VOLUME 87, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 OCTOBER 2001

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Control of Spin in Quantum Dots with Non-Fermi-Liquid Correlations

Alessandro Braggio,1 Maura Sassetti,1 and Bernhard Kramer2

1Dipartimento di Fisica, INFM, Università di Genova, Via Dodecaneso 33, 16146 Genova, Italy2I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany

(Received 5 June 2001; published 13 September 2001)

Spin effects in the transport properties of a quantum dot with spin-charge separation are investigated.It is found that the nonlinear transport spectra are dominated by spin dynamics. Strong spin polarizationeffects are observed in a magnetic field. They can be controlled by varying gate and bias voltages.Complete polarization is stable against interactions. When polarization is not complete it is power lawenhanced by non-Fermi-liquid effects.

DOI: 10.1103/PhysRevLett.87.146802 PACS numbers: 73.63.Kv, 71.10.Pm, 72.25.–b

Spin phenomena in transport properties of low dimen-sional quantum systems have become a subject of increas-ing interest [1,2]. Several fundamental effects have beenpredicted when controlling transport of electrons one byone in quantum dots as, for instance, spin blockade due toselection rules [3] and parity effects in the Coulomb block-ade [4,5]. There are also perspectives of applications inspin electronics, quantum computing, and communication[6]. Previous works have been focusing on spin transportin two dimensional (2D) quantum dots connected to non-interacting leads and in the presence of a magnetic field[7], including also an oscillating magnetic electron spinresonance component [8]. Spin transport in circuits withferromagnetic elements and in the presence of a Luttingerliquid interaction [9–11] have been considered. Of funda-mental interest is the spin control of electron transport inthe presence of correlations since in nanoscale devices thelatter are very important.

In the present paper, we derive a general theory for spinand charge transport through a quantum dot formed in aLuttinger liquid. We consider spin effects in the presenceof a magnetic field. Specifically, we investigate to whatextend non-Fermi-liquid behavior influences spin polariza-tion. We find that spin-charge separation strongly affectsthe current-voltage characteristics. The spin leads to richstructure in the nonlinear differential conductance that re-flects both the collective spin density excitations and theorientations of the total spin in the quantum dot. A mag-netic field in the quantum dot can spin polarize the cur-rent strongly. This can be controlled by varying gateor bias voltages. Full spin polarization can be achieved.Noncomplete polarization is power law enhanced by thenon-Fermi-liquid correlations.

We start from a clean Luttinger liquid with spin.The charge interaction parameter is gr � �1 12V �0��pyF�21�2 with yF the Fermi velocity andV �q� the Fourier transform of the electron interaction�h̄ � 1�. For the exchange interaction, we assumegs � 1. The interacting 1D electrons are mapped, viabosonization, to a harmonic Hamiltonian [12]. Thelow-energy excitations are charge and spin densitywaves, with dispersions vn�q� � yFjqj�gn � yn jqj.

1 0031-9007�01�87(14)�146802(4)$15.00

Here, n � r,s label charge �r� and spin �s�. Spin-charge separation implies yr fi ys. The slowly vary-ing (on the scale of 2pk21

F ) parts of the densitiesare given in terms of field operators Qn�x�, n�x� �r"�x� 1 pnr#�x� � n0 1

p2�p ≠xQn�x�, with pr � 1,

ps � 2, and the mean charge and spin densitiesr0 � 2kF�p and s0 � 0, respectively.

The quantum dot is formed by barriers �Ut�r0�d�x 2

xi� at positions x1 , x2 given by the tunneling Hamilto-nian

Ut

Xa�0,1

Yn

cosp

2�N2

n 1 n0d 2 a� cosp

2�N1

n 2 a�

(1)

with N6n � �Qn�x2� 6 Qn�x1��

p2�p. Physically, N2

r isthe deviation of the number of electrons from the meanvalue, n0 � dr0, in the interval d � x2 2 x1. The ex-cess charge then is Q � 2eN2

r , and the component ofthe total spin of the electrons parallel to the quantizationaxis (assumed parallel to the 1D system) S � N2

