Valley Seebeck effect in gate tunable zigzag graphene nanoribbons

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by  Zhizhou Yu

Valley Seebeck effect in gate tunable zigzag graphene
nanoribbons
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1
0 Zhizhou Yua,b, Fuming Xua,c,∗, Jian Wanga,b
2
aDepartment of Physics and the Centerof Theoretical and Computational Physics, The
n
Universityof Hong Kong, Pokfulam Road, Hong Kong, China
a bThe Universityof Hong Kong Shenzhen Institute of Research and Innovation, Shenzhen,
J
China
6 cCollege of PhysicsScience and Technology, Shenzhen University,Shenzhen, 518060, China

]
l
l
a
h
- Abstract
s
e
m We propose, for the first time, a valley Seebeck effect in gate tunable zigzag
t. graphenenanoribbonsasaresultoftheinterplaybetweenthermalgradientand
a
m valleytronics. A pure valley current is further generated by the thermal gradi-
-
d ent as well as the external bias. In a broad temperature range, the pure valley
n
current is found to be linearly dependent on the temperature gradient while it
o
c
increaseswith the increasingtemperature of one lead for a fixed thermal gradi-
[

ent. Avalleyfieldeffecttransistor(FET)drivenbythe temperaturegradientis
1
v
proposedthat can turn on and off the pure valley current by gate voltage. The
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2 thresholdgatevoltageandonvalleycurrentareproportionaltothetemperature
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1 gradient. When the systemswitchesonatpositivegatevoltage,the purevalley
0
. current is nearly independent of gate voltage. The valley transconductance is
1
0 upto30µSifwetakeAmpereastheunitofthevalleycurrent. ThisvalleyFET
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1 may find potential application in future valleytronics and valley caloritronics.
:
v
i
X 1. Introduction
r
a
Thermoelectricity has been known since the observation of Seebeck effect
in 1821 which revealed the interplay between thermal gradient and electric po-
tential, opening a way for power generation and refrigeration[1, 2]. Recently,
∗Correspondingauthor
Email address: [email protected] (FumingXu)
Preprint submitted toCarbon January 9, 2017
spincaloritronics,anewfieldcombiningthermoelectronicswithspintronicsthat
describes heat and spin transport,has attracted increasing attention[3, 4]. The
spin Seebeck effect referring to the generation of spin voltage as a result of a
temperature gradient has been observed experimentally in both ferromagnetic
metals and magnetic insulators[5, 6, 7]. Spin voltage, the potential between
different spins, leads to a pure spin current which can be measured by the
inverse spin Hall effect. On the other hand, the discovery of graphene[8, 9],
a two-dimensional atomically thin sheet of carbon atoms, opens a new path
for the next generation green electronics[10, 11]. Zigzag graphene nanorib-
bons (ZGNRs), one-dimensional narrow stripes of graphene showing metallic
characteristics,areparticularlyinterestingforits potentialapplicationsinspin-
tronics and thermoelectronics due to its unique electric properties and high
thermal conductivity[12, 13, 14]. The spin Seebeck effect has been proposed
in ZGNR based materials from the first-principles calculation, which shows
controllable thermal induced spin-polarized currents for graphene-based spin
caloritronics[15, 16].
Apart from the spin degree of freedom, graphene can also be character-
ized by its valley index, namely, the K and K′ Dirac point in the Brillouin
zone[10,11]. Accordingtothetimereversalsymmetry,theelectroncarryingone
valley index shows the different direction of propagation from that of another
valley[17,18,19]. Itisfoundthattheintervalleycouplinginsuspendedgraphene
is very weak[20, 21]. Therefore, being a good quantum number, the valley de-
gree of freedom can be used in ’valleytronics’for the application of information
