Finite U thermoelectrical transport in graphene based quantum dots

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by  Jose Ramon Isern-Lozano

Finite U thermoelectrical transport in graphene based quantum dots
Jos´e Ram´on Isern-Lozano
Instituto de F´ısica Interdisciplinar y Sistemas Complejos IFISC (UIB-CSIC), E-07122 Palma de Mallorca, Spain
Ioan Grosu and Mircea Crisan
Department of Theoretical Physics, University of Cluj, 400084 Cluj-Napoca, Romania
Rosa L´opez
Instituto de F´ısica Interdisciplinar y Sistemas Complejos IFISC (UIB-CSIC), E-07122 Palma de Mallorca, Spain and
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA
6
1
Jong Soo Lim
0
2 School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea

n
We study the thermoelectrical transports for an interacting dot attached to two graphene elec-
a
trodes. Graphene band structure shows a pseudogap density of states that affects strongly the
J
transport properties. In this work, we focus on the Coulomb blockade regime and derive the ex-
0 pression for Onsager matrix {O } that relates the electrical and heat currents with electrical and
ij
1
thermal biases in the linear response regime. Our findings show double peak structures for the

electrical and thermal conductances versus the dot level in accordance with the Coulom blockade
]
l phenomenon. Remarkably, however, the thermal conductance is much smaller than the electrical
l conductance,resultinginhighfigureofmeritvalueforsomegatevoltage. Finally,wereportalarge
a
h departure from the Wiedemann-Franz law caused mainly by the pseudogap density of states in the
- contacts and weakly affected by interactions.
s
e
m
I. INTRODUCTION in graphene based quantum dots occur at much higher
. temperaturesthanintraditionalsemiconductordots[15].
t
a Besides, under certain circumstances many-body effects,
m
Graphene, a mono-layer of carbon atoms disposed on suchastheKondoeffect[16]canbeobserved[17,18]. In
- an hexagonal lattice was firstly synthesized a decade ago realitygraphenealtersquitestronglyKondophysics[19–
d
n [1, 2]. However, graphene was not an unknown mate- 28]. The Kondo effect in graphene has been intensively
o rial, but it was theoretically investigated long time ago discussedinconnectionwiththemagneticimpurityprob-
c by P. R. Wallance and others [3–8] as a two-dimensional lem in massless Dirac fermions. The experimental mea-
[ graphite material. The unusual properties of graphene surements show that the Kondo effect can be induced

1 are owing to its Dirac-like band structure, where con- by lattice vacancies in graphene sheets [25, 26] . The
v duction and valence bands touch at six discrete points Kondo problem in massless Dirac fermion system is in-
5 at the edges of the honeycomb Brillouin zone. The rel- deedaparticularcaseofthepseudo-gapAndersonmodel
5 ativistic (Dirac) character of graphene was pointed out [29]wherethedensityofstates(DOS)ofconductionelec-
2 by D.P. DiVincenzo and E.J. Mele [9] before it was cre- trons is proportional to |ε|r, with ε being the energy of
2
ated in the lab. In graphene band structure, only two conduction electrons and r an exponent specific for the
0
of the six Dirac cones are nonequivalent being currently material. ThemasslessDiracfermionsystemcorresponds
.
1 named as K and K’ points [1]. The quasiparticle ex- to r = 1. There has been a great effort (theory and ex-
0
citations at those points obey linear Dirac-like energy periment) to describe the Kondo effect in Dirac-like ma-
6
dispersion and are responsible for many physical phe- terials. However, much less attention has received the
1
: nomena such as the relativistic quantum Hall, the Klein Coulomb blockade regime in which charge fluctuations
v tunneling among others [1]. Besides, graphene quasipar- arethedominanteventsintransport. Coulomb-blockade
i
X ticles are chiral fermions with potential applications in effects are ubiquitous and govern the transport proper-
the so-called valleytronics [10, 11]. One of the successful ties of a large variety of systems: quantum dots [30],
r
a graphene applications is found when graphene acts as a molecular bridges, carbon nanotubes [31], etc. Coulomb
conductive material attached to a nanostructure to cre- blockade transport has been investigated extensively in
ate a single-electron transistor circuitry [12, 13]. An in- nanostructures attached to metallic contacts. However,
stanceistheparticularcaseofaquantumdotcarvedina when the contacts are made by graphene the Coulomb
graphene sheet [12–15]. Here, typical Coulomb blockade blockade effect is an almost unexplored problem. The
phenomenon is observed associated to the transport of goalofthisworkistoinvestigatehowCoulombblockade
charges across the localized dot level [12, 13]. The linear effects are altered by the presence of Dirac-like fermions.
