Spin stripes in nanotubes

4 Pages·0.3 MB·

by  Alex Kleiner

Spin stripes in nanotubes
Alex Kleiner
Institute of Theoretical Physics
Chalmers University of Technology and G¨oteborg University
S-412 96 G¨oteborg, Sweden
2
0 It is shown here that electrons on the surface of a nanotube in a perpendicular magnetic field
0 undergo spin-chirality separation along the circumference. Stripes of spin-polarization propagate
2 along thetube, with a spatial pattern that can be modulated by theelectron filling.

n
a
J

2
The emerging field of spintronics opens a new tube is then A~ = 0,RBsin x , where (x,y) are the cir-
2 R
paradigm of electronics based on the electron’s spin cumferentialanda(cid:0)xisdirection(cid:1)softhetube,respectively.

] rather than charge [1]. The realization of a spin circuit The Hamiltonian of a cylindrical 2DEG in this field is,
l
l requires the generation,conduction and manipulation of
ha spincurrents. Recently,aspatialcontrolofspincurrents H = ~2∂x2 + ~2 i∂ + eRB sin x 2+µgs B
- was demonstrated by engineering a material with a spa- −2m∗ 2m∗ (cid:18)− y ~ R(cid:19) ·
s
tially varying g-factor [2]. An alternative route would
e (1)
m require a magnetic field with spatial variation on the
. nanometer scale. Strong variations of the field can be where m∗ is the effective mass, µ is the Bohr magneton,
t
a achievedalongthecircumferenceofananotubesubjected g is the gyro-magnetic factor and s is the spin operator.
m toauniformmagneticfield,directedperpendiculartoits
Thelongitudinalwavevectorandspinareconservedsince
axis. Here I suggest that a two dimensional electron gas
- theHamiltonian(1)doesnotcontaintheycoordinatenor
d (2DEG) rolled-up to a nanotube, may form spin-stripes
otherspinoperatorsandsotheoperatorsarereplacedby
n
propagatinginalternatingdirections,asafunctionofthe
o their eigenvalues Ky and gµB/2. The wave functions
filling. This applies to the conduction electrons in fields ±
[c satisfying l . R, where l is the Landau length ~/|eB| feoiKrytyhχe↑s,↓p(ixn)-.upInaunnditsspionf-dEoLw/n2,pwahrteirceleEs Lare=neoBw~ψ/m↑,↓∗ =is
1 and R is the tube radius. At magnetic fields ofpB.10T the Landau level energy spacing, eq. (1) becomes the
for example, the radius should be R & 8nm. Such sizes
v following one dimensional Hamiltonian,
4 frequently occur in
9 multi-wall carbon nanotubes (MWCNT) and in the R x 2 gm∗
2 2
3 newclassofrecentlyproducedrolled-upheterostructures H =−l ∂x+(cid:18)Kyl+ l sinR(cid:19) ± 2m (2)
1 e
[3] [4]. The former, showed magneto-conductance fluc-
0
2 tuations [5] [6], with varying interpretations in connec- The Hamiltonian (2) is a variant of Hill’s equation and
0 tion to the diffusive [7] or ballistic [8] [9] nature of the can be easily diagonalized numerically [11]. We want
/ MWCNT charge conductance. The spin conductance of to work in the regime where all the wave-functions are
t
a a MWCNT however, was shown to be ballistic [10] over confined in the circumferential direction. The weakest
m
fairly large distances (& 130nm). On the other hand, confiningpotentialin(2)isforK =0,whichisadouble-
y
- the cylindrical heterostructures, made of silicon, silicon- well with minima at the poles. This potential gives, to
d
germanium [4] or indium-gallium and indium-arsenic [3] a linear order in x, Landau levels centered at the poles,
n
o havethe advantageof controlledradiusesthat caneasily withaspatialextensionofl√2n+1, wheren=0,1,2 .
··
c satisfy l . R, they can be made clean and without the Thus, the potential is always confining if R&l√2n+1.
v: problemofunknownchiralityandinter-shellcoupling. It Thetypicalenergyspectrumandprobabilitydistribution
i wasfoundnumerically[11]thatacylindricalspinlesstwo in this regime are shown in fig. (1).
