Thermoelectric three-terminal hopping transport through one-dimensional nanosystems

0.87 MB·English

by  Jian-Hua Jiang

Thermoelectric three-terminal hopping transport through one-dimensional
nanosystems
Jian-Hua Jiang,1 Ora Entin-Wohlman,2 and Yoseph Imry1
1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
2Department of Physics and the Ilse Katz Center for Meso- and Nano-Scale Science and Technology,
Ben Gurion University, Beer Sheva 84105, Israel
(Dated: January 24, 2012)
2 A two-site nanostructure (e.g, a “molecule” ) bridging two conducting leads and connected to a
1 phonon bath is considered. The two relevant levels closest to the Fermi energy are connected each
0 to its lead. The leads have slightly different temperatures and chemical potentials and the nanos-
2 tructureis also coupled to a thermal (third) phonon bath. The 3×3 linear transport (“Onsager”)
n matrix is evaluated, along with theensuing new figureof merit, and found to be very favorable for
a thermoelectric energy conversion.
J
PACSnumbers: 84.60.Rb,72.20.Pa,72.20.Ee
2
2

I. INTRODUCTION ties. Although thermal transport properties of meso-
]
l scopic structures have been studied in the past,2,3,10,11
l
a investigations of three-terminal thermoelectric transport
h In thermoelectric transport temperature differences are just at their infancy.12 Our main conclusions are
- can be converted to (or generated by) electric voltages.
s drawn from the simple, but important, two localized-
Such phenomena have already found several useful ap-
e state junction in which hopping is nearest-neighbor.13
m plications. Current research is motivated by the need
Later,webrieflydiscusslarger1Dsystems,whichexhibit
for higher performance thermoelectrics as well as the
. rathersurprisingfeaturesofthethermoelectrictransport.
t pursuit of understanding of various relevant microscopic
a We show that such systems may have a high figure of
m processes (especially the inelastic ones). Theory1–3 pre-
merit,asκ canbecome extremelysmall,while the ther-
dicts that high values of the thermopower follow when e
- mopower, S, remains finite. Some of the thermoelectric
d the carriers’ conductivity depends strongly on energy.
transport coefficients we find correspond to transferring
n Indeed, in bulk systems, the thermoelectric effects ne-
o cessitate electron-hole asymmetry, which is often rather electric/thermal current via temperature difference be-
c tween the electron system and a suitable phonon bath.
small. However, in nanosystems, such asymmetry can
[ Hopefully, such systems can be achieved within current
arise in individual samples in ensembles with electron-
technology and be useful in applications.
2 hole symmetry onaverage. Moreover,inelastic processes
v andinterferenceeffects mayplaynontrivialrolesinther- There are several related ideas in the literature. Ref-
1 moelectric transport.3 It is known that the thermoelec- erence 14 presented an early, ingenious, way to cool a fi-
3 nite 2Delectrongas (which playsthe role of the thermal
tric performance is governed by the dimensionless figure
0 bath) at low temperatures by elastic electron transitions
4 of merit ZT,4 where T is the common temperature of
to/from the leads. All the energies involved are only of
1. the system and Z = σS2/(κe +κph), with σ being the order k T. Reference 15 provided an experimental real-
electrical conductivity, S the Seebeck coefficient, and κ B
0 e izationofsomeofthesuggestionsofRef.14,withfurther
and κ the electronic and the phononic heat conduc-
2 ph
analysis. Reference 16demonstrateda quantum ratchet,
1 tivities, respectively. Both κe and κph can be smaller
converting the nonequilibrium noise of a nearby quan-
: in nanosystems5 than in bulk ones, opening a route for
v tum point contactto dc current. Reference 17 suggested
better thermoelectrics.
i a sophisticated Carbon nanotube structure, designed to
X Mahan and Sofo6 have argued that the best thermo-
extractenergyfromadiscretelocaloscillatoratultralow
r electricefficiencycanbeachievedinsystemswherei)the
a temperatures. The presentworkconsidersthe full three-
energywidthofthemainconductingchannelisverynar-
terminal case, where the energies involved can be larger
row, and ii) the phonon thermal conductivity is as small
than k T, and a real reservoir can be cooled, not just
B
as possible. It was suggested that ii) can be also real-
one or several degrees of freedom.