s �2. Thenumbers of imbalanced electrons and spins between theleads are N1

r and N1s , respectively. The coupling to the

source-drain bias V and the gate voltage Vg is describedby HV � 2e�VN1

r �2 1 VgN2r d�, with d the ratio be-

tween gate and total capacitances. The effect of an exter-nal magnetic field is described by a local Zeeman term inthe region between the barriers, HB � 2gBmBBN2

s �2 �2EBN2

s �2, with the Landé factor gB [7,8].The currents are calculated as the stationary limits

of transferred charges and spins In � I" 1 pnI# ��e�2� limt!`� �N1

n �t�. The brackets include both thermaland statistical averages over the collective modes atx fi x1, x2 with the density matrix reduced to N6

n [5].For obtaining the nonlinear current-voltage characteris-

tics we consider the dynamics of the system described bythe variables N6

n under the influence of the external fieldsin the 4D periodic potential Eq. (1). For high barriers, tun-neling between nearest-neighbored minima dominate, withamplitude D that is related to Ut via the WKB approxima-tion [13]. These correspond to processes N2

r ! N2r 6 1

and N2s ! N2

s 6 1 associated with changes of charge andspin numbers in the dot, respectively, and N1

r ! N1r 6 1

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VOLUME 87, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 OCTOBER 2001

that transfer current. We consider sequential tunnelingwith the temperature much smaller than the dot level spac-ing. This can be described by a master equation for chargesand spins with rates,

J�E� �1X̀

n,m�2`

Wrn �er�Ws

m �es�g�g��E 2 ner 2 mes� ,

(2)

where the tunneling rate of a single barrier

g�g��E� �D2

4vc

µbvc

2p

∂121�g ÇG

µ1

2g1

ibE2p

∂ Ç2

3e2jEj�vc ebE�2

G�1�g�(3)

depends on the effective interaction parameterg�2 � grgs��gr 1 gs� and the frequency cutoffvc. The weights, Wn

n �en�, at the discrete energies en are�b21 � kBT ø en�

Wnn �en��

µen

vc

∂1�2gn

3G�1�2gn 1 n�n! G�1�2gn �

e2enn�vcu�n� . (4)

In order to understand the rather complex behavior ofthe transport spectra it is useful to recall the characteristicenergy scales in (2). These are the discretization energiescorresponding to charge and spin modes in the quantumdot relative to the energy of the ground state,

en � vn�q � p�d� � 2gnEn , (5)

with the addition energies for charge and spin

En � pyF�2g2nd . (6)

Without interaction, the addition energies are Er � Es �pyF�2d � EP fi 0, due to the Pauli principle, and thediscreteness of the dot levels. On the other hand, for strongCoulomb interaction, Er ~ V �q ! 0� ¿ Es [5].

The addition energies determine the ground state energyof n charges with the half-integer total spin S � sn�2,E0�n� � �Er�n 2 ng�2 1 Ess2

n 2 EBsn��2. The refer-ence particle number ng � eVgd�Er 1 n0 is defined bythe gate voltage. The energy differences of the many-bodystates of n 1 1 and n electrons are

m�n, s, l, m� �Er

2�2n 1 1 2 2ng� 1

Es

2�s2

n11 2 s2n�

2EB

2�sn11 2 sn� 1 ler 1 mes . (7)

Positive or negative integers l and m denote the differ-ences of the numbers of charge and spin excitation quantawith the energies (5), respectively. These do not changeeither the number of particles or the total spin in the quan-tum dot. The energy differences m�n, s, l, m� play the roleof chemical potentials of the dot and define the transportregions. For symmetric bias, for instance, the condition

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V�2 . m�n, s, l, m� . 2V�2 defines the allowed trans-port channels. For V ! 0 and EB ø Es one finds theCoulomb blockade peaks at gate voltages Vg that corre-spond to m�n, s, l, m�. However, due to the spin, the sepa-ration of the peaks depends on the parity of n, DVn11,n

g ��Er 1 �21�n11�Es 2 EB���ed. Without interaction, onehas to replace here Er � Es � EP, in order to get theseparation of the linear conductance peaks.

