÞDn þ s ½1 þ 2si Ds 2 2 2 dC eV 71 Dn CS 2
ð31Þ
The signs 7 refers to the right (2) and to the left (þ ) barrier. The correlation function ðlÞ† e2Wl ðtÞ ¼ kcs;r ðxl ; tÞcðs;rlÞ ðxl ; 0Þlleads
ð32Þ
results from the trace over the excitations in the leads as in Section 2 [28]. The correlation function ðdÞ e2Wd ðtÞ ¼ kcðdÞ† s;r ðxl ; tÞcs;r ðxl ; 0Þldot
ð33Þ
contains the thermal average over the collective modes in the dot, performed with respect to H0ðdÞ : The dissipative functions Wl;d ðtÞ have the same form as in Section 2. However, for Wl ; in Eq. (16) the spectral density is now J0 ðvÞ=2: For Wd ; J0 ðvÞ must be replaced by the spectral function of the quantum dot, Jd ðvÞ ¼ v
1 X en X dðv 2 me n Þe2v=vc 2g n n¼r;s m¼1
ð34Þ
The structure of this spectral density of the dot indicates that, although they are infinitely quickly relaxing, charge and spin density waves still contribute to the tunneling dynamics. Even if the relaxation prevents the collective excitations to be initial states for the tunneling, it is still possible to reach an excited state as a ‘final’ state for a given energy when an electron enters the dot. Using the bosonization method one can calculate exactly all of the above averages as in Section 2 [35]. For simplicity, we do not specify in the following the initial and final states. Exploiting the discrete nature of the dot spectral density one rewrite Eq. (30) as a function of the energy difference E ¼ DU X GðlÞ ðEÞ ¼ Gð0lÞ alr als gðE 2 lr 1r 2 ls 1s Þ ð35Þ lr ;ls
where
Gð0lÞ ¼
1r vc
1=2gr
1s vc
1=2gs
2vc Gl e2 Gð1=gÞ
ð36Þ
2 2 2 with Gl ¼ R21 l ¼ pe tl =vc the intrinsic conductances of the barriers. As before, the function gðxÞ is determined by
652
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
peaks is affected by the spin that leads to a parity effect in the distances between peaks
the leads [38] 1 bx=2 1 bx 2 2p ð1=gÞ21 e G 2 gðxÞ ¼ 2p 2g 2pi bvc At T ¼ 0 we have ð1=gÞ21 x g0 ðxÞ ¼ uðxÞ vc
ð37Þ
ð38Þ
The weights aln are due to the quantum dot. At T ¼ 0; they can be exactly evaluated a0ln ¼
Gð1=2gn þ ln Þ uðln Þ Gð1=2gn Þln !
ð39Þ
For finite temperature, they have to be numerically determined. In the presence of spin charge separation the above weights modify crucially the rate Eq. (35) which shows non-integer quantized steps at different energies 1r – 1s : For the calculation of the current voltage characteristic one has to determine the occupation probabilities Pn;s for the charge and spin states ln; sl of the quantum dot from the master equation X X ›t Pn;s ðtÞ ¼ n0 ¼n^1 s0 ¼s^1
h i Pn0 ;s0 ðtÞGln0 ;s0 l!ln;sl 2 Pn;s ðtÞGln;sl!ln0 ;s0 l ð40Þ P ðlÞ where G ; l G : In the stationary limit, ›t Pn;s ¼ 0; one has to solve a homogeneous system P of linear equations. The solution must be normalized, n;s Pn;s ¼ 1: The stationary current is h i X X ðRÞ I¼e Pn;s GðRÞ 2 G ð41Þ ln;sl!lnþ1;sþDsl ln;sl!ln21;sþDsl n;s Ds¼^1
If both eV; kB T , Er ; at most two charge states can enter the dynamics, and the current is given by tunneling events corresponding to transitions of the type ln; sl ! ln þ 1; s0 l ! ln; s00 l: As an example, we consider the linear conductance G at low temperatures, kB T p Es for symmetric barriers ðtR ¼ tL Þ: One can solve the master equation in the subspace of the ground states with n and n þ 1 electrons. The position of the conductance peak at T ¼ 0 is determined by the electrochemical potential of the quantum dot, 1 E ð42Þ md ¼ Er n 2 ng þ þ ð21Þn s ¼ 0 2 2
dn;nþ1 ¼ 1 þ ð21Þnþ1
Es Er
ð45Þ
For T – 0 the conductance peak positions shift linearly with temperature res nþ1 nres fðgÞ g ðTÞ ¼ ng þ ð21Þ
kB T Er
ð46Þ
Without interaction one recover the well-known result fð1Þ ¼ log 2=2 [38]. With interaction, fðgÞ increases monotonically increasing interactions, reaching the maximum value fð0Þ ¼ log 2: The temperature dependence of the conductance maximum, which is around j ¼ 0; is determined only by the excitations in the leads present in GðRÞ ð0Þ since one cannot excite charge and spin modes in the dot ðlr ¼ ls ¼ 0Þ Gmax ðTÞ / T ð1=gÞ22