izedinnanoscalecompositestructureswherephononsare
scattered by large variations in geometry and abundant
interfaces,7,8 while i) leads6 to a very small κ /(S2σ),
e
II. MODEL SYSTEM
as indeed has been confirmed in studies of quantum dot
arrays.9
A. Hamiltonian
Here we consider three-terminal thermoelectric trans-
port in small one-dimensional (1D) nanosystems accom-
plished via inelastic phonon-assisted hopping, and show The Hamiltonian, H = H + H +H , consists
e e−ph ph
that such processes lead to several nontrivial proper- of the electronic and phononic parts and the electron-
2
phonon interaction. The electronic part is (electronic and N is the Bose function
B
operators are denoted by c and c†)
ω −1
q
He =XEic†ici+Xǫk(p)c†k(p)ck(p) NB =hexp(cid:16)kBTph(cid:17)−1i , (6)
i k(p)
+ J c†c + J c†c +H.c. . (1) determined by Tph, the temperature of the local phonon
(cid:16)X i,k(p) i k(p) X i,i+1 i i+1 (cid:17) bath (see Fig. 1). We assume that this phonon bath is
i,k(p) i
strongly coupled to a thermal reservoir and is thermally
Here i labels the localized states, of energies E (includ- isolated as much as possible from the leads, such that
i
ing usual Coulomb-blockade effects; i.e., it is assumed its temperature is determined by that reservoir. On the
implicitly that a large Hubbard interaction confines the other hand, phonons in the leads are in good thermal
occupation of each level to be 0 or 1) and k (p) marks contactwiththeelectronsthereandsharethesametem-
the extendedstates in the left (right)lead,of energiesǫ perature. Theseassumptionsarefurtherelaboratedupon
k
(ǫ ) (all energies are measured from the common chemi- in Sec. IIIC. In Eq. (4), Γ =|M |2ν (|E |), where
p in q,ij ph ij
calpotential). Thematrixelementcouplingthelocalized ν is the phonon density of states. The linear hopping
ph
states to each other is J , and those coupling them to conductanceatlongdistances(|x −x |≡|x |≫ξ)and
i,j i j ij
theleadstatesareJ . Allareexponentiallydecaying, high energies (|E |, |E |≫k T) of the bond ij is18
i,k(p) i j B
with a localization length ξ, e.g.,
e2
J =α exp −|xi−xL(R)| , (2) Gij ∼ k T|αe−ph|2νph(|Eij|)ηi−j1 ,
i,k(p) e (cid:16) ξ (cid:17) B
2|x | |E |+|E |+|E |
with xi and xL(R) being the coordinates of the center of ηij =exp(cid:16) ξij (cid:17)exp(cid:16) i 2kjT ij (cid:17) . (7)
thelocalizedstatesandtheleft(right)boundary,andthe B
prefactor α yielding the coupling energy. The electron-
e AsopposedtoEq. (4),thetunnelingconductionfrom,
phonon interaction is
say, site i to the left lead can be accomplished by elas-
H = M c†c (a +a† )+H.c. , (3) tic tunneling processes with a transition rate ΓiL =
e−ph X q,ij i j q −q γ f [1−f (E )],whereγ =2π|J |2ν (E )andf and
q iL i L i iL ik L i L
ν aretheFermidistributionanddensityofstatesofthe
L
where the phonon modes, of wave vector q and fre-
leftlead. Thecorrespondinglinearinterfaceconductance
quency ωq, are described by the operators a†q, aq. Their is then GiL ≃ e2|αe|2νL(Ei)(kBT)−1exp[−2|xiL|/ξ −
H~am=ilt1o)n.ianTihseHeplhec=troPn-pqhωoqnao†qnaqco(wupeliunsge iusniMtsqw,ijher=e d|Euic|t/a(nkcBeTa)t].thTehrisighcotnldeaudc)tawniclel b(aenadsstuhmeeidntteorfbaceemcuocnh-
α exp(−|x −x |/ξ), with α being the electron- larger than the hopping conductance between the two
e−ph i j e−ph
phonon coupling energy. The transportthrough the sys- localized states.
tem is governed by hopping when the temperature is
above a crossover temperature, T , estimated below in
x
Sec. IIC for the most important two-site case. At lower
C. The two-site case
temperatures the dominant transport is via tunneling.