In the following, we consider the limit T � 0. Resultsfor the differential conductances dI�dV for EB � 0and EB � 0.4Es as functions of V are shown in Fig. 1.Zero bias voltage has been assumed at the position of aconductance peak corresponding to an n(even)-to-�n 1 1�ground-state-to-ground-state transition. The differentialconductance shows sharp peaks at bias voltages Vnlsm

at which a new transport channel enters the above biasvoltage window. Above Vnlsm, the conductance as afunction of V drops according to the interaction-inducednon-Fermi-liquid power law �V 2 Vnlsm�1�g22.

From (7) one can easily identify the spectral origins ofthe peaks in the conductance spectra. At low voltages,the spectra are completely dominated by spin excitationsdue to spin-charge separation. The discretization and addi-tion energies corresponding to the spin are factors gr andg2

r, respectively, smaller than those corresponding to thecharge. For B � 0 peak �e� corresponds to a charge den-sity excitation at er, while � f � is due to the ground-state toground-state transition at Er 2 Es. All of the other peaksin Fig. 1 are spin related. Because gs � 1 the transitionat 2Es is degenerate with the spin density excitation at es

[peak �a�, with multiples �d� and �g�]. A finite exchangewould remove this degeneracy and discriminate betweenspin addition energies and spin density waves. The twosmall features �b� and �c� are combinations of the excita-tions �e� and � f � with Es; they correspond to er 2 2Es

0.0

2.5

5.0

0 0.5 1 1.5 2

0.0

2.0

4.0

0 0.5 1 1.5 2

5.0

2.5

0.0

2.0

4.00.0

0 0.5 1.51 2

FIG. 1. Differential conductance dI�dV as a function ofthe source-drain bias eV�Er for gr � 0.4 [vc � 105Es ,units 1023 �vc�Es��g21��2ge2D2�4v2

c ]. Top panel: EB � 0,ng � 0.58; bottom panel: EB � 0.4Es , ng � 0.548.

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VOLUME 87, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 OCTOBER 2001

and Er 2 3Es, respectively. In the noninteracting limit,the peaks in the nonlinear differential conductance appearat bias voltages that are multiples of 4EP due to the absenceof spin-charge separation. Apart from the peak at V � 0,all of the peaks are absent without electron interaction, forthe bias voltages in Fig. 1 (top panel).

The spin-related features are even more strikingly dis-played in the spectra for EB � 0.4Es (Fig. 1, bottompanel). All of the peaks in Fig. 1 (top panel) acquireZeeman sidebands corresponding to energies Epeak 6 EB.Exceptions are �b� and �c�. They have only sidebandsEpeak 1 EB since the initial states corresponding to thelower sidebands cannot be occupied by electrons enteringthe dot at T � 0.

As an example of the rich spin-related fine structure inthe transport spectrum a density plot of the differentialconductance in the �eV�Er , ng� plane for EB � 0.4Es isshown in Fig. 2. The black regions of the Coulomb block-ade are well developed. With increasing bias voltage thecomplexity of the dot excitation spectrum rapidly increasesand displays a considerable number of spin-related tran-sitions. With nonvanishing exchange interaction, gs fi 1,accidental degeneracies due to spin addition and excitationare lifted and the transport spectrum becomes even morecomplex.

Figure 3 shows the behavior of a current peak for fixedbias, I�ng�, when changing the magnetic field. As Bis changed, peak height and position vary with periods2Es and 4Es, respectively (Fig. 3, bottom panel). Thiscan be understood by considering the processes that con-tribute towards the current. We start by discussing thepeak position for V ! 0. For small B and keeping ngas to match the maximum of the peak (Fig. 3, top panel)one finds from (7) that with increasing B one has to ad-just ng to lower values ~ 2EB�2 since sn11 2 sn � 11which corresponds to the sn � 0 ! sn11 � 1 transition.When EB $ 2Es, the energy of the state sn � 2 becomeslower than that of the sn � 0 state such that transportgets now support from transitions sn � 2 ! sn11 � 1

0

2

4

0

1

2

-0.8 -0.4 0.0 0.4 0.8

FIG. 2 (color). Density plot of the differential conductance inthe (eV�Er , ng) plane for interaction gr � 0.4 and magneticfield EB � 0.4Es [vc � 105Es , color code (left) with units1023 �vc�Es��g21��2ge2D2�4v2

c ].