conductance exhibits peaks at the dot resonances sepa- ExperimentallyCoulombblockadeoscillationshavebeen
ratedbythemean-dotlevelspacingandelectron-electron reported to occur in graphene quantum dots where the
Coulomb repulsion. Interestingly, transport of charges linear conductance oscillates with the dot gate position
2
are characterized by the electrical (G), thermal (K), and
thermoelectrical (L ) conductances, respectively. Our
12
mainfindingsindicatethatgraphenetransistorsofferex-
traordinary high thermoelectrical efficiencies. We report
values of ZT ≈8. Besides, due to the strong energy de-
pendent tunneling rates, graphene interacting quantum
dots do not follow the Wiedemann-Franz law.
The paper is organized as follows. In Sec. II, our
model for an interacting dot coupled to two Dirac-like
contactsisintroduced. TheOnsagermatrixiscomputed
to describe electrical and heat transports when electri-
cal and thermal biases are applied, and the Seebeck and
ZT coefficients are defined in addition. Since the trans-
portcoefficientsdependonthedotGreen’sfunction,itis
furthermorecalculatedemployingtheequation-of-motion
(EOM) technique [34–37]. Our results for various trans-
ports are thoroughly explained in Sec. III. Finally, Sec-
tion IV summarizes the main achievements of this work.
FIG.1. (a)Twographeneflakesattachedtoaconfinedinter-
acting site. Different temperatures and electrical potentials
areappliedtotheleftandrightgrapheneelectrodes. (b)En-
ergy band diagram for the system under consideration. The II. THEORETICAL MODEL
dotlevelisdenotedbyε . Fordoubleoccupancythedotlevel
dσ
isshiftedbyanamountU thatdescribestheCoulombrepul- We employ an Anderson-like model to describe a spin
sion. ThehybridizationΓ(ε)=γ0|ε−µ|dependsonenergyε. degenerate localized level with strong on-site Coulomb
We consider undoped graphene electrodes with µ = εF = 0 interaction which is coupled to two Dirac-like electrodes
located at the Dirac point.
asshowninFig.1. ThefullHamiltonianconsistsofH=
H +H +H . Thelocalizedlevel(dot)isdescribedby
D G V
(cid:88)
H = ε d†d +Un n , (1)
[12, 13]. D dσ σ σ ↑ ↓
σ
Graphenehasbeenrevealedasasingularmaterialthat
exhibits amazing properties in a plenty of fields, namely, where dσ(d†σ) annihilates (creates) an electron with spin
electronics, spintronics, optics, etc. [1]. A great portion σ = {↑,↓} in the localized level, εdσ denotes the spin-
ofworkshasbeendevotedtothestudyofitspurelyelec- resolved energy level, and U represents the strength of
tronic properties. In comparison, its thermal properties theCoulombinteraction. Here,nσ =d†σdσ istheparticle
arepoorlyinvestigated. Recentmeasurementsshowthat number operator for the localized level. The graphene
graphene offers high thermal conductance values [32]. In contact Hamiltonian reads
this respect, graphene based materials offer new chances (cid:88) (cid:90) +kc
to the progress of the cross-field of thermoelectricity. HG = dk(εk−µ)c†αskσcαskσ (2)
Goodthermoelectricdevicesatnanoscalearethosewhere α=L/R,s,k,σ −kc
the heat dissipated or wasted is efficiently transformed where ε = (cid:126)v k is relatively measured with respect to
k F
into useful electricity and vice-versa [33]. The figure of the chemical potential µ with v ≈ 106m/s being the
F
merit ZT is a coefficient that quantifies the efficiency
grapheneFermivelocity. Thechemicalpotentialµcanbe
in the heat-to-electricity (electricity-to-heat) conversion
tuned by doping techniques and, hereafter, we consider
process. When the heat is only carried out by electrons
µ = ε = 0. We recall that the linear dispersion ε =
F k