processing similar to spin used in spintronics[19, 22, 23, 24, 25, 26, 27, 28]. A
valleyfilter wasfirstlyproposedinZGNRbasedballisticpointcontactinwhich
the occupation of a single valley was achieved and such a valley polarization
could be inverted by a local gate voltage[24]. By introducing the line defect in
graphene,a controllable100%valley polarizationhas beenreportedandwidely
studied theoretically[25, 26, 27]. Moreover,the generationofa pure bulk valley
current without net charge current through quantum pumping induced by me-
chanicalvibrationshasbeendemonstratedingraphenebyusingthewell-known
2
Dirac Hamiltonian[28].
In this paper, we explore the valley degree of freedom in the Seebeck effect
of ZGNRs, namely, how to generate a valley current by temperature gradient.
SimilartothespinSeebeckeffect,wecallitvalleySeebeckeffect. Anovelwayof
generatingpurevalleycurrentisfurtherproposedbyapplyingbothtemperature
gradientandbiasvoltage. Wefindthatthepurevalleycurrentislinearwiththe
thermal gradient in a broad temperature range while it increases significantly
with the increasing temperature of leads under a same thermal gradient. We
also present a gate tunable field effect transistor (FET) for the pure valley
current driven by the temperature gradient, revealing a new perspective for
valley caloritronics device applications.
2. Model and formalism
The previous theoretical work shows that the lowest propagating mode for
ZGNRshasafixedvalleyindexwhileforarmchairnanoribboneachpropagating
channel is contributed by the mixed state of both valleys. As a result, the
valley index or valley current for armchair nanoribbon is not well defined [24].
Therefore, ZGNRs are chosen to study the valley effect. In order to control
the valley current, two different gate regions are introduced into ZGNRs. The
first gate with a constant voltage is applied in region I and the second gate
with tunable gate voltage is applied in region II as indicated in Fig. 1(a). In
thetight-bindingapproximation,the HamiltonianforZGNRscanbewrittenas
(here we set ¯h=q =1 for simplicity),
H =t c†c +H.c.+ v c c† + v c c†, (1)
i j g1 m m g2 n n
X X X
hi,ji m∈I n∈II
wherec (c†)annihilates(creates)anelectrononsiteiofZGNRs. v ,v denote
i i g1 g2
the gatevoltageintroduced inregionI and regionII, respectively. h...i refersto
thenearest-neighboringsitesandtisthenearestneighborhoppingenergywhich
is setto be 2.7eV[11]. Inourcalculation,the width andlengthofthe proposed
3
ZGNRsbasedvalleySeebeckdeviceare28.4nmand99.7nm,respectively. The
length of each gate region is set to be 25.8 nm.
The electric current can be obtained from the Landauer-Bu¨ttiker formula,
dE
I = (f −f )T(E), (2)
Z 2π L R
where f is the Fermi-Dirac distribution defined as,
1
f (E,T)= , (3)
α
exp[(E−E )/k T ]+1
F B α
with the Fermi energy E , the Boltzmann constant k , and the temperature
F B
T in lead α. T(E) is the transmission coefficient,
α
T(E)=Tr[Γ GrΓ Ga], (4)
L R
whereGr(a) istheretarded(advanced)Green’sfunctionandΓ isthelinewidth
α
function of lead α. Denoting I and I′ as the particle current of electron
K K
carrying valley index K and K′, respectively, we define the valley current as,
Iv =IK −IK′. (5)
From the band structure of ZGNRs, it can be found that the momentum and
valleyindex ofelectroninthe firstsubbandarelockedtogethersothatthe left-
movingelectronhasvalleyindexK whileelectronwithvalleyindexK′movesto
the right as shown in Fig. 1(a). Such a unique electronic property of ZGNRs is
independent ofthe ribbonwidth. Becauseata givenenergythe signoff −f
L R
determinesthe directionofelectronflowandhence valleyindex,wecanexpress
the valley current of ZGNRs as,
dE
I = sgn(f −f )(f −f )T(E). (6)
v Z 2π L R L R
whichwillbe usedinthe calculationofvalley current. Tomakethe comparison
withelectriccurrent,inthefollowingcalculation,wewilltaketheunitofelectric
current as the unit of valley current.
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(a)

