Accessing nanotube bands via crossed electric and magnetic fields

Wade DeGottardi,1 Tzu-Chieh Wei,2 Victoria Fernández,3,4 and Smitha Vishveshwara1

1Department of Physics, University of Illinois at Urbana–Champaign, 1110 W. Green Street, Urbana, Illinois 61801-3080, USA2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1

3Instituto de Física La Plata, CONICET 1900 La Plata, Argentina4Departamento de Física, Universidad Nacional de La Plata, CC 67 1900, La Plata, Argentina

�Received 19 August 2010; published 7 October 2010�

We investigate the properties of conduction electrons in single-walled armchair carbon nanotubes in thepresence of mutually orthogonal electric and magnetic fields transverse to the tube’s axis. We find that thefields give rise to an asymmetric dispersion in the right- and left-moving electrons along the tube as well as aband-dependent interaction. We predict that such a nanotube system would exhibit spin-band-charge separationand a band-dependent tunneling density of states. We show that in the quantum dot limit, the fields serve tocompletely tune the quantum states of electrons added to the nanotube. For each of the predicted effects, weprovide examples and estimates that are relevant to experiment.

DOI: 10.1103/PhysRevB.82.155411 PACS number�s�: 71.10.Pm, 71.20.Tx

I. INTRODUCTION

A highlight of graphene-based nanotubes is the presenceof a band �valley� quantum degree of freedom which offers anew facet to a range of strongly correlated low-dimensionalphenomena. In armchair nanotubes, the bands, which owetheir existence to the two-atom basis of the underlying hon-eycomb lattice, form a pair of gapless linearly dispersingmodes.1 While a direct consequence of their presence is themeasurable quantum conductance of 4e2 /h, richer effects canarise, for instance, in field-induced orbital moments,2

Coulomb-blockade profiles,3 and in Kondo physics.4 An abil-ity to precisely control the band sector would not only shedlight on these effects but also access other band-dependentphysics. For instance, in the recently reported nanotube Mottphase,5 the nature of this ordered phase could be probed bytuning interband interactions. In the Luttinger liquid-phasetypical of interacting one-dimensional systems, spin andcharge sectors have been shown to decouple in quantumwires;6 nanotubes could extend such fractionalization intoyet another sector. Recently, attention has turned toward“valleytronics” as a potential application for quantum infor-mation and quantum devices in the parent system ofgraphene;7 the ability to manipulate spin and band quantumstates in nanotubes would offer an attractive alternative.However, save for some exceptions,2,8 the band degree offreedom has thus far remained relatively resilient to any con-trolled coupling.

In this work, we explore the possibility of coupling to theband sector via applied transverse fields. Electric and mag-netic fields individually have been shown to alter the low-energy properties of conduction electrons.9,10 Here, buildingon our prior work,11 we find that the transverse field configu-ration shown in Fig. 1 in which both fields are present offersa powerful means of accessing the band sector. The fieldsbreak time-reversal, particle-hole, and band symmetries, andin fact, render the dispersion asymmetric in the right- andleft-moving electrons. Furthermore, the effective Coulombinteraction is modified from its field-free form to includecouplings between the charge and band sectors. As a conse-

quence, nanotubes in transverse fields could provide a real-ization of a tunable asymmetric Luttinger liquid.12–14 Mostpertinently, the system displays spin-band-charge separation,acts as a band selector for electrons tunneling into the nano-tube and exhibits a variety of tunable quantum dot shell-filling configurations.

II. MODEL

The setup of interest, as shown in Fig. 1, consists of anarmchair nanotube �a tube described by a chiral vector of theform �n ,n�, see Ref. 1 for definition� lying along the x-axissubject to an applied magnetic field B� =−By and electric fieldE� =Ez �see Ref. 15 for a possible experimental implementa-tion�. For a tube of diameter d=3nac /�, where ac�0.142 nm is the carbon-carbon bond length, the effect ofthe fields can be characterized by the parameters b=�3B�e�d2 /4� and u= �e�Ed /2t, where t�2.7 eV �Ref. 1� isthe electron hopping strength between neighboring sites.Near half-filling, the field-free low-energy electronic disper-

FIG. 1. �Color online� An armchair carbon nanotube in the pres-ence of transverse magnetic �pointing in the −y direction� and elec-tric fields �pointing in the z direction�. The carbon atoms belongingto the A and B sublattices are indicated by dark and light shading,respectively.