thentheZT isproportionaltothesquareoftheSeebeck (cid:126)v k leads to the linear density of states
F
(thermopower) coefficient or the ratio between the elec-
tricalandthermoelectricallinearconductancesunderthe ρ(ε)∝|ε| (3)
open-circuit condition. The Hamiltonian then corresponds to the so-called An-
The purpose of our work is to analyze the thermo- derson pseudogap model [38]. Here, c (c† ) anni-
αskσ αskσ
electric transport through a quantum dot (or interacting hilates (creates) a relativistic electron in the contact α
localized level) coupled to Dirac-like electrodes. We are (L/Rfortheleft/rightcontact)withvalleyindexs,wave-
interestedinatemperatureregimewherechargefluctua- vectork,andspinσ. k isacutoffofthemomentumsuch
c
tions are important and therefore we discard the Kondo that D = ±(cid:126)v k is the cutoff of the energy. Dirac-like
F c
physics. Below we develop the theoretical model to de- quasi-particles are tunnel coupled to the localized level
scribe this system. We will consider the simplest case by means of the hybridization Hamiltonian
where different voltages and temperatures are applied
across the quantum dot. Then, pure electrical, ther- H =V˜ (cid:88)(cid:90) +kcdk(cid:112)|k|(cid:16)c† d +d†c (cid:17), (4)
V αskσ σ σ αskσ
mal, and cross-conductance responses (thermoelectric) iσ −kc
3
√
where V˜ = V πΩ /2π being Ω the area of the of the heat-to-electricity (or vice-versa) conversion pro-
0 0 0
graphene unit cell and V the tunneling amplitude. cesses is measured by the figure of merit ZT
0
S2GT
ZT = . (12)
A. Onsager matrix: transport coefficients K
All transport coefficients are expressed in terms of the
In the next step, we compute the transport coeffi- transmission coefficient T(ε)
cients by employing the Onsager matrix [39] that con-
(cid:88)
nects linearly charge and heat currents with the applied T(ε)=2πΓ(ε) ρ (ε), (13)
dσ
forces,whichinourcasearetheelectricalandthermalbi- σ
ases. Weaimtoinvestigatesuchtransportcoefficientsin
where Γ(ε) = γ |ε| with γ = πV˜2ρ (here, ρ =
Dirac-like systems when Coulomb blockade phenomenon 0 0 0 0
2/((cid:126)v )2). ρ (ε) denotes the local DOS for the inter-
takes place. For such purpose we write down all the On- F dσ
acting level which can be found as
sager matrix elements in terms of the local density of
sdteastcersip(tDioOnSw)ilolfdtehpeenldocoanliztehdeaimpppruoraitcyh.emThpeloryefeodret,ooduer- ρdσ(ε)=−π1(cid:61)(cid:0)Grσ,σ(ε)(cid:1). (14)
rivetheimpurityDOS.Inthelinearresponseregime,the
Onsager matrix reads Grσ,σ is the retarded Green function for the interacting
localizedlevel. Inthefollowing,wederiveaGreen’sfunc-
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
I L L ∆V tionsuitablefortheCoulombblockaderegime. Toattain
= 11 12 . (5)
J L L ∆T such a goal, we employ the EOM technique [35–37] fol-
21 22
lowed by a decoupling procedure (see below). We men-
Here, I and J are the charge and heat currents, respec- tionthatKondocorrelationisnotincludedinourdecou-
tively generated when an electrical bias ∆V = VL−VR pling scheme. However, our approach yields an excellent
and a thermal bias ∆T = TL − TR are applied to characterization of the transport properties of strongly
the left and right contacts. For definiteness, we take interacting quantum dots for temperatures larger than
VL/R =V ±∆V/2andTL,R =T±∆T/2whereV andT the Kondo temperature or when the localized level is
arethebiasandbackgroundtemperaturecommontothe weakly coupled to the electrodes [16]. Besides, it is
two graphene flakes. In an explicit manner, the elements knownthatforzerodopedgrapheneKondoeffectisnever
of the Onsager matrix elements are [40, 41] built.