ð47Þ
A sequence of conductance peaks obtained for two temperatures which show the characteristic low temperature features is shown in Fig. 3.
4. Non-linear transport In this section, we discuss some of the non-linear transport features. The master equation for obtaining the stationary probability distribution Pn;s is solved numerically. From this, the current – voltage characteristic and the differential conductance are calculated. We consider asymmetric tunnel barriers. The asymmetry is described by the ratio of the conductances of the left and right tunnel barrier, A ¼ GL =GR : Note that for A . 1; V . 0 and kB T , Es ; the electrons traverse the higher right barrier into the dot, and the lower left barrier out of the dot. Thus, states ðn; sÞ with n ¼ even will have a higher occupation probability as compared to the states ðn þ 1; s þ 1Þ: This will eventually create a ‘trapping’ phenomenon at sufficiently large
The conductance for kB T , Es is
be2 GðRÞ ðjÞe2bj=2 G ¼ pffiffi 8 cosh{½bj þ ð21Þn log 2=2}
ð43Þ
where j ¼ Er ðng 2 nres g Þ corresponds to the deviation from the resonance peak position at T ¼ 0; nres g ¼nþ
1 E þ ð21Þn s 2 2Er
ð44Þ
The rate GðRÞ ðjÞ is given in Eq. (35). The position of the
Fig. 3. Linear conductance (unit e2 G0ðRÞ =Es ) of a quantum dot as a function of the number gate voltage ng for symmetric tunnel barriers at temperatures kB T equal 0:02Es (full curve) and 0:5Es (dotted), with Er ¼ 5Es ; g ¼ 0:9; gr ¼ 0:6:
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
asymmetries, inducing the rich features that will be described below. Fig. 4 shows the conductance G in the ðV; ng Þ-plane including several regions of Coulomb blockade for A ¼ 100: The parity effect mentioned above is clearly observed. The rich structure for V – 0 provides in principle detailed information about the spin addition features and the collective spin and charge excitations of the quantum dot. It is obvious that in order to extract this information a detailed analysis of the possible transitions including the behavior of the corresponding tunnel rates is necessary. Of particular importance are the conductance features indicated by the white lines. These correspond to negative differential conductance. The non-linear transport features are rather stable against changes in the temperature as can be seen by comparing the left and right panels. As an example we discuss in some detail the behavior of the negative differential conductance indicated by the white transition lines in Fig. 4. They are more clearly shown in the magnification of the region near the Coulomb peak between charge numbers n, n þ 1 (n even) shown in Fig. 5. The physical origin of the negative differential conductance is here related to the dynamical occupation of excited states with higher spin (states with spin s and 2s are degenerate, Eq. (26)). As a consequence of the spin selection rule they can act as traps in certain regions of parameters. To understand the phenomenon let us identify the different lines in Fig. 5. The two fundamental black transition lines starting at V ¼ 0 with slope Er ðng 2 nres g Þ ¼ ^eV=2 are the boundaries of the stable Coulomb blockade regions with n and n þ 1 charges and correspond to transitions from the lowest spins ðn þ 1; 1Þ ! ðn; 0Þ; and ðn; 0Þ ! ðn þ 1; 1Þ; respectively. Here, the resonant value of the gate voltage corresponds to nres g ¼ 0:52: Once the state ðn þ 1; 1Þ is occupied, transition to higher spin as ðn þ 1; 1Þ ! ðn; 2Þ becomes
Fig. 4. The differential conductance G (arbitrary unit), as a function of bias voltage V (unit Es =e) and number of charges ng : System parameters: n even, Er ¼ 5Es ; g0 ¼ 1; gr ¼ 0:63; asymmetry A ¼ kB T ¼ 7:5 £ 1023 Es ; right kB T ¼ 100; CL ¼ CR ; left: 22 7:5 £ 10 Es : Right: gray scale.