The two-site example of our system is depicted in the
The thermopower in the hopping regime has been
upper panel of Fig. 1.
discussed by Zvyagin.19 The simplest example is that
of a two-site system (i,j = 1,2) depicted in Fig. 1,
which describes, e.g. a diatomic molecule13 or a series-
B. Hopping and interface resistors
connected double quantum dot.9 In such a case trans-
port is accomplished by nearest-neighbor hopping. As
The system described above bridges two electronic
site 1 (2) is in a good contact with left (right) lead, we
leads, held at slightly different temperatures and chemi-
may assume that the local chemical potential and tem-
calpotentials,T ,µ ,andT ,µ ,suchthatthecommon
L L R R perature there are µ and T . The transport is
temperatureisT ≡(T +T )/2. Thegolden-ruletransi- L(R) L(R)
L R dominated by the hopping from 1 to 2 when the tem-
tion rate Γ , between two localized states, located at x
ij i perature is higher than T . This temperature is esti-
and x and having energies E < 0 < E ,18 necessitates x
j i j mated from the requirement that the elastic tunneling
the inelastic electron-phonon scattering (3), and reads
conductance across the system, G , is comparable to
tun
Γ =2πΓ f (1−f )N (E ) , (4) the hopping one. The former is given by the trans-
ij in i j B ji
mission Γ (E)Γ (E)/[(E − E )2 + (Γ (E) +
where E ≡ E −E , the carriers’ local Fermi function Pi=1,2 iL iR i iL
ji j i Γ (E))2/4], where the tunneling rates are Γ (E) =
is iR iL(R)
2π|J |2ν (E). Since site 1 (2) is coupled mostly
E −µ −1 i,k(p) L(R)
f = exp i i +1 , (5) to the left (right) lead, we use their perturbation-theory
i h (cid:16) kBTi (cid:17) i mixtures, governed by the small parameter J12/E21. At
3
For an electron transferred from left to right, the bath
SSSSSooooouuuuurrrrrccccceeeee jjjjj JJJJJjjjjj,,,,,ppppp DDDDDrrrrraaaaaiiiiinnnnn gives an energy −E1 (E2) to the left (right) lead, and
MMMMM thusthephononstransfertheenergyE totheelectrons.
µµµµµ qqqqq,,,,,iiiiijjjjj µµµµµ 21
TTTTT11111 JJJJJiiiii,,,,,kkkkk iiiii TTTTT22222 A net energy of E ≡[E1+E2]/2 is transferred from left
11111 qqqqq 22222 to right. Hence, the net electronic energy current, Ie,
Q
and the heat current21 exchanged between the electrons
and the phonons, Ipe, are
Q
aaaaa PPPPPhhhhhooooonnnnnooooonnnnn bbbbbaaaaattttthhhhh TTTTT
ppppphhhhh
Ie =EI , and Ipe =E I . (11)
Q N Q 21 N
The linear-response transport coefficients are obtained
SSSSSooooouuuuurrrrrccccceeeee GGGGG DDDDDrrrrraaaaaiiiiinnnnn by expanding Eqs. (10) and (11) to first order in δT ≡
22222RRRRR
µµµµµ GGGGG1111122222 µµµµµ TL−TR, δµ≡µL−µR, and ∆T ≡Tph−T,
11111 GGGGG xxxxx,,,,, EEEEE 22222
11111LLLLL 22222 22222
TTTTT TTTTT
11111 22222 I G L L δµ/e
xxxxx11111,,,,, EEEEE11111  Iee = L K10 L2  δT/T  , (12)
Q 1 e 3
IQpe   L2 L3 Kpe ∆T/T 
bbbbb PPPPPhhhhhooooonnnnnooooonnnnn bbbbbaaaaattttthhhhh TTTTT
ppppphhhhh
where I = eI is the charge current. All transport
e N
coefficients in Eq. (12) are given in terms of the linear
FIG. 1. (Color online) a. A two localized-state (i and j, hopping conductance G [given by Eqs. (7)],
graypoints)systemcoupledtotwoleads,oftemperaturesTL
and TR, and chemical potentials µL and µR (with the choice G G
µL > µR and TL > TR). The phonon bath temperature is L1 = eE , L2 = eE21 ,
Tcopnht.inTuuhme loocfasltizaetdessitnattehsealreeadcso,upanledda(rdeotatlesdo clionueps)ledto(tthhee K0 = GE2 , L = GEE , K = GE2 . (13)
wavyline)tothephononbath;b. Theeffectiveresistors rep- e e2 3 e2 21 pe e2 21
resentingthesystem: Thestraight(blue)arrowsindicatethe
Note that L , L , and K are related to E , and Ipe
netelectroniccurrentsandthewavy(brown)onethephonon 2 3 pe 21 Q
vanishes linearly with the latter.