X
dimensional electron gas (2DEG) under a perpendicular
r
magnetic field, forms Landau level like states at the top
a
and bottom and chiral states, similar to the edge states
in the Hall bar, at the sides.
The magnetic field B is taken here to be perpendicular
to the surface at the lines x = 0 and x = πR here-
after the north and south ‘poles’. The ‘equators’ are at
x = πR/2 and x = 3πR/2, and states located anywhere
above or bellow the equators are called here ‘north’ or
‘south’ states. The vectorpotential onthe surface ofthe
1
g = 2 in Eq. (2). The first two terms in Eq. ( 3) are
theenergiesofaharmonicoscillatorwithananharmonic
correctionofasingle-wellpotentialV(K )ataminimaof
y
Eq. (2). Since Eq. (2)hastwo minimas for K <R/l2,
y
| |
thesetermsalonewouldgiveatwo-folddegeneracy,with-
out counting the spin. The third term largely removes
this degeneracy by mixing the north and south states,
1
and the last term is the Zeeman splitting, with s = .
±2
Since the conduction properties are determined only by
electrons at the Fermi-energy, having the map between
the energy-momentum-spin state and the spatial distri-
bution of that state (fig. 1), we can now find the spatial
distribution of the conduction electrons and their spins.
The spin polarization density at a given Fermi-energy E
is defined as
FIG.1. Energy-Probabilitydensityphasespace. (A):En- P(E,x)=(P↑(E,x) P↓(E,x))/(P↑(E,x)+P↓(E,x)),
−
ergyspectrumofeq. (2),calculatednumericallyforR=2.75l wherethespin-uporspin-downpolarizationP (E,x)=
↑,↓
(pRro=ba5b0ilnitmy danisdtrBibu=tio2nT)a,lomn∗g=thmeeciarcnudmgfe=re2n.tial(Bco):orSdpiantaitael PityKydegn(Esit,iKesy)w|χit↑h,↓(tEh,eKcyo,rxr)e|s2ponfadcitnogrsdenthsiety-pofr-osbtaatbeisl-,
of the lowest band (n= 0) states in the spectrum. Here the
summed over all states K at the energy E. Fig. (2)
y
electronsarewell confinedtotheproximityoftheirpotential
shows the energy dependent spatial spin-polarization. It
minima. At K =0 the potential in (2) is a double-well, one
y is dominated by states at energies with a divergent den-
awtaredaschonpeoloef.tAhes|eKquya|tionrcsr,eaanseds,wthheentwKowellsRm/lo2vtehcelyosmerertgoe- sity of states g(E) = dE −1 at some proximity. As
to one well, at the equator. This po|inty|is≥illustrated in (C) (cid:16)dKy(cid:17)
evident from fig. (1), these are either the Landau-like
wherewezoomonthepotentialsoftwostates,K =0.5R/l2
y states at the poles having K = 0, or states centered
and 1.1R/l2 marked in (B) with white lines, having a dou- y
around the equators, to be reffered to as pole and equa-
ble-well and a single well potentials, respectively.
tor singularities, respectively.
Theeigenfunctions(seefig. 1B)areconfinedinthecir-
cumferentialdirectiontotheirpotentialminima,depend-
ingonK . SincetheHamiltonian(2)issymmetricunder
y
a simultaneous sign inversion of x and K , states with
y
oppositeK arecenteredatoppositesidesofthecircum-
y
ference [11],and states with K =0 arethus centeredat
y
thepoles. Inthelimitofavanishingmagneticfield,each
bandisfour-folddegenerate,i.e: twiceduetospindegen-
eracyandtwice due toclock-wiseandcounterclock-wise
propagatingmodes. The magneticfieldremovesthe four
degeneracies, as shown in fig. (1A), except at K 0,
y
≈
where a two-fold degeneracy remains. Higher magnetic
fieldswillnotremovethisdegeneracybutratherincrease
it,sincehere,intheconfinementregime,thepotentialfor
K 0 has two deep and isolated potential wells at the
y
two≈poles. Only as K R/l2 the two potential wells FIG. 2. Spin polarization distribution along the circum-
y
→
getclosetoeachotheracrossoneoftheequatorsfortheir ference vs. energy, with parameters as in fig. (1). The
corresponding states to mix and remove the degeneracy. pronounced polarization densities are mainly at energies in
theproximityofsingularitiesinthedensity-of-statesofeither
The total energy can be approximated analytically (see
spin-uporspin-downbands(redandblue,respectively). Each
note [12]) for small K ’s to give
y singular spin state is either centered at a pole or around an
1 ~ 2 equator. It is marked on the right with P or E respectively.