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sion, which can be traced to bonding and antibonding statesof the underlying graphene sublattices, consists of four de-generate gapless linear modes each having a Fermi velocity

vF=3tac

2� �8�105 m /s. These modes correspond to right andleft movers �indexed by r= � =R /L� at two Fermi points�indexed by �=�� which define the band degrees of free-dom.

In this Hall-type setup, classically, each of the fieldscauses a spiraling motion of electrons, and thus for fixedkinetic energy, reduces their linear velocity along the tube’saxis. Moreover, the nonvanishing Poynting vector E� �B� af-fects right and left movers differently. Indeed, a detailed per-turbative band-structure calculation11 for large enough tubes�10�n�� /u� shows that the right- and left-moving speedsare given by vr�vF�1− 1

3b2− n2

�2 u2�n�bu�. The spectrum re-

mains gapless,16 but the two Fermi points reside at energiesdiffering by �F�2�tbu / ��3n�, indicating a mild band-degeneracy breaking even at the band structure level. Thecorresponding low-energy noninteracting asymmetric Hamil-tonian can be expressed as

H0 = − i��r� dxrvrr�

† �xr�, �1�

where the band shift and Zeeman term are left implicit. Theoperator denotes the annihilation of a fermion and =�correspond to electronic spin components. For tubes of di-ameter d=6.78 nm �n=50� and experimentally accessiblefield strengths B=6.4 T and E=0.02 V /nm, we find vR=0.89vF and vL=0.77vF, yielding a pronounced asymmetryin right- and left-moving velocities. For the remainder of thiswork, we consider tubes of this diameter, corresponding toultralarge single-walled tubes.17

Transverse fields significantly alter not only the low-energy band structure but also the nature of Coulomb inter-actions within the tube. The usual forward scattering contri-bution, HC1= 1

2dxV�c+2 �x�, remains and continues to

dominate for larger tubes �n�10�.18 Here �c+�x�=�R,c++�L,c+ with �r,c+=��r�

† r� is the net charge densityand V� 2e2

ln� L�d � is the effective one-dimensional

interaction,11,19 where e is the charge of an electron, is thedielectric constant, and L is the length of the tube. Moreinterestingly, a second contribution becomes manifest due tothe uneven distribution of charge in the circumferential di-rection induced by the presence of the applied fields. A care-ful accounting of circumferential modes shows that this ad-ditional effective Coulomb term along the tube has the form

HC2 =��

4 dx��R,c− + �L,c−���R,c+ − �L,c+� , �2�

where �=4e2buh /� and h is a numerical factor which de-pends on the details of the geometry of the underlyinggraphene lattice; for n�20 we have h��1.46n−4.60�.11

Here, �r,c−=���r�† r� is the difference in charge density

between the two bands. The presence of this interaction termhas two origins: �i� the magnetic field couples to the crystalmomentum, and thus the Fermi points, giving rise to thecontribution involving the charge in the c− sector and �ii� the

electric field couples to the net charge imbalance in bondingand antibonding states, or alternatively, to the density differ-ence in the right and left movers in the c+ sector. This inter-action, which is key to the physics studied here, goes a stepbeyond the usual spin-charge-separating Coulomb term bydirectly coupling to the band degrees of freedom.

III. BOSONIZATION

As with the field-free case, the asymmetric gapless modesand Coulomb interactions can be studied viabosonization,12–14,20 which renders the total Hamiltonian forthe tube to be quadratic. The one-dimensional fermionic op-erators can be bosonized as

r� =�r�

�2�ac

exp�i�kFx + i�r�� , �3�

where the �r�’s are the so-called Klein factors which en-force anticommutation relations between differentchannels.19 The chiral bosonic fields satisfy the commutationrelations ��r��x� ,�r�����x���=−i�r�rr������� sgn�x−x��.The density associated with each sector takes the form �r�

=r�x�r� /2�. Given the asymmetric dispersion and currentdependent interaction, we explicitly employ these chiralfields to ensure that this algebra is preserved. It is convenientto introduce a spin and channel decomposition for the chiralfields, �r,�= 1

2 ��r,c++��r,c−+�r,s++��r,s−�. The total ki-netic energy takes the bosonized form

H0 =1

4��

a=c/s� dx�vR��x�R,a�2 + vL��x�L,a�2� �4�

and the total interaction term, Hint�HC1+HC2, takes theform

Hint =V

2�2 dx��x�R,c+ − �x�L,c+�2

+�

4� dx��x�R,c+ + �x�L,c+���x�R,c− − �x�L,c−� .