653
Fig. 5. The differential conductance G (arbitrary unit) as a function of bias voltage V (unit Es =e) and ng ; for kB T ¼ 7:5 £ 1023 Es : Other parameters as in Fig. 4, apart Er ¼ 25Es : Right: gray scale.
available by increasing the voltage. The activation threshold in this case is given by Er ðng 2 nres g Þ ¼ eV=2 2 2Es starting, in the above figure, at ðV; ng Þ ¼ ð2Es ; 0:48Þ: Along this line negative differential conductance regions occur. By iterating the procedure with increasing voltage even higher spin states become occupied. Because of the asymmetry A . 1 the lines with negative conductances always correspond to transitions from odd s to s þ 1 even. So far, we only have considered the excited states of the dot with higher spins. In order to complete the picture, we must include all of the transitions involving collective charge and spin excitations. This enhances considerably the complexity of the spectrum since at sufficiently high bias voltage each transition of the type ðn; sÞ ! ðn0 ; s0 Þ can also occur via the channels ðn; sÞ ! ðn0 ; s0 ; lr ; ls Þ ! ðn0 ; s0 Þ: For example, consider charge and spin density waves for the transition ðn; 0Þ ! ðn þ 1; 1Þ: These correspond to the lines Er ðng 2 nres g Þ ¼ ð2eV=2Þ þ lr 1r þ ls 1s : They are parallel to the fundamental line ðn; 0Þ ! ðn þ 1; 1Þ: Analogously, including charge and spin collective modes in the transition ðn þ 1; 1Þ ! ðn; 0Þ give rise to Er ðng 2 nres g Þ ¼ ðeV=2Þ þ lr 1r þ ls 1s with transition parallel to the fundamental line ðn þ 1; 1Þ ! ðn; 0Þ: Now, in order to better understand the first transitions ðs ¼ 0; ^1; ^2Þ and the corresponding occupation probabilities, we consider Fig. 5 at fixed gate voltage ng ¼ 0:49 as a function of V: Starting in the Coulomb blockade region with charge n and V ¼ 0; by increasing the bias one eventually populates the ðn þ 1; 1Þ-state via a tunneling process through the right barrier near V ¼ 1:5Es : At this voltage a current starts to flow since this state can easily be emptied again via tunneling through the left barrier. When the bias is further increased, the transition ðn þ 1; 1Þ ! ðn; 2Þ becomes also possible at V ¼ 2:5Es : The state ðn; 2Þ can decay into the ground state ðn; 0Þ only via the ðn þ 1; 1Þ-state, i.e. if first another electron enters the quantum dot. This could increase the current further. However, if the occupation of the
654
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
where
Fig. 6. Occupation probabilities Pn;0 (dashed), Pn;2 (dashed – dotted), Pnþ1;1 (dotted), and current I (unit eG0ðRÞ ; full line) as a function of V (unit Es =e); system parameters as in Fig. 5.