heatcurrent,withG1L,G2R,andG12beingtheconductances
of the tunnelingand the hoppingresistors, respectively. The transport coefficients L2 (L3) correspond to,
e.g., generating electronic current (energy current) via
the temperature difference ∆T.12 When reversed, this
low temperatures and for |E1|,|E2| ≫ Γ1,Γ2, where process performs as a refrigerator: Electric current
Γi ≡ΓiL+ΓiR., we estimate pumps heat current away from the phononic system
2W and cools it down. In analogy with the usual two-
G ∼e2E−2|α |6ν (0)ν (0)E−2exp − , (8)
tun 1 e L R 2 (cid:16) ξ (cid:17) terminal thermopower S, here we use the three-terminal
thermopower12 of this process,
where W is the system length between the leads. The
hopping conductance is given by Eqs. (7) (with i,j = L k E
S = 2 = B 21 . (14)
1,2). Comparing those two, with exponential accuracy, p TG e k T
theelastictunnelingmechanismcanbeimportant20 only B
when NotethatS ofourmodelcanbeverylargeastheenergy
p
|E |+|E |+|E | W −|x | taken from the phonons per transferred electron can be
1 2 12 12
exp − ≪exp −2
h 2k T i h ξ i several times kBT.
B
→ η ≫exp(2W/ξ) , (9)
12
giving k T ∼(|E |+|E |+|E |)ξ/[4(W −|x |)]. B. Two-terminal figure of merit, for ∆T =0
B x 1 2 12 12
Asignificantfeature ofoursetup is thatthe electronic
III. THREE-TERMINAL THERMOELECTRIC
heat conductance can vanish while the thermopower
LINEAR TRANSPORT
stays finite
A. Transport equations L2 L k E
K =K0− 1 =0 , S = 1 = B . (15)
e e G TG e k T
B
The electronic particle current through the system is,
as in Eq. (4) allowing for the temperature and chemical According to Ref. 6, the largest two-terminal figure of
potential differences, merit is achieved in systems with the smallest κe/σS2
(provided that S stays finite). Here this ratio vanishes,
I =Γ −Γ . (10)
N 12 21 andthenZ islimitedbyκph. Thelattercanbeminutein
4
nanosystems.5Moreover,itcanbereducedbymanipulat- This has a straightforward physical interpretation: The
ing phonon disorder and/or phonon-interface scattering wasted work is due to the elastic conductance and the
(avoidingconcomitantlydrasticchangesintheelectronic unwanted heat diffusion, and Z˜T is limited by the ratio
system). Our system is then expected to possess a high of the waste to the useful powers. In nanosystems K
pp
figure of merit. canbelimitedbythecontactbetweenthesystemandthe
leads. Hence the ratios can be made small and Z˜T can
stillbelarge. Thethree-terminaldevicecanalsoserveas
C. Three-terminal figure of merit aheaterandasathermoelectricbattery,wherethesame
figure of merit describes the efficiency.4
The three-terminal geometry suggests novel possibili-
ties for thermoelectric applications. For example, when 3 100
tre∆haleTtecetl<oroicfc0aatlhlapewnhdohorenδkaµotinn>pvsuye0mss,ηttptee=hdemde,,IsfQwpereoth/umo(psIeetsδheeµerffiv)pce.hiseonanscoyanirsseygfsritvigeemenrabttoyo(rtt1hh6oee)f ln(G/G) and eS/k0B ---- 0124321 ln(G/G0)+1S0 Relative change in S 111000---321
-5 10-4
Consider first the special situation with E = 0, where 0 2 4 6 8 10 12 14 16 1 1.5 2 2.5
L1 = Ke0 = L3 = 0. For a given ∆T, δµ is adjusted to a Ncom b W / ( 4 LM )
optimize the efficiency, yielding
η =η (2+Z˜T −2 Z˜T +1)/(Z˜T) , (17) FIG. 2. (Color online) a. The conductance, ln(G/G0), and
0 p thermopower,eS/kB;theabscissagivesthenumberofcompu-
where η =T/|∆T|is the Carnotefficiency, andthe new tations, Ncom. The parameters are W = 800,LM = 20,ξ =
