Strong Nernst-Ettingshausen effect in folded graphene

0.7 MB·English

by  Friedemann Queisser

Strong Nernst-Ettingshausen effect in folded graphene
Friedemann Queisser and Ralf Schu¨tzhold∗
Fakult¨at fu¨r Physik, Universita¨t Duisburg-Essen, Lotharstrasse 1, 47057 Duisburg, Germany
(Dated: January 18, 2013)
We study electronic transport in graphene under the influence of a transversal magnetic field
B(r) = B(x)ez with the asymptotics B(x ) = B0, which could be realized via a folded
→ ±∞ ±
graphene sheet in a constant magnetic field, for example. By solving the effective Dirac equation,
we find robust modes with a finite energy gap which propagate along the fold – where particles
3
and holes move in opposite directions. Exciting these particle-hole pairs with incident photons
1
would then generate a nearly perfect charge separation and thus a strong magneto-thermoelectric
0
(Nernst-Ettingshausen) or magneto-photoelectric effect – even at room temperature.
2

n PACSnumbers: 72.80.Vp,78.67.Wj,85.80.Fi.
a
J

7 Introduction The Nernst-Ettingshausen effect [1] de-
1 scribes the generation of an electric current (or voltage)

by a temperature gradient in the presence of a magnetic
] B
l field. Suchthermoelectriceffectsfacilitatethedirectcon-
l
a version of thermal into electric energy and thus are of
h
general interest. Obviously, the C (charge), P (parity),
- FIG. 1. Sketchof the considered set-up.
s and T (time reversal) symmetries must be broken for
e
such an effect to occur. One way to achieve this is a
m
magnetic field in a suitable geometry: trajectories of op-
Eigen-modes We consider length scales (e.g., curva-
at. posite charge carriers are bent to antipodal directions. ture radius of fold) far above the lattice spacing of
However, the mean free path in usual materials is too
m graphene 0.25 nm and energies of 1 eV or below. In
short to generate an efficient charge separation in that ≈
thislimit,wemaydescribethelow-energybehaviorbyan
-
d way – at least at room temperature. For example, the effective Dirac equation in 2+1 dimensions (~=q =1)
e
n classical cyclotron radius r = mev/(qeB) of a free elec-
o tron at room temperature in a magnetic field B of one iγµ(∂ +iA )Ψ=0, (1)
µ µ
c Tesla r = (µm) is much larger than the typical mean
[ O with xµ = [v t,x,y], where v 106m/s is the Fermi
free path (in the nanometer range). Thus, the Nernst- F F
1 Ettingshausen effect is strongly suppressed by multiple velocity [14]. The Dirac matrice≈s γµ = [σz,iσy, iσx]
−
v acting on Ψ = [ψ ,ψ ] are related to the Pauli matrices
scattering events and dissipation etc. 1 2
2 σx,y,z. In the Landau gauge, the vector potential A =
4 This motivates the study of graphene [2–6], since this µ
[0,0,A(x)] generates the magnetic field B(x) = ∂ A(x)
1 system offers a comparably long mean free path and x
with the asymptotics B(x )= B .
4 a large electron mobility, a linear (pseudo-relativistic) →±∞ ± 0
. Inviewofthetranslationsymmetryintandy,wecan
1 dispersion relation at low energies (i.e., near the Dirac
make the separation ansatz for the modes
0 points), anda verylargeFermivelocityv 106m/s [3],
F
3 see also [7, 8]. In this case, the pseudo-re≈lativistic cy- Ψ(t,x,y)=exp iEt+iky ΨE,k(x), (2)
1 {− }
clotron radius at room temperature in a magnetic field
:
v of one Tesla is much smaller (some tens of nanometers). arriving at the two coupled equations
i
X In this regime, quantum effects should be taken into ac- iv [∂ +k+A(x)]ψE,k(x)=EψE,k(x)
count – even at room temperature [4]. F x 2 1
ar In the following, we consider folded graphene in a ivF[∂x−k−A(x)]ψ1E,k(x)=Eψ2E,k(x). (3)
transversal magnetic field, see Fig. 1. In principle, the Hence, we can choose ψE,k(x) to be real, for example,
1
foldingofgraphenehasalreadybeenrealizedexperimen- while ψE,k(x) is imaginary. We observe a particle-hole
tally, see, e.g., [9, 10]. This set-up is advantageous since 2
symmetry since replacing E E and ψE,k ψE,k
we avoid real edges in graphene which are typically not → − 2 → − 2
yields a new solution Ψ−E,k =σzΨE,k =(ΨE,k)∗.