�5�

As discussed in previous work on asymmetricbosonization,12–14,20 in diagonalizing the full HamiltonianH=H0+Hint, care needs to be taken to ensure that the diag-onal basis preserves the chiral commutation rules respectedby the original fields, �r�. Diagonalization requires care dueto the unusual commutation relations �the so-called Kac-Moody algebra� among chiral fields.21 Here, we circumventunwieldy manipulations involving Bogoliubov transforma-tions by employing the trick of converting left-handed fieldsto right-handed fields via the transformation �La� i�La,which then allows for standard rotations.

This method makes it clear that as far as the diagonaliza-tion of the Hamiltonian is concerned, the asymmetric disper-sion simply shifts the values of the plasmon velocities buthas no effect on the eigenvectors. Consider a generic Hamil-tonian of the form H=H0+Hint with n channels a=1, . . . ,nwith

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H0 =1

4��

a dx�vR��x�R,a�2 + vL��x�L,a�2�

=1

4� dx�� TH0�� , �6�

where �� = ��R,1 , . . . ,�R,n ,�L,1 , . . . ,�L,n�, H0=diag�vR ,vR , . . . ,vL ,vL , . . . ,�, and Hint is a bilinear of thevarious densities �R/L,a. Now, after the transformation�L,a� i�L,a we have H0�H0� with

H0� = diag�vR,vR, . . . ,− vL,− vL�

= diag�v,v, . . . ,− v,− v, . . . ,� + �I , �7�

where v= �vR+vL� /2, �= �vR−vL� /2, and I is the 2n�2nidentity matrix. It is clear from Eq. �7� that � serves tomerely shifts the resulting eigenvalues and that the eigenvec-tors are independent of �. While this fact limits the role thatthe asymmetry plays in certain bulk quantities �such as thetunneling density of states�, the asymmetry can still play acrucial role in properties such as boundary and impurity scat-tering and in particular quantum dot physics.

IV. PHYSICAL PROPERTIES

The coupling between the c� sectors �Eq. �2�� plays acentral role in the altered physical properties of the tube. Theplasmon modes for the coupled charge sectors c� move withfour different velocities

vR/L,1/2 = � � + v1/2,

v1/2 =v

�2Kc+�1 + Kc+

2 ���1 − Kc+2 �2 + 2Kc+�

�v�2�1/2

,

�8�

where v= �vR+vL� /2 is the average velocity of the noninter-acting fermions, �= �vR−vL� /2 the asymmetry, and Kc+�1 /�1+4V /��v is the standard Luttinger parameter. Thespin sectors s� each retain their band structure velocitiesvR/L.

The existence of the two different plasmon velocities v1/2demonstrates that transverse fields can access the band sec-tor, which is normally resilient to any bulk coupling. Thusfar, momentum resolved bulk tunneling experiments onquantum wires6 have revealed two distinct plasmon disper-sions corresponding to spin and charge modes; analogousexperiments on nanotubes in transverse fields ought to unveilthe three pairs of velocities predicted above. For example,for a tube d=6.78 nm and experimentally accessible fieldstrengths B=6.4 T and E=0.02 V /nm, the plasmon veloci-ties are given by vR,1=4.07vF, vL,1=3.95vF, vR,2=0.32vF,and vL,2=0.20vF for the charge sectors and vR=0.89vF andvL=0.77vF for the spin sectors. The observation of three dis-tinct modes would reflect complete splintering of the nano-tube electron into its spin, charge, and band sectors. Figures2 and 3 show the plasmon velocities of a nanotube in thepresence of a 12 T magnetic field as a function of electricfield strength.