e2 (cid:90) +∞ (cid:20) ∂f(ε)(cid:21)
L = dε T(ε) − , (6)
11 h ∂ε B. Green’s function
−∞ T,µ
e (cid:90) +∞ (cid:20) ∂f(ε)(cid:21)
L = dε T(ε)(ε−µ) − , (7) A retarded Green’s function Gr(t) for fermionic oper-
12 hT ∂ε
−∞ T,µ ators A and B is defined as
1 (cid:90) +∞ (cid:20) ∂f(ε)(cid:21)
L22 = hT dε T(ε)(ε−µ)2 − ∂ε , (8) GrA,B(t)≡(cid:104)(cid:104)A,B(cid:105)(cid:105)rt =−iθ(t)(cid:104){A(t),B(0)}(cid:105), (15)
−∞ T,µ
whose EOM in energy space takes the following form
where T(ε) is the transmission probability. The On-
sager’s coefficients (L ) are now connected to the linear ε+(cid:104)(cid:104)A,B(cid:105)(cid:105)r+(cid:104)(cid:104)[H,A],B(cid:105)(cid:105)r =(cid:104){A,B}(cid:105), (16)
ij ε ε
electrical (G), thermal (K), and thermoelectrical (L )
12
with H being the Hamiltonian under consideration and
conductances as
ε+ = ε+i0+. For the dot Green’s function Gr (ε) =
σ,σ
G =L , (9) (cid:10)(cid:10)d ,d†(cid:11)(cid:11)r, it is easy to show that
11 σ σ ε
L2
K=L22−TL1121. (10) (cid:0)ε+−εdσ(cid:1)Grσ,σ(ε)=1+U(cid:10)(cid:10)dσnσ¯,d†σ(cid:11)(cid:11)rε
It is worthy to define the Seebeck coefficient or the- +V(cid:101)(cid:88)(cid:90) kc dk (cid:112)|k|(cid:10)(cid:10)cαskσ,d†σ(cid:11)(cid:11)rε . (17)
mopower in terms of the Onsager transport coefficients. α,s −kc
The Seebeck coefficient reads
The equation for (cid:10)(cid:10)c ,d†(cid:11)(cid:11)r is found to be
∆V (cid:12) L αskσ σ ε
S =− (cid:12) = 12 , (11)
∆T(cid:12)I=0 L11 (cid:10)(cid:10)c ,d†(cid:11)(cid:11)r = V(cid:101)(cid:112)|k| Gr (ε) (18)
An important remark here is in order. Note that for αskσ σ ε ε+−εk σ,σ
sufficient low temperatures the thermal conductance is
such that the dot Green’s function becomes
dominated by the electronic contribution and therefore
we can neglect the phonon contribution. The efficiency (ε−ε −Σr(ε))Gr (ε)=1+U(cid:10)(cid:10)d n ,d†(cid:11)(cid:11)r, (19)
dσ 0 σ,σ σ σ¯ σ ε
4
where Σr(ε) = V(cid:101)2(cid:80) (cid:82)kc dk |k| is the self-energy localized level. The self-energy is evaluated as
0 α,s −kc ε+−εk
due to the hybridization between graphene contacts and
Σr(ε)=−η(cid:20)εln(cid:12)(cid:12)(cid:12)D2−ε2(cid:12)(cid:12)(cid:12)+iπ|ε|Θ(D−|ε|)(cid:21) (20)
0 (cid:12) ε2 (cid:12)
where η =2(V˜/(cid:126)v )2 =γ /π.