perfect and contain cracks or other defects which might
Thetwofirst-orderequations(3)canbecombinedinto
inducescattering,couplingtovibrationaldegreesoffree-
one second-order equation
dom, or further unwanted effects. Form a theoretical
pointofview,these edgescanonlybe describedinideal- v2[k+A(x)+∂ ][k+A(x) ∂ ]ψE,k =E2ψE,k,(4)
ized cases, e.g., via effective boundary conditions which F x − x 1 1
then depend on the concrete realization (e.g., zigzag or and analogously for ψE,k with ∂ ∂ . This equa-
2 x ↔ − x
armchair structure [11–13]). tion can be cast into the form of a one-dimensional
2
Schr¨odingerequation ψE,k =E2ψE,k withtheHamil- where we have used the normalization Ψ Ψ = 1.
Hk 1 1 h E,k| E,ki
tonian = v2( ∂2 + ) containing the effective po- Together with Eq. (7) we find that particles with E > 0
Hk F − x Vk
tential = [k +A(x)]2 +A′(x). Since this Hamilto- and holes with E < 0 have the opposite current (and
k
V
nian is self-adjoint Hk = Hk† and the potential Vk has groupvelocity),i.e.,allparticles(withk >−Amin)move
the asymptotics (x ) = , we get a complete to the right and all holes move to the left. In this way,
k
V → ±∞ ∞
set of discrete, orthonormal, and localized (in x) eigen- one obtains a (nearly) perfect charge separation.
functions ψE,k(x) for every value of k. These modes are Asymptotics It is illustrative to study the two limit-
1
non-degenerateforeachk,i.e.,theenergybandsE(k)do ing cases k . For large and positive k, the po-
→ ±∞
not cross [15]. Due to the particle-hole symmetry, each tential k can be approximated by k k2 +2kA(x).
V V ≈
of these eigen-functions ψE,k(x) corresponds to a pair of Thus, to lowest order in k, we obtain E vFk, i.e.,
1 ≈ ±
modes Ψ±E,k(x) of the original problem (3) with oppo- these modes propagate with a speed close to the Fermi
velocity. Going to the next order in k, we may expand
site energies. Furthermore, with = k +A(x) ∂ ,
k x
we may write = v2 † whDich shows that − is A(x)arounditsminimumatx0 wherethemagneticfield
Hk FDkDk Hk B(x ) = 0 vanishes A(x) A +A′′(x )(x x )2/2
non-negative (and thus E is real). In addition, k can- 0 ≈ min 0 − 0
H and obtain harmonic oscillator eigen-functions centered
not have a zero eigen-value E = 0 since the correspond-
ing ψ1E=0,k(x) must satisfy Dkψ1E=0,k = 0, which gives athtexp0o[taesnstuimalibneghtahvaetsAas′′k(xB0′)(x6=)0,]t.hSeimncoedtehsearsetiffstnreosnsgolyf
ψE=0,k(x) exp kx+ dxA(x) and analogously for 0
ψ1E=0,k(x).∝Due to{ the aRsymptotic}s B(x )= B localized around x0 for large k and basically propagate
2 →±∞ ± 0 along the x0-line where the magnetic field vanishes. For
and thus A(x ) B x, this solution is not nor-
→ ±∞ ∼ 0| | fixedandlargek,these modes haveequidistantvaluesof
malizableandthus isstrictlypositiveforanyk. Ergo,
Hk E where the distance scales with B′(x0).
the modes do always have a finite energy gap E =0.