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with sn11 2 sn � 21, while sn11 1 sn � 3 instead of1. When increasing B further, ng has now to be adjustedto higher values ~ 1EB�2 in order to compensate for theZeeman shift. The original position is reached after a totalchange of B corresponding to EB � 4Es. For V fi 0, thepeak position remains unchanged as long as energies of thetransitions are inside the interval �2V�2, V�2�.

For understanding the peak height we consider V fi 0.The bias V gives the width of the current peak. First, weremember that for EB ø eV , the current is due to twotransitions, namely, sn � 0 ! sn11 � 61. This leads tothe asymmetry of the (nonspin polarized) peak for EB � 0that can be observed in Fig. 3 (top panel). When EB � eVthe contribution of the ground-to-excited-state transitionsn � 0 ! sn11 � 21 is suppressed. Further increasingB, the current peak becomes symmetric, completely polar-ized (cf. Fig. 4), its height is reduced and remains constant.For Es ø EB , 2Es transport gets support also fromtransition sn � 2 ! sn11 � 1 such that the peak heightstarts to become again asymmetric and to increase until,exactly at EB � 2Es, both contributions are equally im-portant. Then, the current peak acquires the same shapeas for EB � 0, but reflected at ng � 1�2. Increasing Bproduces oscillatory behavior due to further changes ofthe values of sn and sn11. The current has its maximumvalue at ng � 1�2 whenever EB is an odd multiple of Es .These fully polarized current states are also reflected in thedigital behavior [14] displayed in Fig. 3 (bottom panel).The latter becomes less stable when increasing the biasvoltage V .

Figure 4 shows the behavior of the current polarization,P � Is�Ir . The top left panel shows P as a function ofthe source-drain voltage for a given magnetic field anddifferent interaction strengths, with ng at the maximumof the linear conductance. The polarization is completefor eV # 2EB, independently of the interaction. When

0

0.5

1

0.4 0.5 0.6 0.7

0.00.20.40.61.01.41.61.82.0

FIG. 3 (color). Top panel: Current I , normalized to the valueat B � 0, as a function of the gate voltage ng for differentmagnetic fields EB�Es (inset) for gr � 0.4 and at eV � 0.1Er

(vc � 105Es). Bottom panel: position of maximum of peak,n0 (dashed line, right scale) and current at peak maximum, I0(full line, left scale) as a function of EB�Es .

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VOLUME 87, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 OCTOBER 2001

0.40.50.60.71.0

FIG. 4 (color). Current polarization P � Is�Ir for differentinteractions gr (inset) as a function of eV�EB for EB � 0.4Es

and ng at the maximum of the conductance peak (top left panel);P in the plane �eV�Er , ng� for gr � 0.4 and magnetic fieldEB � 0.5Es (top right panel), EB � Es (bottom left panel) andEB � 1.5Es (bottom right panel, includes color code).

eV . 2EB, P decreases as a function of V according toa non-Fermi-liquid power law and is higher for strongerinteraction. Thus, correlations enhance the polarizationwhen the latter is not complete.

The other three panels of Fig. 4 show density plots ofthe polarization for fixed interaction. Details of the behav-ior can be understood from the discussion related to Fig. 3.Varying the magnetic field, complete polarization, P � 1(yellow), is transferred between different regions of the�eV , ng� plane. The top right panel corresponds to EB �0.5Es where the dominating sn � 0 ! sn11 � 1 chan-nel leads to complete spin polarization near the left-handedge of the region of nonzero current. Near EB � Es

(bottom left panel) the current peak is symmetrically spinpolarized (for small V , yellow diamond). Increasing themagnetic field fully polarizes the right-hand edge of thecurrent region (EB � 1.5Es, bottom right panel). Exactlyat EB � 2Es the two involved transport channels haveequal weight and spin polarization is exactly zero, as forEB � 0. By further increasing B, the behavior of the po-larization is reversed, P � 21. It can be displayed by thesame panels, but in opposite direction. The periodicity ofP with respect to EB corresponds to 4Es .