higher-spin state ðn; 2Þ is favored at the expense of the ground state ðn; 0Þ; it will accumulate occupation probability (Fig. 6). This will eventually lead to a reduction of the current, and to a negative differential conductance. Asymmetric tunnel barriers provide only one possible mechanism for preferring the higher-spin state. In the parameter region considered here, the right tunnel barrier is higher that the left which implies that the n-electron states will be more frequently occupied on the average. There are also other mechanisms that could lead to such asymmetry in the occupation, for instance due to spin dependent tunneling [39]. However, it is important to realize that in the present case trapping alone is not sufficient to induce negative differential conductance. The detailed analysis of the master equation in the limit of large asymmetry, which is the most favorable case to induce negative conductance, shows that the crucial ingredient for the phenomenon discussed here is the spin charge separation in the quantum dot. Without spin– charge separation ðgr ¼ gs ¼ 1Þ; no negative differential conductance can occur. This is intimately related to the behavior of the rate Eq. (35) which, in the absence of spin – charge separation, has equal steps centered at equidistant energy. This behavior will never strongly promote higher spin at the expense of the lower ones, giving always an increasing current and a positive conductance. Only non-equal steps, induced by gr – 1; provide this possibility. In order to make this crucial ingredient more transparent, we have reconsidered the parameter region taking into account only the five dominant spin states. In this case the master equation is solved analytically. We quote here the result for the transition line ðn þ 1; 1Þ ! ðn; 2Þ for voltages low enough not too excite charge modes (first white line in Fig. 5). We assume non-interacting leads, kB T p Es and spin charge separation ðgr , 1Þ: One gets
G ¼ eL
ðLÞ 2GR0!1 GR2!1 ›G1!2 ðVÞ 2 ›V D
ð48Þ
ðLÞ ðRÞ L ¼ 2GR0!1 GR2!1 þ G1!0 ðG2!1 2 2GðRÞ 0!1 Þ
ð49Þ
ðLÞ ðRÞ ðRÞ ðLÞ D ¼ G1!0 G2!1 þ 2GðRÞ 0!1 ðG2!1 þ G1!2 Þ
ð50Þ
with the rates evaluated at the transition energies U0!1 ¼ eV 2 2Es ; U2!1 ¼ eV; U1!0 ¼ 2Es ; U1!2 ¼ 0: The condition for negative differential conductance is determined solely by L , 0: The latter, when evaluated along the first part of the line ðn þ 1; 1Þ ! ðn; 2Þ has rates values GðLÞ 1!0 ¼ ðRÞ ðRÞ ð3=2ÞG0ðLÞ ; G0!1 ¼ G0ðRÞ ; while GðRÞ 2!1 ¼ ð3=2ÞG0 : We then have negative differential conductance if A $ 8: If gr ¼ 1 ðLÞ ðRÞ the values of the rates change as G1!0 ¼ 2GðLÞ 0 and G2!1 ¼ ðRÞ 2G0 yielding L . 0 for any asymmetry. Similarly, one can understand other regions of negative differential conductances in Fig. 5. To conclude we show in Fig. 7 the behavior of the differential conductance with temperature. The negative peak corresponds to the above discussed transition line ðn þ 1; 1Þ ! ðn; 2Þ: The positive peak is contributed by the first collective charge excitation above the ðn þ 1; 1Þ-ground state. With increasing temperature, the conductance peaks are broadened due to the Fermi distributions in the left and right leads. The negative conductance features look quite robust against increasing temperature, despite the position of the peak is shifted linearly towards the positive one.