(b)
1.0
n 0.8
o
si
s
mi 0.6
s
n
a
Tr 0.4
0.2
0.0
-0.4 -0.2 0.0 0.2 0.4
Energy (eV)
Figure 1: (a) Schematic diagram of ZGNRs with two semi-infinity leads (blue shadow), two
static gate regions with vg1 = 0.5 V (red shadow) and vg2 is tunable in the central region
(orangeshadow). (b)TransmissionspectrumofZGNRswithvg1=0.5Vandvg2=0.
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3. Results and Discussion
It is well known that there is a conductance plateau of 1G around the
0
Fermi level for pristine ZGNRs which is symmetrical about the Fermi level,
i.e., T(E) = T(−E) with respect to the Fermi level. We also know that the
difference of the Fermi-Dirac functions f (E)−f (E) = f (−E)−f (−E) is
L R R L
anoddfunctionofenergymeasuredattheFermilevel. Asaconsequence,there
is no electric current due to the temperature gradient since the contribution of
current below and above the Fermi level exactly cancel each other in Eq.(2),
whileapurevalleycurrentcanbegeneratedfromthedefinitionofEq.(6). This
is the simplest but trivial way to achieve the valley Seebeck effect.
InordertoachievethevalleySeebeckeffectthatcanbeefficientlycontrolled,
one has to break the symmetry of T(E)=T(−E). We first apply a static gate
voltage in region I with v = 0.5 V so that the system becomes a double
g1
barriertunnelingstructure. Thetransmissionspectrumofthesystemisplotted
in Fig. 1(b). We see that the transmission plateau below the Fermi level still
remains while its transmission coefficient oscillating around 0.99G instead of
0
1G . On the other hand, only several resonant peaks exist above the Fermi
0
level. Therefore, the symmetry of the transmission coefficients with respect to
energy is broken so that the non-trivial valley Seebeck effect could be studied.
Figure2(a)presentstheelectricandvalleycurrentsasafunctionoftemper-
aturegradient. Thethermalinducedelectriccurrentiscausedbyatemperature
gradient(∆T)betweenthe left electrode (T ) andrightelectrode (T ) without
L R
the external bias voltage. We first set the temperature of the right lead to be
zero. Wefindthattheelectriccurrentdependslinearlyonthetemperaturegra-
dient and the differential thermoelectric conductance dI/dT is 5.6 nA/K. The
dependent of valley current on the temperature gradient is also linear with a
slightly larger slope. The valley current at ∆T = 200 K is only 10 nA higher
thantheelectriccurrentatthe sametemperaturedifferenceandthedI /dT ra-
v
tio is 5.7 nA/K. To further study the valley Seebeck effect in the linear regime,
namely, ∆T < T, we calculate the electric and valley current at different tem-
6
(a) 1.2 0.3
A)
1.0 current ( 00..12
A)
ent ( 0.8 0.00 10 20T (K3)0 40 50
urr 0.6
c
0.4 electric current
valley current
0.2
50 100 150 200
T (K)

0.115
(b)
0.114
)
nt ( 0.113
e
curr 0.112
0.111
electric current
valley current
0.110
50 100 150 200
temperature (K)
Figure 2: (a) Electric and valley currents as a function of ∆T with fixed TR = 0 K. Inset:
Electricandvalleycurrentsasafunctionof∆T withfixedTL=200K.(b)Electricandvalley
currentsasafunctionofTL withfixed∆T =20K.
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peraturegradientsby fixingT =200K while keepingT >T togeneratethe
L L R
positive electric current, as shown in the inset of Fig. 2(a). Both the electric
and valley current show linear dependence on the temperature gradient with
the same differential thermoelectric conductance as those of T =0 K, namely,
R
5.6 nA/K and 5.7 nA/K, respectively.
Wealsostudytheelectricandvalleycurrentforafixedtemperaturegradient
while varying the temperatures of both leads. Fig. 2(b) shows the current as a
functionofthetemperatureofleftleadwith∆T =20Kasanexample. Wefind
that the electric current first increases quickly with the increasing temperature
and reaches a maximum about 112.2 nA at T = 131 K. It then decreases
L
slightly due to the contribution of the transmission peak at 0.05 eV and the
electriccurrentreducesto111.9nAatT =200K.Thevalleycurrentishigher
L
than the electric current for all temperatures of left lead with the same fixed
temperature gradientsince alltransmissioncoefficients contribute positively on
itanditenhancessignificantlywiththeincreasingtemperatureofleftlead. The
valley currentis about 111.1nA at T =50 K and raisesto 114.5 nA when the
L
temperature of left lead increases to 200 K.
In order to obtain a pure valley current without the accompanying electric
current,anexternalbiasisthenappliedtobalancetheelectriccurrent. Fig.3(a)
presentsthepurevalleycurrentandtheappliedbiasasafunctionoftemperature
gradient with T = 0 K. An external bias is applied to make electric current
R
vanish while the valley current is nonzero. Obviously, this bias depends on the
temperature gradient linearly. We find that the bias increases from 3 mV at
∆T = 50 K to 12.1 mV at ∆T = 200 K with a dV /dT ratio of 61 µV/K. In
b
contrastto the vanished electric current, the valley currentis enhanced slightly
compared with the case without external bias as shown in Fig. 2(a) with a
dI /dT ratioof6.6nA/K.Itis1316nAunderthetemperaturegradientof200K,
v
whichis186nAhigherthanthatwithnoexternalbias. Thepurevalleycurrent
and the corresponding external bias at different temperature gradients with
T = 200 K are also plotted in the inset of Fig. 3(a). Both valley current and
L
applied bias exhibit nearly linear characteristics with the dI /dT and dV /dT
v b
8
(a) 1.4 14
1.2 12
A)
valley current ( 00001.....24680 valley current (A) 000...123 246 bias (mV) 246810 bias (mV)
0.0 0
0 10 20 30 40 50
T (K)
0.0 0
50 100 150 200
T (K)