In conclusion, we have shown how one can control thespin properties of the transport through a 1D quantum dotembedded in a non-Fermi liquid by changing magneticfield, and gate and source-drain voltages. This is due to theinteraction that separates the energy scales of the chargeand the spin excitations such that Es ø Er . Completespin polarization can be achieved in spite of the presenceof correlations. Once it is achieved, it is not influencedby the interaction. When polarization is not complete, itis enhanced by the non-Fermi-liquid correlations. Thisshows that the electron spin is crucial for understandingnonlinear transport in 1D quantum dots.

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The above results have been obtained for T � 0. WhenT . 0, we expect temperature-induced Luttinger liquidpower-law broadening of the conductance peaks, and cor-respondingly a smearing of the spin polarization featureswhich is the subject of future work. At very low tempera-tures, one would expect that coherent tunneling processesdominate. Including spin, these lead to the well-knownquantum-dot Kondo physics which is not considered in ourapproach [15]. Therefore, the above results apply to tem-peratures higher than the Kondo temperature.

Our results for the transport spectra are consistent withseveral of the nonlinear features observed recently in a onedimensional (1D) quantum dot formed by two impuritiesin a cleaved-edge overgrowth quantum wire [16]. Moredetailed experiments, however, are needed in order to testthe above predictions, especially concerning the control ofthe spin. We expect that the effects can be observed in thetransport through double barriers formed in cleaved-edge-overgrowth quantum wires and in carbon nanotubes [17].

This work has been supported by Italian MURST viaPRIN 2000, by the EU within the TMR and RTN pro-grammes, and by the DFG.

[1] G. A. Prinz, Phys. Today 48, No. 4, 58 (1995); Science282, 1660 (1998).

[2] J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80,4313 (1998); D. D. Awschalom and J. M. Kikkawa, Phys.Today 52, No. 6, 33 (1999).

[3] D. Weinmann, W. Häusler, and B. Kramer, Phys. Rev. Lett.74, 984 (1995).

[4] S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage,and L. P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 (1996).

[5] T. Kleimann, M. Sassetti, B. Kramer, and A. Yacoby, Phys.Rev. B 62, 8144 (2000).

[6] A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVin-cenzo, D. Loss, M. Sherwin, and A. Small, Phys. Rev. Lett.83, 4204 (1999).

[7] P. Recher, E. V. Sukhorukov, and D. Loss, Phys. Rev. Lett.85, 1962 (2000).

[8] H. A. Engel and D. Loss, Phys. Rev. Lett. 86, 4648 (2001).[9] A. Brathaas, Y. Nazarov, and G. E. W. Bauer, Phys. Rev.

Lett. 84, 2481 (2000).[10] Q. Si, Phys. Rev. Lett. 81, 3191 (1998).[11] L. Balents and R. Egger, Phys. Rev. Lett. 85, 3464 (2000).[12] J. Voit, Rep. Prog. Phys. 58, 977 (1995).[13] A. Braggio, M. Grifoni, M. Sassetti, and F. Napoli, Euro-

phys. Lett. 50, 236 (2000).[14] M. Ciorga, A. S. Sachrajda, P. Hawrylak, C. Gould,

P. Zawadzki, S. Jullian, Y. Feng, and Z. Wasilewski, Phys.Rev. B 61, R16 315 (2000).

[15] A. Furusaki, Phys. Rev. B 57, 7141 (1998).[16] O. M. Auslaender, A. Yacoby, R. de Picciotto, K. W. Bald-

win, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 84,1764 (2000).

[17] H. W. Ch. Postma, M. de Jonge, Z. Yao, and C. Dekker,Phys. Rev. B 62, R10 653 (2000).

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