5. Conclusion We have reviewed results of extensive studies of the electron transport through tunnel barriers connecting
Fig. 7. Middle: two differential conductance peaks, one negative (white) and one positive (black), as a function of voltage (unit Es =e) and temperature (unit 1023 Es ), for ng ¼ 0:5: Right: gray scale. Top: the conductance as a function of V (unit Es =e) for kB T ¼ 2 £ 1022 Es : Bottom: as in the top but with kB T ¼ 2 £ 1023 Es : System parameters: Er ¼ 25Es ; g0 ¼ 1; gr ¼ 0:63; A ¼ 100; CL ¼ CR ; conductance unit: e2 GðRÞ 0 =Es :
M. Sassetti et al. / Solid State Communications 131 (2004) 647–655
Luttinger liquids with spin. We have considered the temperature region of sequential tunneling. Special emphasis has been devoted to the signatures of non-Fermi liquid correlations. For a single barrier connecting two semiinfinite leads the well-known non-analytic temperature behavior has been recovered. For two barriers connecting a Luttinger liquid of finite length with leads the temperature behavior of the linear conductance peaks has been briefly mentioned. In the non-linear transport region, the energy scales of the excited states have been detected in the differential conductance. Negative differential conductance features have been found that are related to the occupation of states with higher spins in the quantum dot. The physical origin of these features is the presence of the non-Fermi liquid correlations which provide not only different energy scales for charge and spin excitations but also influence the magnitude and temperature behavior of the tunnel rates. A strong asymmetry enhances the chance to create a negative differential conductance but it would be not sufficient alone. In the figures described above we have used high asymmetries in order to display the effect more clearly. However, as shown by analytic analyses, negative differential conductances can be found also at smaller asymmetries. Most importantly, the phenomenon is rather stable against increasing the temperature (Figs. 4 and 7). In order to see whether or not such phenomenon could be observed experimentally we consider a one dimensional quantum dot of the length 0.2 mm. The corresponding energy scale is Es ¼ ðpvF =aÞ < 1 meV if we assume a Fermi velocity of order 105 m/s. This corresponds to a temperature of about 10 K and the temperatures used in obtaining the above results kB T < 1022 Es is roughly 0.1 K. It might be more difficult to obtain sufficient asymmetry in the tunnel resistances still fulfilling the conditions for sequential tunneling. However, it seems to us that asymmetric barriers are the genuine situation in experiment, rather than the exception. So one can hope that in experiment on semiconductor quantum dots as well as in Carbon nanotubes the effect predicted above might be observable.
References [1] [2] [3] [4] [5] [6]
S. Tomonaga, Prog. Theor. Phys. 5 (1950) 544. J.M. Luttinger, J. Math. Phys. 4 (1963) 1154. F.D.M. Haldane, J. Phys. C 14 (1981) 2585. H.J. Schulz, Int. J. Mod. Phys. B 5 (1991) 57. J. Voit, Rep. Prog. Phys. 58 (1995) 977. S. Tarucha, T. Honda, T. Saku, Solid State Commun. 94 (1995) 413. [7] A. Yacoby, H.L. Stormer, N.S. Wingreen, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 77 (1996) 4612. [8] A. Yacoby, H.L. Stormer, K.W. Baldwin, L.N. Pfeiffer, K.W. West, Solid State Commun. 101 (1997) 77. [9] D.H. Cobden, M. Bockrath, P.L. McEuen, A.G. Rinzler, R.E. Smalley, Phys. Rev. Lett. 81 (1998) 681.