6 Forlargeandnegativek-values,pthe minima ofthe po-
Current The current density of the modes reads
tential are given by A(x ) + k = 0 and thus the
k ±
jµ =v Ψ γµΨ =v Ψ† γ0γµΨ . (5) modes aVre localized at large and nearly opposite values
E,k F E,k E,k F E,k E,k
of x k/B due to A(x ) B x. In this
The zeroth component j0 = vFρ is simply given by the regim±e∼,A±(x|)isa0p|proximatelyl→ine±ar∞and∼thu0s|w|erecover
density ρ = ψE,k 2 + ψE,k 2. As one would expect,
| 1 | | 2 | the harmonic oscillator eigen-functions corresponding to
jx vanishes identically since ψE,k(x) is real and ψE,k(x)
1 2 the usual (pseudo-relativistic) Landau levels in a con-
imaginary, cf. Eq. (3). Using the same argument, the
stant magnetic field [16]. Note, however, that the eigen-
current density in y-direction simplifies to
functions ψE,k(x) are linear superpositions of the Lan-
∗ 1
jy =ivF(cid:16)ψ2E,k(cid:17) ψ1E,k−h.c.=−2ivFψ1E,kψ2E,k. (6) dEa.uInlevtehlisscleimntiet,retdheateixg+en-aennderxg−ieswEithdothneostadmeepeennderogny
Fromthe triangleinequality (2ab a2 + b2 ), we may k anymore (En = v √2B n with n N) and thus the
infer jy vFρ, i.e., the speed| of|t≤he| as|soc|iat|ed charge current Jy alsLo van±ishFes. H0ence these∈modes are not so
| | ≤
carriers is at most the Fermi velocity vF (as expected). interesting for our purpose.
The total current in y-direction can be obtained by Matrix elements Now we arein the positionto study
2v2 the excitation of particle-hole pairs by incident photons
Jy = dxjy = F dxψE,k[k+A(x)]ψE,k, (7)
Z − E Z 1 1 (in the infra-red or optical regime). In second quantiza-
tion, the interaction Hamiltonian reads
where we have used v ψE,k = iEψE,k from Eq. (3).
For the lowest E2 modFeDsk(fo1r a given k2), i.e., the upper- Hˆ = dxdy ΨˆγµAˆ Ψˆ . (9)
int Z µ
most negative mode and the lower-most positive mode,
the wave-function ψ1E,k(x) corresponds to the ground where the photon field operator Aˆµ contains the cre-
srteamte[1o5f])H.kSianncdeohnenecceanitriespneoant-tzheerosafmoreallilnexo(fnaordgeutmheenot- afrteiqonuenancyd ωa,nnwihaivleantiuomnboeprekra,toarnsdaˆp†ωo,kla,σrizaantdionaˆωσ,k.,σTfhoer
for ψ2E,k(x), the integrand jy = −2ivFψ1E,kψ2E,k is non- Dirac field operator Ψˆ is a linear combination of the an-
zeroforallxandhencethecurrentJy isfinite. Butother nihilation operators for particles cˆ Ψ and the
E>0,k E>0,k
lmarogdeesencoouugldhhkav>e JyA=0=at sommine kA-v(xal)ue,.thHeowinetveegrr,afnodr creation operators for holes cˆ†E′<0,k′ΨE′<0,k′.
− min − { } If we now consider the transition matrix elements
ainlltxheanadbotvheuseqtuhaetciounrrψen1Et,kh[kas+aAfi(nxi)t]eψv1Ea,lkueis. positive for hout|Uˆint|ini with an initial photon |ini=aˆ†ω,k,σ|0i and
a final particle-hole pair out = cˆ† cˆ† 0 , we
Furthermore, the current Jy is related to the slope | i E>0,k E′<0,k′| i
get to first order in perturbation theory
dE/dk ofthe dispersionrelation,i.e., the groupvelocity:
Writing Eq. (3) as HˆE,k|ΨE,ki=E|ΨE,ki, we find AEω,,kk,;σE′,k′ = √12ω Z dtdxdy ΨE,kγµAσµΨE′,k′ ×
dHˆ dE
Jy = ΨE,k E,k ΨE,k = , (8) e+iEt−ikye−iωt+ik·re−iE′t+ik′y, (10)
−h | dk | i −dk ×
3
where Aσ encodes the polarization of the photon. As Now, the integrand in the matrix elements (11) be-
µ
usual, the t-integral gives δ(ω E + E′), i.e., energy haves as ψE,k(x)ψE′,k(x) ψE,k(x)ψE′,k(x) for the two
conservation. Since the wavelen−gth of the photons un- photon pol1arizatio2ns. Ins±erti2ng Eq.1(12) and integrat-
der consideration (in the optical or infra-red regime) is ing over x, we see that the matrix elements (11) be-
much larger than the typical length scales of the elec- tweenmodesofthesamepseudo-parityvanishforphoton
tronic modes in graphene, we may neglect the photon polarizations in x-direction whereas the transition be-
wavenumberk. Therefore,they-integralyieldsδ(k k′), tween modes of opposite pseudo-parity is forbidden for
−
i.e.,weexciteparticle-holepairswiththesamewavenum- the other polarization.