The effects of transverse fields on the low-energy nano-tube physics, particularly in the band degree of freedom, areprominently manifested in physical observables. The tunnel-ing density of states, an ubiquitous, experimentally acces-sible quantity, retains its power-law form ��E��E� charac-teristic of Luttinger liquids but exhibits a modified exponent.Using standard procedures22 that now account for field-dependent effects, we find that the tunneling exponent, tolowest order in �, is given by

�r� =1

8 1

Kc++ Kc+ − 2� −

�

4�vr� 1 − Kc+

1 + Kc+� . �9�

The first part of the exponent, also present in the field-freecase, reflects the suppression in tunneling due to interactions�where Kc+ is now tunable�. The second part, which dependson the tunable coupling � of HC2, further suppresses or en-hances tunneling depending on the sign of r�. As an esti-mate, for a d=6.78 nm tube in a 6.4 T B field and 0.02 V/nmE field with a field-free value of Kc+=0.23 �corresponding toa 1-�m-long tube�, we have distinctly different exponents�R+=�L−=0.22 and �R−=�L+=0.53. The form of Eq. �9� re-flects band �valley� selection; for example, a right-movingelectron would preferentially tunnel into the �=+ Fermipoint for the field configuration shown in Fig. 1.

0.005 0.010 0.015 0.020E �in V�nm�

0.85

0.90

0.95

v1,R�L�vF and vR�L�vF

FIG. 2. �Color online� Plasmon velocities v1,R and v1,L and bandstructure velocities vR and vL �solid, dotted, dashed, and dotted-dashed, respectively� for a 1-�m-long, 4.75-nm-diameter armchairtube in a 12 T magnetic field as a function of electric field strength.

0.005 0.010 0.015 0.020E �in V�nm�

4.44

4.46

4.48

4.50v2,R�L�vF

FIG. 3. �Color online� Plasmon velocities v2,R and v2,L �solidand dashed, respectively� for a 1-�m-long, 4.75-nm-diameter arm-chair tube in a 12 T magnetic field as a function of electric fieldstrength. The field-free charge plasmon velocity of this tube wouldbe v�=4.52vF.

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A physical consequence of transverse fields yielding sucha band-dependent exponents would be the presence of twodifferent power-law contributions to the non-Ohmic conduc-tance of the nanotube.23 On a related note, the presence of animpurity would distinguish these exponents; conductanceproperties would be sensitive to whether or not an electronimpinging on the impurity switched band index. Band-dependent effects similar to those in the density-of-states ex-ponent would be manifest in any susceptibilities involvingthe sublattice degree of freedom. Tendencies toward differentorderings, such as charge or spin density waves, which aregoverned by associated susceptibilities, would in turn reflectband dependence. For instance, charge density waves have

contributions from operators OCDW�� ��r�rr�

† −r��, re-flecting like �+� and staggered �−� band correlations at A andB sublattice sites. While these operators both come on anequal footing in the field-free case, we find that fields dis-criminate between these two ordering tendencies. In particu-lar, to first order in �� /v�, OCDW�

+ is unaffected by the pres-ence of fields, whereas OCDW�

− ’s scaling dimension becomes

�− =3 + Kc+

0

4−

2Kc+0

1 + Kc+0

���v

, �10�

where Kc+0 is the field-free value of the Luttinger parameter.

Thus, the operator OCDW�− is rendered more relevant. Fields

can thus differentiate between two types of band-dependentordering at the sublattice level.

V. QUANTUM-DOT PHYSICS

In the quantum-dot limit achieved by high resistance con-tacts or sufficiently low temperatures, we find that transversefields enable controlled tuning of Coulomb blockade physicsand nanotube quantum states. Here, as in previoustreatments,18,24,25 we describe the dot as a finite-sized versionof the net nanotube Hamiltonian and focus on the topologicalsectors as relevant for standard quantum dot experiments in-volving adiabatic tuning. Field-induced interactions give riseto a Coulomb blockade structure in which the charge andband degrees of freedom are coupled. The coupling betweenthe c+ and c− sectors arises due to the interaction HC2 �Eq.�2�� and the unusual form of the current operators. In particu-lar, the charge current differs from the naïve form Jc+=vR�R,c+−vL�L,c+ due to HC2. The form of the current iseasily derived by noting that the current Ja for every sectorobeys a continuity equation �t�a=−�xJa. For the c+ sector,we obtain

�t�c+ = i�H,�R,c+ + �L,c+�

=− �x�vR�R,c+ − vR�L,c− + ��c−� �11�

thus giving Jc+=vR�R,c+−vR�L,c−+��c−. For a=c− ,s� wehave Ja=vR�R,a−vL�L,a.