F 0
InordertogettheCoulombblockadesolution,weneed
to calculate the EOM for (cid:10)(cid:10)d n ,d†(cid:11)(cid:11)r. It is given by
σ σ¯ σ ε
(ε+−εdσ−U)(cid:10)(cid:10)dσnσ¯,d†σ(cid:11)(cid:11)rε =(cid:104)nσ¯(cid:105)+V(cid:101)(cid:88)α,s (cid:90)−kkccdk (cid:112)|k|(cid:104)(cid:10)(cid:10)cαskσnσ¯,d†σ(cid:11)(cid:11)rε+(cid:68)(cid:68)d†σ¯cαskσ¯dσ,d†σ(cid:69)(cid:69)rε
(cid:68)(cid:68) (cid:69)(cid:69)r(cid:105)
− c† d d ,d† . (21)
αskσ¯ σ¯ σ σ
ε
Wekeeponlythecorrelation(cid:10)(cid:10)c n ,d†(cid:11)(cid:11)r ontheright understand the transport properties of our system, in
αskσ σ¯ σ ε
hand side and calculate its EOM which can be approxi- Fig. 2 we briefly discuss the behavior of the dot DOS
mated as and the transmission coefficient. It is shown that for
ε = −U the DOS displays two resonances located at
(ε+−εk)(cid:10)(cid:10)cαskσnσ¯,d†σ(cid:11)(cid:11)rε ≈V(cid:101)(cid:112)|k|(cid:10)(cid:10)dσnσ¯,d†σ(cid:11)(cid:11)rε . (22) εd≈σ εdσ+(cid:60)eΣ0(ε) and ε≈εdσ+U +(cid:60)eΣ0(ε). The res-
onance around the Fermi energy (ε =0) is very narrow
F
Gathering all the results, we obtain
in comparison with the high energy resonance which is
(cid:20) (cid:21) more wider. This is explained by the fact that the hy-
1−(cid:104)n (cid:105) (cid:104)n (cid:105)
Gr (ε)= σ¯ + σ¯ . bridization depends strongly on energy. In addition, the
σ,σ ε−ε −Σr(ε) ε−ε −U −Σr(ε)
dσ 0 dσ 0 dotDOSattheFermienergybecomesmaximum. Never-
(23)
theless,evenifthelocalDOSreachesthehighestvalueat
The poles of Gr (ε) are located around ε and ε +U
σ,σ dσ dσ ε , the transmission coefficient, evaluated at the Fermi
such that this solution properly describes the Coulomb F
energy, vanishes. This is illustrated in Fig. 2(b). The
blockaderegime. Thedotoccupationmustbecalculated
transmissioncoefficientistheproductofthecontactand
self-consistently using
dot density of states. The former vanishes exactly at the
(cid:104)n (cid:105)=(cid:90) +Ddε f(ε)(cid:20)−1(cid:61)(cid:0)Gr (ε)(cid:1)(cid:21). (24) FeveernmitheonuergghytghievidnogtrDisOeStoaattnauinllsttrhaenshmigihsseisotnvcaolueffie.cTiehnet
σ π σ,σ
−D fact that the transmission coefficient gets a zero value at
ε yields a two-dip structure for the electrical conduc-
where f(ε) = 1/(exp(ε/k T) + 1). In the absence of F
B tance versus the dot level position as shown below.
a magnetic field, we expect the paramagnetic solution
(cid:104)n (cid:105)=(cid:104)n (cid:105).
σ σ¯
Inthenextsection,wepresentourresultsforthetrans-
portcoefficientsasafunctionofthedotlevel(IIIA)and A. Transport coefficients versus the gate voltage
the background temperature (IIIB). The Seebeck coeffi-
cient and the ZT figure of merit are also analyzed. Fi- Now we discuss the electrical and thermal conduc-
nally, we illustrate that Dirac-like setup exhibits strong tances when the dot level is tuned. We start by show-
departures from the Wiedemann-Franz law. ing the evolution of the electrical conductance G as a
function of the dot level for several values of the on-site
interaction U (see Fig. 3). For the noninteracting case
III. RESULTS (U/D = 0), the conductance exhibits a dip (antireso-
nance) at the contact Fermi energy ε . In general, the
F
Our results for the transport properties of a graphene resonance happens when the localized level alignes with
based quantum dot are shown. Unless it is indicated, ε . Since the contact DOS vanishes at ε = ε , we ob-
F F
we consider the strong Coulomb blockade regime (U = serve the antiresonance. In the presence of U, there are
0.1D) and investigate how the transport properties de- twoeffectivelevelslocatedatε andε +U. Whenthese
d d
pend on when the dot level is varied and when the back- two levels align with ε , the transmission again vanishes
F
ground temperature is tuned. In these two cases, we due to the nonavailability of states of the graphene con-
explore the electrical and thermal conductances, and tactsatε . Thetransmissionshowsadipthatresembles