0 3.3, in units average nearest-neighbor distance, T0 = 3000,
figure of merit is
T = 20, and the energy-band width is 1090 (T0 and LM are
Z˜T =L2/(GK −L2) . (18) the Mott19 temperature and length, defined in the text). b.
2 pe 2 The relative change in S as a function of the system length
Inserting here Eqs. (13) yields Z˜T → ∞ upon neglect- W/(4LM),obtainedbyaveraging over106 randomconfigura-
tions. Parameters (except W)are thesame as in a.
ingthe“parasitic”conductances,discussedbelow. When
E 6= 0, such an optimization can be achieved by setting
δT =0.
In reality, Z˜T must be finite. To calculate a more re-
D. Longer 1D systems
alistic efficiency, we generalize Eq. (12) by adding the
elastic transmission, the tunneling conductance G , to
el
the hoppingconductanceG, andthe elasticcomponents, For a chain of localized states, the picture is similar
L and K0 , to L and K0. (The elastic transmission though slightly more complex. Consider first nearest-
1,el e,el 1 e
doesnotcontributetoL ,L ,andK ,whicharerelated neighbor hopping, where the system is a chain of resis-
2 3 pe
to the heattransferbetweenthe electronicandphononic tors. The same considerations as in the two-site case
systems.12)Wealsoincludethephononheatconductance [in which the energy transferred is determined by site
K , replacing Ie by I , the total heat current from the 1 (2) and the left (right) lead] hold here for the left-
p Q Q
most,ℓ,(rightmost,r)localizedstateandthe left(right)
left to the right lead (δT is now also the temperature
lead. Hence, the thermoelectric transport is described
difference for the phononic systems in the left and right
by Eqs. (11)-(14), with E (E ) replaced by E (E ).
leads). Due to the absence of phonon-drag effects in lo- 1 2 ℓ r
In particular, the thermopower coefficients S and S are
calizedelectronicsystems,thetemperaturedifferenceδT p
completely determined at the left and right boundaries,
should not contribute to other currents beside I . Fi-
Q
despite the fact that transport coefficients are usually
nally, there are phononic heat flows from the two leads
determined by both the boundaries and the “bulk”.
to the system being cooled. Hence the numerator of Eq.
(16)is replacedbyIpe−K ∆T/T,whereK describes Inthevariable-rangehoppingregime,theresultissimi-
Q pp pp lar: SandS aredeterminedbytheresistorsclosedtothe
the phononic heat conductance in such processes. p
left and right boundaries, within a distance comparable
Following the same procedure as above, the efficiency
totheMottlength. Thisobservationisconfirmedbynu-
is optimized by adjusting δµ at δT = 0. The result is
mericalsimulations. In Fig. 2a we plot the thermopower
similar, except that the figure of merit is modified,
S andtheconductanceGfordifferentrandomconfigura-
L2 tionsinwhichtheenergiesE andlocationsx ofthesites
Z˜T = 2 i i
(G+Gel)(Kpp+Kpe)−L22 are random: The Ei are chosenfrom a uniform distribu-
tioninthe range[−E ,E ], withE =545(units
G K G K −1 max max max
= el + pp + el pp . (19) are defined in the figure caption) being larger than the
h G Kpe GKpe i hopping energy, ∼ (T0T)1/2, determined by the Mott22
5
temperature T which is of the order of the levelspacing der of [ξ/(νk T)]1/(d+1), where ν is the density of states
0 B
on scale ξ). The locations x are chosen from a uniform at the Fermi level.