E =~ω(n+ )+3λ (2n2+2n+1) ∆E +2s e.g: the arrow labeled 1P marks a singular state in the first
2 (cid:18)2mω(cid:19) ± n sub-band,centered at the↑Pole having a spin-up.
(3)
whereω,λand∆E arefunctionsofK ,giveninthenote We can follow, for example, the spatial distribution of
n y
[12]and m m , having set for simplicity m =m∗ and the equator singular states carrying spin-up. There are
e e
≡
2
four such states in fig. (2), marked at the right as 1E map (fig. 2) may be observedby sweeping the gate volt-
↑
to 4E ,where the correspondingwave-functionsaround age with the conventional two or four point contact set-
↑
each equator have, one to four peaks, respectively . A up. For short tubes, K in fig. (1) becomes discreet,
y
similarobservationcanbemadeforthepolesingularities with most of the allowed states fall at energies were the
(marked with P in fig. 2), which are the Landau-like density-of-states diverges. In other words, the polariza-
states. Their energycanbe foundanalyticallyby simply tion map will converge to the discreet levels marked by
setting K =0 in Eq. (3), giving the arrowson the rightin fig. 2. Only one third of these
y
levels, the pole states (eq. 4) can be considered as mod-
1 1
E =E (n+ ) E (2n2+2n+1), (4) ified levels of the ‘flat’ quantum dot. The other levels
n L R
2 ± 2 − areintrinsictothe tube. Thiscouldbe supportedbythe
experimental observation [14] that the spin dependent
wherethefirsttermistheusualZeemansplitLandaulev-
energy levels of a short carbon nanotube ‘dot’, do not
elsandthesecondtermisthecurvaturecorrectiondueto
~2 follow the simple, flat, quantum dot level filling. How-
the lateral energy, E = . The spin-polarization
R 8mR2 ever, the filling pattern in [14] can not be expected to
P(E,x) is dominated by the spin of the singular state
follow the polarizationmap (fig. 2) since it was not con-
with the closest energy to E. However, when the Fermi
ducted in the confinement regime so that other reasons,
energy lies between the energies of singular states with
suchaselectron-electroninteractions,mayhaveplayeda
opposite spins, such as between 2E and 2P , or be-
↑ ↓ more important role, as suggested by the authors. The
tween 4E and 3P in fig. (2), there is a coexistence
↑ ↓ confinement condition for the experimental resolution of
of spin-up and spin-down with different spatial distribu-
spin-stripes requires either large fields or large radiuses.
tions. This gives rise to the formation of spin-polarized
e.g: if the radius is in the range 5nm < R < 25nm,
stripesonthesurfaceofthetube. Fig. (3)showsthespin
at the lowest filling n = 0, the confinement condition
stripesatthatenergy,withtheadditionalinformationon
R & l√2n+1 gives B & 20T and B & 1T, where the
the chirality of these stripes.
higher field corresponds to the lower radius. These con-
ditions, as already noted, can be easier achieved using
the cylindrical heterostructures.
Inconclusion,itwasshownthatwhenananotubeissub-
jectedtoaperpendicularmagneticfield,underthespeci-
fiedconditions,thereisaformationofspin-stripesonthe
surfaceofthe tube withdifferentpropagationdirections.
The sensitivity of the spin pattern to the filling energy
opens a potentially new wayto generateandmanipulate
spin currents with a gate. Finally, the stripe formation
maybe tested directlyby the recently demonstrated[13]
spin-polarizedscanningtunnelingmicroscope(SP-STM).