655
[10] D.H. Cobden, J. Nyga˚rd, Phys. Rev. Lett. 89 (2002) 046803. [11] M.R. Buitelaar, A. Bachtold, T. Nussbaumer, M. Iqbal, C. Scho¨nenberger, Phys. Rev. Lett. 88 (2002) 156801. [12] W. Liang, M. Bockrath, H. Park, Phys. Rev. Lett. 88 (2002) 126801. [13] H.W.Ch. Postma, M. de Jonge, Z. Yao, C. Dekker, Phys. Rev. B 62 (2000) R10653. [14] T.A. Fulton, P.L. Gammel, D.J. Bishop, L.N. Dunkleberger, Phys. Rev. Lett. 63 (1989) 1307. [15] L.J. Geerligs, V.F. Anderegg, P.A.M. Holweg, J.E. Mooij, Phys. Rev. Lett. 64 (1989) 2691. [16] U. Meirav, M.A. Kastner, S. Wind, Phys. Rev. Lett. 65 (1990) 771. [17] For timely overviews of the field, see, e.g. (a), in: B. Kramer (Ed.), Quantum Coherence in Mesoscopic Systems, NATO ASI Series B 254, Plenum, New York, 1992. (b) in: M. Devoret, H. Grabert (Eds.), Single Charge Tunneling, NATO ASI Series B 294, 1992, p. 21. (c) in: B. Kramer (Ed.), Quantum Transport in Semiconductor Submicron Structures, NATO ASI Series E 326, Kluwer, Dordrecht, 1996. (d) in: L.L. Sohn, L.P. Kouwenhoven, G. Scho¨n (Eds.), Mesoscopic Electron Transport, NATO ASI Series E 345, Kluwer, Dordrecht, 1997. [18] G.-L. Ingold, Y. Nazarov, Single Charge Tunneling, NATO ASI Series B 294, 1992, p. 21. [19] D. Weinmann, W. Ha¨usler, B. Kramer, Phys. Rev. Lett. 74 (1995) 984. [20] K. Jauregui, W. Ha¨usler, B. Kramer, Phys. Rev. B 53 (1997) R1713. [21] (a) C.L. Kane, M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 1220. (b) C.L. Kane, M.P.A. Fisher, Phys. Rev. B 46 (1992) 15233. [22] A. Furusaki, N. Nagaosa, Phys. Rev. B 47 (1993) 3827. [23] O.M. Auslaender, A. Yacoby, R. de Picciotto, K.W. Baldwin, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 84 (2000) 1764. [24] M. Sassetti, F. Napoli, U. Weiss, Phys. Rev B 52 (1995) 11213. [25] H. Maurey, T. Giamarchi, Europhys. Lett. 38 (1997) 681. [26] R. Egger, H. Grabert, Phys. Rev. B 55 (1997) 9929. [27] M.P.A. Fisher, L.I. Glazman, Mesoscopic Electron Transport, NATO ASI Series B 345, Kluwer, Dordrecht, 1997. [28] A. Furusaki, Phys. Rev. B 57 (1998) 7141. [29] A. Braggio, M. Grifoni, M. Sassetti, F. Napoli, Europhys. Lett. 50 (2000) 236. [30] Yu.V. Nazarov, L.I. Glazman, Phys. Rev. Lett. 91 (2003) 126804. [31] D.G. Polyakov, I.V. Gornyi, Phys. Rev. B 68 (2003) 035421. [32] T. Kleimann, M. Sassetti, B. Kramer, A. Yacoby, Phys. Rev. B 62 (2000) 8144. [33] A. Braggio, M. Sassetti, B. Kramer, Phys. Rev. Lett. 87 (2001) 146802. [34] T. Kleimann, F. Cavaliere, M. Sassetti, B. Kramer, Phys. Rev. B 66 (2002) 165311. [35] M. Fabrizio, A.O. Gogolin, Phys. Rev. B 51 (1995) 17827. [36] (a) S. Eggert, H. Johannesson, A. Mattsson, Phys. Rev. Lett. 76 (1996) 1505. (b) A.E. Mattsson, S. Eggert, H. Johannesson, Phys. Rev. B 56 (1997) 15615. [37] U. Weiss, Quantum Dissipative Systems, Second ed., World Scientific, Singapore, 1999. [38] G. Usaj, H.U. Baranger, Phys. Rev. B 64 (2001) 201319. [39] M. Ciorga, A. Wensauer, M. Pioro-Ladriere, M. Korkusinski, J. Kyriakidis, A.S. Sachrajda, P. Hawrylak, Phys. Rev. Lett. 88 (2002) 256804.