F
the Seebeck and ZT coefficients. In order to better a Fano singularity originated when an interference event
5
( 10 2 4e2 /h)
(a) −
80 G ×
k T/D=0.001
B 6
60
) k T/D=0.00082
ε B 4
( U/D= 0.1
σ
d 40
ρ 2
(a)
20 0
0 6
-0.10 -0.08 -0.06 -0.04 -0.02 0.00
4
ε/D U/D= 0.05
2
(b)
40 (b) 0
)
2 6
− 30
0
1× 4 U/D= 0.025
( 20
ε) 2 (c)
(
T10 0
10
0
8
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20
6
ε/D U/D= 0
4
2 (d)
0
FIG. 2. (a) Dot density of states ρ (ε) and (b) transmis-
dσ
sioncoefficientT(ε). Weconsidertwodistincttemperatures: -0.1 -0.075 -0.050 -0.025 0.00
kBT/D=0.001(solidline)and8.2×10−5(dottedline). Rest ε /D
ofparameters: ε =−U,η=0.1,D∼7eV,andU/D=0.1. dσ
dσ
FIG.3. ElectricconductanceGasafunctionofε forU/D=
dσ
occurs. However, notice that the origin of the dip in 0.1 (a), 0.05 (b), 0.025 (c), and 0 (d). Rest of parameters:
η=0.1, D∼7eV, and k T/D=0.001.
our setup is due to the lack of electronic states of the B
graphene contacts instead of being produced by an in-
terference between different paths. Fano resonances are
more dramatic behaviors as a function of ε . The first
usually found in double dot systems [42]. It is remark- dσ
remarkablefactistheabsenceofthedipsattheresonance
able that elevated values for the Seebeck coefficient are
points. Even though electrical transport is blockaded at
encountered in double dots where Fano antiresonances
the resonance points, the heat transports do not follow
take place [43]. Here, as shown later, we find similar be-
such behaviors.
havior, i.e., the occurrence of dips at the transmission
Importantly, K is three orders of magnitude smaller
leads to high thermopower values.
than G. This is a notable feature when compared with
Another aspect of G is that the peaks near to the dips
normal conductors that possess high electrical and ther-
are not symmetric. Around ε = −U, that is, the two
dσ mal conductances at the same time. Good thermo-
maxima are different in their heights. This results from
electrical devices are those that display poor thermal
the fact that the localized level DOS is not symmetric
conductances and high electrical conductances just like
with respect to the resonance points. Notice that this
our graphene transistor. In metallic conductors, the
does not happens in normal contacts.
Wiedemann-Franz law
WefocusonthestronginteractingcasewhereCoulomb
blockade phenomenon is better exhibited. We choose K π2 (cid:18)k (cid:19)2
= B (25)
U/D = 0.1. Our purpose is to analyze the temperature GT 3 e
dependence of the electrical G and thermal K conduc-
tances as shown in Fig. 4(a) and (b), respectively. It is is satisfied. By utilizing graphene contacts, such a rela-
observed that G has moderate temperature dependences tion between G and K does not hold anymore. Conse-
as a function of the gate voltage. In the middle of the quently, the Seebeck coefficient attains much higher val-
valley, the thermal activation is not sufficient to draw ues. This fact is precisely illustrated in Fig. 5(a) where
a quantitative change in G. On the contrary, K shows the Seebeck coefficient is displayed as a function of ε
dσ
6
7.0
4
(a) (a)
) 6.0
h
/ 2
5.0 )
24e 4.0 /e
B 0
2 k
− 3.0 (
0
1×2.0 S 2
(
G1.0 4
0.0
-0.11 -0.10 -0.09 -0.009 0.0 0.007 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02
ε /D ε /D
dσ dσ
5.0 8
) (b) k T/D=0.001 7 (b) kBT/D=0.001
h B
/ 4.0 k T/D=0.00082 6 kBT/D=0.00082
B B
4k 3.0 kBT/D=0.00064 5 kBT/D=0.00064
T
5− kBT/D=0.00046 Z 4 kBT/D=0.00046
0 2.0 3
1
×
2
( 1.0
K 1
0.0 0
-0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 -0.12 -0.10 -0.08 -0.02 0.00 0.02
ε /D ε /D
dσ dσ
FIG. 4. (a) Electric conductance G and (b) thermal conduc- FIG. 5. (a) Seebeck coefficient S and (b) figure of merit ZT
tance K versus ε for various k T/D. Rest of parameters: versus ε for various k T/D. Rest of parameters: η = 0.1,
dσ B dσ B
η=0.1, D∼7eV, and U/D=0.1. D∼7eV, and U/D=0.1.