FIG.3. Spin and chirality polarization. The Fermi-energy
lies between the 2E↑ and the 2P↓ singularities in fig. (2). [1] S. A. Wolf et al., Science 294, 1488 (2001)
Red and blue colors represent, as in fig. (2), spin-up and [2] G. Sallis et al., Nature 414, 619 (2001)
spin-down polarizations, respectively. The arrows give the [3] V. Ya.Prinz et al. Physica E 6, 828 (2000)
direction of propagation (chirality), taken from the sign of [4] Oliver G. Schmidt,Karl Eberl, Nature 410, 168 (2001)
ddKEy. Theblackorwhitecolorofsomearrowsisforvisibility. [5] C. Sch¨onenbergeret al. Appl. Phys.A 69, 283 (1999)
[6] J. O.Lee et al. Phys.Rev B 61, R16362 (2000)
[7] C. Sch¨onenberger and A. Bachtold, Phys. Rev. B 64,
There is a rather complex pattern of left moving and
157401 (2001)
right moving spin ‘lanes’. The fact that spin distribu- [8] J. Kim et al., Phys. Rev.B 64, 157402 (2001)
tion of the highest occupied electron states is entirely [9] S. Roche and R. Saito, Phys. Rev. Lett. 87, 246803
dependent on the filling, as shown in the polarization (2001)
map(fig. 2)suggeststhatagatevoltagemaycontrolthe [10] K. Tsukagoshi, B. W. Alphenaar and H. Ago, Nature
spatial distribution of spins. Experimentally, it appears 401, 572 (1999).
feasible to observe the spin-stripes spatial structure by [11] H.AjikiandT.Ando,J.Phys.Soc.Jpn.62,1255(1993)
[12] For R & l√2n+1 all states are confined to the prox-
a spin-polarized STM (SP-STM) [13], at temperatures
imity of their potential minima, which are at x =
kT E . The vertical structure of the polarization min
L
≪
3
Rsin−1 |KRy|l2 and πR−xmin for −R/l2 ≤Ky ≤0, and
for states with 0 K R/l2 at 2πR x and πR+
y min
≤ ≤ −
x . The potential at a small distance ǫ from the min-
min
ima can beexpandedas V(K ,ǫ)= 1mω2ǫ2+λǫ4, with
y 2
ω = 1 El(R2 K2l2) and λ = El 7K2l2 4R2 .
Rqm l2 − y 24R4 (cid:16) y − l2 (cid:17)
The potential at K R/l2 is a double-well, sym-
y
| | ≤
metric about the equator, described by V(K ,ǫ) at the
y
vicinity of its two minima x and πR x . De-
min min
−
noting ψ for the two single well wave functions, the
1/2
total wave function of state K , to a zeroth order is
y
Ψ = 1 (ψ ψ ), where s/a corresponds to the
s/a √2 1 ± 2
symmetric and antisymmetric product respectively. Due
to the non-zero tunneling probability across the equa-
tor, the degeneracy of the symmetric and antisymmet-
ric states is lifted. In the semi-classical approximation,
theenergyissplitby 4m~2ψ(xeq)ψ′(xeq)whereψ(xeq)and
ψ′(xeq) are either one of the single well wave functions
andtheirderivative,attheequator.Takingψ(x )tobe
eq
theeigenfunctionsoftheharmonicpartoftheenergy,the
first two energy splittings are ∆E1 = 2m~2√π1a3be−(b/a)2
abn=d ∆RE(π2/2=−4m~si2n√−π11aK5byR3le2−).(bT/ah)2is, rwehsuerlte aagr≡eesqwmi~tωhatnhde
numericsfor K . R .
| y| 2l2
[13] HeinzeS. et al., Science288, 1805 (2000)
[14] SanderJ. Tans et al., Nature 394, 761 (1998)
ACKNOWLEDGMENTS
I am indebted to Sebastian Eggert for many discus-
sions and to Kim Jaeuk, Mikael Fogelstro¨m and Paata
Kakashvili for helpful comments on the manuscript.
4