ber k=k′. The remaining x-integral reads Yetanothersetofselectionrulescanbeobtainedinthe
asymptotic regimes. For k we only get transitions
AEω=,kE;E−′E,k′′,=kk≈0,σ ∝Z dxΨE,kγµAσµΨE′,k′. (11) between modes of opposite→ene∞rgies(due to the orthogo-
nality of the harmonic oscillator eigen-functions). In the
Let us first assume Aσ = const and consider the tran- opposite limit k , we recover the well-known [16]
µ → −∞
ssuitciohnabsetthweeeunppmeor-dmesosotfnthegeastaivmeemEo2de(i(.efo.,rEa =giv−enEk′)), pnropNertwiehseroefwtheeonLlayngdeatutrleavneslistioEnLns =for±nvF√n2B01n.with
∈ → ±
and the lower-most positive mode, cf. Fig. 2. In this Polarization dependence So far, we have discussed
case, we may use the aforementioned particle-hole sym- the caseAσ =constinEq.(11). Thisis certainlyagood
µ
metry Ψ = σzΨ and simplify the integrand via approximation if the polarization of the incident photon
−E,k E,k
ΨE,kγµΨE′,k = ΨE,kγµσzΨE,k. Inserting γ1 = iσy and pointsinydirection,i.e.,isalignedwiththesymmetryof
γ2 = iσx and using the properties of the Pauli ma- ourset-up. However,fortheother(x)polarization,Aσ in
µ
−
trices, we see that the matrix element for the photon Eq.(11)shouldbereplacedbythelocalprojectionofthe
polarization in x-direction Ax yields the same expres- photon wave function Aσ onto the graphene plane, i.e.,
µ
sion as in the current Jy, cf. Eq. (5), and vice versa. become x-dependent Aσ(x). The profile of Aσ(x) then
µ µ
Consequently, the matrix elements (11) vanish for the depends on the incidence angle of the photon. If the
photon polarization in y-direction, but yield a non-zero photon is incident from top, i.e., propagates parallel to
contributionforthephotonpolarizationinx-direction,at theexternalmagneticfieldk B,thetwographenesheets
k
leastifk islargeenough[cf.thediscussionafterEq.(7)]. (topandbottom)haveoppositeprojections. ThusAσ(x)
µ
Moreover, the modes with large currents J and thus is anti-symmetric Aσ( x) = Aσ(x) and the above se-
y µ − − µ
large group velocities dE/dk do also have large matrix lection rules are reversed. If the photon propagates per-
elements, which enhances the magneto-thermoelectricor pendicularlythroughthefold(k B),wegetasymmet-
⊥
magneto-photoelectric effect we are interested in. ric projection function Aσ( x) = Aσ(x) which vanishes
µ − µ
Pseudo-parity Further selection rules arise if we as- far away from the folding region (i.e., for large x). In
| |
sume reflection symmetry B( x) = B(x) and thus this case,the aboveselectionrules do stillapply, but the
− −
A( x)=A(x) which yields the additional symmetry matrix elements might be reduced a bit.
− Example profile In order to visualize the behavior of
ψE,k( x)= iψE,k(x)=i ψE,k(x), (12) themodesbymeansofaconcreteexample,letusconsider
1 − ± 2 PE,k 2
a magnetic field of the following form
wherewecall = 1thepseudo-parityofthismode.