forvariousbackgroundtemperaturesT. Itisshownthat
the Seebeck coefficient reaches quite high values S (cid:38) 4.
ZT
Asexpected,S vanishesattheresonancepointsε =ε
dσ F 1.2
andε +U =ε . Aroundthesepoints,S isanoddfunc-
dσ F
tion of εdσ. Related with S is the figure of merit ZT. In 1.0 kBT/Γ=1.0
Eq. (12), the thermal conductance K contains the elec- k T/Γ=0.5
B
tronic and phonon contributions (i.e., K = K +K ).
e ph 0.8
Throughthiswork,Kph isneglectedsinceweassumethe kBT/Γ=0.1
low temperature limit (T <80K). Besides, in graphene,
0.6
the thermal conductance due to phonons can be tailored
to get very low values even at room temperature by in-
0.4
cluding anti-dots or nanoribbons in it [44]. Therefore,
ourcalculationsdonotconsiderthephononcontribution
0.2
to the thermal conductance. The reported large values
ofZT indicatethatheat-to-electricityconversionprocess
0.0
is performed with high efficiency. Good thermoelectrical
-20 -15 -10 -5 0 5 10 15 20
conductors can exhibit ZT (cid:38) 1. Our device shows ZT
ε /Γ
values close to 8 as illustrated in Fig. 5(b) which is a dσ
remarkable fact for practical applications. These results
for the figure of merit ZT can be compared with those
FIG. 6. Figure of merit ZT as a function of ε for various
achieved for the case of a localized level tunnel coupled dσ
k T/D. Metalliccontactsareconsideredinsteadofgraphene
tonormalcontactswithanenergyindependenttunneling B
contacts. Restofparameters: Γ=1,D/Γ=100,andU/Γ=
rate Γ. Such comparison is performed in Fig. 6 where
2.
the ZT values for the normal dot case are displayed. We
observe that much smaller ZT values are reached.
7
5 7
(a) (a)
TG4 h) 6
2 / 5
) 2
kB 3 4e 4
π 2
/( 2 0− 3
K 1
×2
2e 1 (
3 G1
0
0
-0.04 -0.02 0.00 0.02 0.04
5 10 15 20 25 30
ε /D
dσ
(b)
5 ) 15
(b) h ε /D= 0.1025
G / dσ −
T 4 B ε /D= 0.1
2 4k 10 dσ −
) ε /D= 0.0975
B 3 5− dσ −
πk kBT/D=0.001 10
/( 2 kBT/D=0.00082 ×5
K (
k T/D=0.00064 K
23e 1 kBT/D=0.00046 0
B
0 5 10 15 20 25 30
-0.10 -0.05 0.00
k T/D ( 10 4)
ε /D B × −
dσ
FIG.8. (a)ElectricalconductanceGand(b)thermalconduc-
FIG. 7. Violation of the Wiedemann-Franz law as a func- tanceKversusthebackgroundtemperaturek T fordifferent
B
tion of εdσ for U/D = 0 (a) and 0.1 (b). As shown, several values of εdσ. At εdσ/D = −0.1025 and −0.0975, maximum
temperatures are considered. Rest of parameters: η = 0.1, values are reached in G, while for ε /D = −0.1 the dip is
dσ
D∼7eV. observed in G (see Fig. 3(a)). Rest of parameters: η = 0.1,
D∼7eV, and U/D=0.1.