i
distributionintherange[0,W],whereW isthelengthof
thesampleinunitsofthenearest-neighbordistance. The
conductance of the whole network is calculated by solv-
ingtheKirchhoff’sequations,22whichyieldsthecurrents
through each bond for a given source-drain bias. The IV. SUMMARY AND CONCLUSIONS
thermopowerS isobtainedfromtheparticlecurrent,I ,
N
andthe heatcurrentIQe viatherelationS =IQe/(INeT). In this work we studied the three-terminal thermo-
The heat current Ie (particle current) is calculated electric transport and thermopower mainly in simple
Q
by summing over the heat currents (particle currents) 1D hopping systems in the linear-reponse regime. We
flowing through all the bonds connected with the leads. worked out the figure of merit for the three-terminal
In each such bond (iL) [(iR)] which connects the i−th thermopowerandexpressedit asa function ofthe three-
localized state to the left (right) lead, the heat current terminal thermoelectric transport coefficients. We ob-
flow is I(iL) =E I(iL) [I(iR) =E I(iR)] with I(iL) [I(iR)] tained expressions for the thermoelectric transport coef-
Q i N Q i N N N ficients in the simple two-site case. The system studied
being the particle current in that bond. The total heat
exhibits a large thermopower and high figure of merit in
and particle currents are Ie = 0.5 (I(iL)+I(iR)) and
Q Pi Q Q theappropriatecases. Weanalyzetheconditionsforhigh
I = I(iL) = I(iR), respsectively. figure of merit in reality.
N Pi N Pi N
At each N −th computation, with N being an For longer 1D chains, we found that, contrary to in-
com com
oddnumber,anewrandomresistornetworkisgenerated. tuition based on the usual conductances, the thermo-
At the subsequent, N +1−th , computatio only the electric transport coefficients in hopping systems are
com
middle half part [W/4,3W/4] of the network is replaced solely determined by the states which are close to the
by a new random configuration, while the parts close to interfaces (approximately, within the relevant hopping
the left and right boundaries, [0,W/4] and [3W/4,W], length). Moredetailsonthesesurprisingresultsandtheir
are not modified. It is seen from Fig. 2a that the con- generalizationswillbegivenelsewhere. Finally,itshould
ductance G changes dramatically when the central part be emphasized that all the results obtained here are in
is modified, whereas the thermopower is practically im- agreement with the systematic microscopic derivations
mune to modifications of the central part. To further which will appear in consequent work. Both Eq. (8) and
studythesensitivityofthethermopowertothesitesthat Eq. (13) agree with the results derived from the non-
are a distance larger than W/4 away from the two inter- equilibrium Green function method.
facesasafunctionofW,weplottherelativechangeofthe
thermopowerS as a function of W in Fig. 2b. This rela-
tivechangeisdefinedas|S −S |/|S +S |, ACKNOWLEDGMENTS
2n+1 2n+2 2n+1 2n+2
where S and S denote the thermopowers cal-
2n+1 2n+2
culated in the 2n + 1 and 2n + 2-th computation, re- We thankM.Pollak,A.Rosch,P.W¨olfle andA.Amir
spectively. The results are obtained by averaging over forilluminatingdiscussions. OEWacknowledgesthesup-
106 random configurations. It is seen that the relative port of the Albert Einstein Minerva Center for Theoret-
changeinS decaysexponentiallywithincreasingW,im- ical Physics, Weizmann Institute of Science. This work
plying that sites located several Mott hopping distances was supported by the BMBF within the DIP program,
L awayfromthe boundaries haveanegligible effect on BSF, by the ISF, and by its Converging Technologies
M
the thermopower S. The Mott length, L , is of the or- Program.
M
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