(b) 0.075 2.9
2.8
A)0.070
nt ( 2.7 V)
e m
ey cur0.065 2.6 bias (
all
v0.060
2.5
0.055 2.4
50 100 150 200
temperature (K)
Figure3: (a)Valleycurrent(blacksolidline)andthecorrespondingexternalbias(bluedotted
line)asafunctionof∆T withfixedTR=0K.Inset: Valleycurrent(blacksolidline)andthe
corresponding external bias (blue dotted line) as a function of ∆T with fixed TL = 200 K.
(b)Valleycurrent(blacksolidline)andthecorrespondingexternalbias(bluedottedline)as
afunctionofTL withfixed∆T =20K.
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ratio of 4.1 nA/K and 107 µV/K, respectively. We find that the pure valley
current is a little bit smaller than the valley current without external bias as
shownin the inset of Fig. 2(a) at the same temperature gradient. For instance,
at∆T =20K,thepurevalleycurrentis77nAwithanexternalbiasof2.28mV
while the valley current without bias is 114 nA.
For a fixed temperature gradient of 20 K, the pure valley current and the
corresponding applied bias as a function of the temperature of left lead are
plotted in Fig. 3(b). We find that the valley current shows significant increase
from 56.8 nA at T =50 K to 70.9 nA at T =200 K, while the corresponding
L L
externalbias reduces from 2.82mV to 2.45 mV because the contributionof the
transmission coefficients above the Fermi level on the current becomes smaller
at higher temperatures of both leads with the same temperature gradient.
In order to control the pure valley current, we then introduce another gate
voltage in region II described in Fig. 1(a), namely, various values of v is used
g2
in the Hamiltonian as defined in Eq. (1). This setup is a prototype of valley
FET. The pure valley current as a function of gate voltage v under several
g2
temperature gradients with T = 0 K is plotted in Fig. 4(a). We find that
R
under a negative gate voltage such as -0.05 V, the transmission plateau below
the Fermi level shifts by 0.05 eV to the lower energy compared with that for
the case of v = 0 while the resonant transmission peaks above the Fermi
g2
level shift to the higher energy as shown in the inset of Fig. 4(a). Therefore,
there is a transmission gap of 0.06 eV around the Fermi level leading to a very
smallvalley currentunder a gate voltageof-0.05V. When we increasethe gate
voltage, the transmission plateau moves towards the Fermi level which reduces
the transmission gap, resulting in a significant increase of valley current. The
threshold gate voltage and on valley current of the system are proportional to
thetemperaturegradientandthevalleycurrentincreasestoamaximalvalueat
the neutralgatevoltage,whichshowsthe potentialapplicationas a valleyFET
driven by the temperature gradient. The valley transconductances dI /dv
v g2
of such a prototypical thermoelectrical valley FET are 27 µS for ∆T = 20 K
and around30 µS for the cases of ∆T =40,60,80and100 K, which are almost
10