InFig.7,weinvestigatetheWiedemann-Franzlaw. As
anticipated,theWiedemann-Franzlawisnotfulfilled. To
pinpoint the main source of such violation, we consider
U =0 and U (cid:54)=0 cases. It is observed that the violation
recall that δ is a function of ε . On the other hand,
dσ
of the Wiedemann-Franz is nothing to do with U. We
K(T) increases linearly with T irrespective of ε .
dσ
thus conclude that the departure from the Wiedemann-
Franz law is due to the Dirac-like energy dispersion rela-
We end up by exploring the behavior of the See-
tion of the contacts.
beck and ZT coefficients. The Seebeck coefficient [see
Fig. 9(a)] vanishes for the resonance point as expected
due to the particle-hole symmetry without respect to
B. Transport properties versus background temperature. Fortheothertwolocalizedlevelvaluesthe
temperature Seebeck coefficient takes opposite signs. This is due to
the fact that for ε /D = −0.1025 the DOS has more
dσ
We now examine G, K, S, and ZT as a function of weight at the left of the Fermi energy (so-called hole-
the background temperature T. Figure 8(a) displays the like transport) whereas for ε /D = −0.0975 the situa-
dσ
electricalconductanceG whenk T isvariedforthreedot tionisreversed(electron-liketransport). Hereisremark-
B
level positions (ε /D = −0.1025, −0.1, and −0.0975). able the fact that at low and moderate temperatures S
dσ
At the resonance point (ε /D = −0.1), G(T) is almost reaches considerable high values (above S > 1). Finally
dσ
flatgettingaverysmallvalueduetothelackofelectronic the ZT coefficient is proportional to S2 and therefore is
statesatthecontactsforenergiesclosetotheDiracpoint. not sign sensitive. For ε /D = −0.1 the ZT is always
dσ
Fortheotherdotlevelpositions,G(T)showsamaximum null whereas for the other two level positions ZT shows
withitspositionthatslightlydependsonthevalueofε . a similar behavior. The ZT has an optimized value at
dσ
The maximum occurs when k T ≈ δ which denotes the certain temperature scale k T∗ reaching the value close
B B
peak width of the dot DOS for each level position. We to ZT ≈8.
8
localized level coupled to Dirac-like electrodes made by
6
graphene. Our results support the fact that Dirac-like
4 (a) electrodes lead to much better thermolectrical devices
withveryhighvaluesfortheSeebeckandZT coefficients.
) 2 ThelattercanreachvaluesofZT ≈8. Indeed,duetothe
e
/ strong energy dependence of the density of states of the
B 0
k electrodesourgraphenedevicedisplayaratherlowther-
(
malconductanceincontrastwiththehighandmoderate
S-2
values reached for the electrical one. These two facts
-4 yield high thermoelectrical efficiencies. Besides, such en-
ergy dependence is responsible for the violation of the
-6
Wiedemann-Franz law. In addition, we report results
5 10 15 20 25 30
for the conductances as a function of the background
temperature. Remarkably, G(T) and ZT(T) shows an
8
(b) optimal value which depends on the dot level position.
On the contrary, K(T) grows monotonically with tem-
6 ε /D= 0.1025 perature. Such growing behavior depends on the dot
dσ −
level position. Finally, as expected the Seebeck coeffi-
ZT 4 εdσ/D=−0.1 cient reverses its sign when transport occurs via quasi-
εdσ/D=−0.0975 holes and changes to quasi-electrons (from εdσ +U < 0
to ε +U >0). We believe that the reported values for
2 dσ
theZT ingraphenebasedquantumdotswillmotivateits
experimental verification using the present technology.
0
5 10 15 20 25 30
k T/D ( 10 4)
B × −
FIG. 9. (a) Seebeck coefficient S and (b) figure of merit ZT
asafunctionofk T whenε /D ischanged. Restofparam-
B dσ
eters: η=0.1, D∼7eV, and U/D=0.1.
V. ACKNOWLEDGEMENT
IV. CONCLUSIONS
Work supported by MINECO Grant No. FIS2014-
Inclosing, wehaveanalyzedthelinearelectrical, ther- 52564 and by the Kavli Institute for Theoretical Physics
mal, and thermoelectrical transports for an interacting through NSF grant PHY11-25915.
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