The Coulomb blockade Hamiltonian follows from requir-ing ����r�r,�=N� /L, where the N�’s are integers andthe no-current condition Ja=0. The channel occupation num-bers are defined Nc�= �N+↑+N+↓�� �N−↑+N−↓� and Ns�

= �N+↑−N+↓�� �N−↑−N−↓�. In the presence of a gate voltage

VG, the resulting Hamiltonian characterized by quantumnumbers Na for each topological channel is given by

HT = �a=c/s�

�a

8Na

2 − �Nc+ −��

4L vR − vL

vR + vL�Nc+Nc− +

1

2�bNc−

− �ZNs+, �12�

where �a=�0+4Ea, �0=2��vRvL

L�vR+vL� , and Ea is the interactionstrength in a given mode. Here, Ec+=V /L, Ec−=−��2 /4�vL, and Es�=0. The term � is proportional toeVG, and �Z=�BB accounts for the Zeeman splitting. Theband splitting �b will depend on boundary conditions; in theabsence of Fermi point mixing at the tube ends, it reduces tothe band structure Fermi point mismatch �F.11 As typicalparameter values, for a 1 �m tube of diameter 6.78 nm, andmagnetic and electric field strengths of 4.93 T and 0.0242V/nm, we find �0=1.36 meV, Ec+ /�0=5.93, Ec− /�0=−0.23,�Z /�0=0.21, and � /L�0=0.24.

By virtue of their field dependence, several parameters inHT can be varied to access a wide variety of shell fillingconfigurations of the nanotube quantum dot in Coulombblockade experiments. The configurations correspond to setsof electron occupation numbers which minimize the energyassociated with Eq. �12�.11,18 While the associated parameterspace is too extensive for an exhaustive study, a few salientcharacteristics of shell filling are as follows. �i� In actualexperiments,3,26 the band degeneracy is often naturally bro-ken due to physical attributes such as the confining potentialcreated by the leads, yielding patterns such as in Fig. 4�a�.Here, the field dependence of �b enables controlled tuning ofband degeneracy breaking as well as probing the extent ofFermi point mixing at the tube ends.11,27 �ii� A fourfold pe-riodicity has been observed in some experiments reflecting

FIG. 4. �Color online� Examples of shell filling. Figure indicatesthe order, � point, and spin of tunneled electron with increasingchemical potential � �not drawn to scale�. �a� Filling order for atube in the absence of fields and with �b�0. �b� Shell filling forNc+=Nc− exhibits twofold periodicity and complete polarizationinto band �=+. This condition is approximately met for a 1 �mtube of diameter 6.78 nm in the presence of magnetic and electricfields 4.93 T and 0.0242 V/nm, respectively �see text for specificvalues of the Coulomb blockade parameters for this case and otherdetails�.

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the band and spin degeneracy of the tube.3 �iii� As a directdemonstration of band tuning, the parameter values quotedabove yield the twofold periodic shell-filling pattern in Fig.4�b�. In particular, these values are chosen such that Eq. �12�is minimized by the condition Nc+=Nc−, entirely restrictingthe tunneling into a given � point. We remark that this con-dition requires a fine tuning of fields, and also that other fieldvalues can even render �c− negative, resulting in an instabil-ity toward complete polarization into one band. �iv� Period-icity can also be entirely obliterated by choosing an irrationalratio between two of the relevant shell-filling parameters. Asdemonstrated above, transverse fields provide a precisemeans of preparing and manipulating the electronic spin andband quantum numbers of the nanotube quantum dot.

VI. CONCLUSION

We have presented transverse fields as powerful probes toaccess and explore a rich range of physics in armchair nano-

tubes that directly addresses the band degree of freedom. Wehave found that these fields induce an unusual interactionthat couples the charge and band sectors. We have predictedthat signatures of these field effects will be apparent in avariety of measurements including those probing Luttingerliquid parameters and plasmon structure, tunneling density ofstates and impurity scattering. We have shown that transversefields can be used for controlled manipulation of nanotubequantum-dot states, making this proposed setup a potentialbuilding block for nanoscale quantum devices.

ACKNOWLEDGMENTS

We would like to thank N. Mason and M. Stone for theirperceptive comments. This work is supported by NSF underGrant No. DMR09-06521 �W.D. and S.V.�, NSERC andMITACS �T.-C.W.�, and UNLP and CONICET �V.F.�

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