E,k
Recalling thePparticle±-hole symmetry Ψ = σzΨ , B(x)=B0tanh(αx), (13)
−E,k E,k
we find −E,k = E,k. The pseudo-parity of a given where 1/α measures the width of the fold. For α ,
mode caPn be dete−rmPined easily for large and positive we get a step function B(x)=B sign(x) with the v→ect∞or
0
kS,incwehetrhee wweavhea-fvuencitψio2En,kψ≈1E,kv(Fxk)ψo1Ef,kth/Ee lofrwoemstEpqo.sit(i3v)e. peqouteantitoianl(A3()xc)a=n bBe0|sxo|l,vcefd. [e1x7a].ctIlyn(tphiiescleiwmiiste,)thinetmeromdes
mode (for k ) corresponds to the ground state of parabolic cylinder functions, cf. [18]. Incidentally, the
→ ∞
of a harmonic oscillator, it is Gaussian and symmet- spectrum for such a step function B(x)=B sign(x) can
0
ric ψ1E,k(−x) = ψ1E,k(x). Hence this mode has an also arise for some edge states [12, 13].
even pseudo-parity E,k = +1. The wave-function However, such a step function can only be a good ap-
P
ψE,k(x)ofthenextmodecorrespondstothefirstexcited proximation if k is not too large and if the curvature
1
state of aharmonic oscillatorand thus is anti-symmetric radiusofthegraphenefoldismuchsmallerthanthetyp-
ψE,k( x)= ψE,k(x),whichgivesanoddpseudo-parity ical magnetic length scale ℓ = 1/√B. For one Tesla,
1 − − 1 B
= 1 and so on. Together with the above result we get ℓ 26 nm while the radius of curvature cannot
E,k B
P − ≈
= we find that, for a fixed k, the pseudo- be too small since it should be much largerthan the lat-
−E,k E,k
P −P
parity of the modes alternates if we go up and down in tice spacing 0.25 nm. Thus, let us consider a finite α
≈
energy. Assumingthatthe modesdeformcontinuouslyif and take α = 1/ℓ as an example. The spectrum can
B
k changes[i.e.,thatA(x)issufficientlywell-behaved],we then be obtainednumerically andis giveninFig.2. The
may deduce an alternating pseudo-parity for all k. spectraforothervaluesofαarequalitativelysimilar. As
4
demonstratedabove,thetwolowestmodesaremonoton- incidenceangleofthephotons,whichshouldenableusto
ically increasing/decreasing, whereas the higher modes distinguishthis effect from otherphenomena experimen-
can have dE/dk = 0 at some small k-values. For large tally. Furthermore, we find that those modes with com-
k , we recover the asymptotics discussed above. parably large group velocities (i.e., large currents) tend
| |
to have large matrix elements (at least for low-energy
transitions) and thus are more strongly coupled to the
incident photons (i.e., “nature favors our goal”).
Outlook: electric field If we apply an additional elec-
2
tricfieldperpendiculartothefoldandthemagneticfield,
we get an electrostatic potential Φ(x) = βv A(x) with
F
γ some constant β. If we have β < 1 (i.e., if the elec-
ε 0 tric field is sub-critical), we m|ay| transform Φ away by
an effective Lorentz boost in y-direction with a veloc-
ity v = βv where v plays the role of the speed of
boost F F
2 light[19]. IntheLorentzboostedframe,wegetthesame
− modes as discussed above, but with a reduced magnetic
fieldB′ =B 1 β2. Sincethisfieldentersthecharac-
4 2 0 2 4 0 0 −
− − teristic energypscale via vF√2B0, the dispersion relation
κ
after transforming back to laboratory coordinates reads
FIG.2. Dispersion relationofthelowest bandswithκ=kℓB E E′ =E(1 β2)3/4 kvFβ, (14)
and E =εvF√2B0 and sketch of thephoto-absorption. → − −
i.e., the spectrum in Fig. 2 is compressed and tilted.
Conclusions Via the effective Dirac equation (1), we Acknowledgements FruitfuldiscussionswithA.Lorke
studied the low-energy behavior of electronic excitations and M. Schleberger are gratefully acknowledged. This
ingrapheneundertheinfluenceofatransversalmagnetic work was supported by DFG (SFB-TR12).
field B(x) with the asymptotics B(x ) = B .
0
→ ±∞ ±
Such a field profile B(x) arises within a folded graphene
sheetinaconstantmagneticfield,forexample,seeFig.1.
Based on general arguments, we find a discrete set of
∗
modes (see also [20]) which are localized near the fold [email protected]
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