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2021

A driven fractal network: Possible route to efficient thermoelectric application

Kallol Mondal,1, ∗ Sudin Ganguly,1, † and Santanu K. Maiti1, ‡

1Physics and Applied Mathematics Unit, Indian Statistical Institute,

203 Barrackpore Trunk Road, Kolkata-700108, India

An essential attribute of many fractal structures is self-similarity. A Sierpinski gasket (SPG)triangle is a promising example of a fractal lattice that exhibits localized energy eigenstates. In thepresent work, for the first time we establish that a mixture of both extended and localized energyeigenstates can be generated yeilding mobility edges at multiple energies in presence of a time-periodic driving field. We obtain several compelling features by studying the transmission and energyeigenvalue spectra. As a possible application of our new findings, different thermoelectric propertiesare discussed, such as electrical conductance, thermopower, thermal conductance due to electronsand phonons. We show that our proposed method indeed exhibits highly favorable thermoelectricperformance. The time-periodic driving field is assumed through an arbitrarily polarized light, andits effect is incorporated via Floquet-Bloch ansatz. All transport phenomena are worked out usingGreen’s function formalism following the Landauer-Buttiker prescription.

I. INTRODUCTION

Deterministic fractals are neither a perfectly orderednor a completely disordered structure but somewhat inbetween them. Unlike the Anderson localization1, in de-terministic fractals, the localization occurs due to the fi-nite ramifications and self-similar structures such as Sier-pinski gasket (SPG). The existence of highly degeneratelocalized states and the Cantor set energy spectrum arethe hallmarks of the SPG structures2. These localizedstates become delocalized in the presence of a magneticfield3. Owing to such distinctive properties, SPG struc-tures have been studied extensively over the years inmany contexts, and several other unique features havebeen observed4–12.

Recently, it has been suggested that a spatialanisotropy can be tuned using a time-periodic drivingfield13 and thus, it is possible to manipulate the mate-rial and topological properties14. Such a possibility ledus to think about the structure-induced localization phe-nomenon in SPG structures. What will be the nature ofthe energy eigenstates in the presence of a driving fieldis still an open question to the best of our concern. Weare particularly interested in finding a mobility edge thatseparates localized energy eigenstates from the extendedones.

Usually, in 1D and 2D systems, localization to delocal-ization transition does not occur in the presence of un-correlated disorder since all the states are localized1,15.However, quasi-periodic lattices, such as one-dimensional(1D) Aubry-Andre (AA) chains16 exhibit a delocalizationto localization transition at a critical disorder strength.Below this critical value, all states are extended, and be-yond that, all the states become localized. For such asituation, mobility edge does not occur as the mixtureof extended and localized states is no longer available.However, mobility edge has been observed in 1D AAchains17 and ladder networks18 in the presence of higher-order hopping integral(s). So far, no attempt has beenmade to detect mobility edge in fractal lattices, which

essentially motivates us to probe into it. It is well knownthat all the states of an SPG lattice become localized inthe asymptotic limit due to the structure-induced local-ization.As the mobility edge is directly associated with the

metal-insulator transition, the electronic transmissionfunction will be highly asymmetric around the mobil-ity edge. The asymmetry in the transmission probabilityis the key requirement to get a favorable thermoelectric(TE) response19. Hence given the existence of mobilityedge, SPG structure could be a potential candidate forthe TE applications, and we investigate such a promis-ing aspect in the present work. SPG structures alreadyhave been fabricated experimentally with several mate-rials, such as submicrometer-width Al wires20, and alsovery recently from aromatic compounds21, metal-organiccompounds22, and by manipulation on CO molecules ofCu(111) surface23. Therefore, with the recent experi-mental realizations of SPG structures, we believe thatour proposition can be substantiated in a suitable labo-ratory.The SPG is driven by an arbitrarily polarized light, and

its effect is incorporated through the standard Floquet-Bloch ansatz in the minimal coupling scheme13,24–30.The mobility edge is detected by superimposing theenergy eigenvalues of the non-interacting electrons andthe two-terminal transmission probability. We computethe later one by using the well-known Green’s func-tion formalism, based on Landauer-Buttiker prescrip-tion31,32. The TE performance is studied by evaluat-ing the electrical conductance, thermopower, and ther-mal conductance due to electrons utilizing Landauer pre-scription19,33. Since at finite temperature, the effect ofphonons cannot be ignored, we also give an estimationof the thermal conductance due to phonons for a precisemeasurement of the TE efficiency employing the non-equilibrium Green’s function formalism34–36.The key findings of our work are: (i) generation of mul-

tiple mobility edges in the presence of a driving field, and(ii) achieving of high thermoelectric performance due tothe existence of asymmetric transmission function around

2

S D

FIG. 1: (Color online). Schematic view of an irradiated 3rdgeneration SPG fractal network. The atomic sites are locatedat the vertices of each equilateral triangle, as shown by redsolid spheres. The SPG is attached to two electrodes (sourceS and drain D). The electrodes are kept at two different tem-peratures T+∆T/2 and T−∆T/2, where ∆T is infinitesimallysmall.

the mobility edges. Our analysis can be utilized to designefficient thermoelectric devices at the nanoscale level andto study some fascinating phenomena in similar kind offractal lattices and other topological systems.The rest of the work is organized as follows. In Sec.

(2), we present our model Hamiltonian for the SPG sys-tem in the presence of an arbitrarily polarized light. Inthis section, we also present a brief theoretical descriptionfor the calculations of two-terminal transmission proba-bility and different TE quantities, including the thermalconductance due to phonon. All the results are criticallyinvestigated in Sec. (3). Finally, in Sec. (4), we concludeour essential findings.

II. SPG NETWORK AND THEORETICAL

FORMULATION

A. SPG and Hamiltonian

The SPG network is perfectly self-similar with threenon-overlapping copies of the previous generation, andevery triangular plaquette is a replica of the full struc-ture as shown in Fig. 1 where a third-generation SPGis depicted schematically. To evaluate the transmissionprobabilities and investigate TE performance, we clampthe SPG network between to perfect, reflectionless, semi-infinite, and 1D electrodes, namely the source (S) anddrain (D). A temperature difference ∆T is set amongthese electrodes S and D, and this ∆T is chosen to besmall enough such that we can work in the linear responseregime. An arbitrarily polarized light (magenta curve) isincident on the SPG perpendicular to the lattice planewhile the electrodes are free from any kind of irradiation.To describe the system, we use the tight-binding frame-

work, which can potentially capture the essential physics

of quantum transport. In this framework, the modelHamiltonian consists of four parts, as described below

H = HSPG +HS +HD +HC (1)

where HSPG, HS(D) and HC represent the sub-parts ofthe Hamiltonian associated with the SPG network, thesource (drain), and the coupling between semi-infiniteleads and the SPG network, respectively. The couplingpart of the Hamiltonian consists of two terms; one is thecoupling between the source and SPG network, and theother one is the coupling between the drain and SPGnetwork. In the absence of light irradiation, these sub-Hamiltonians are expressed as follows.

HSPG =∑

n

ǫnc†ncn +

∑

〈nm〉

tnm(c†ncm + h.c.) (2a)

HS = HD = ǫ0∑

n

d†ndn + t0∑

〈nm〉

(d†ndm + h.c.

),(2b)

HC = HS,SPG +HD,SPG

= τS(c†pd0 + h.c.

)+ τD

(c†qdN+1 + h.c.

). (2c)

The annihilation operators c, d and their hermitian coun-terparts c†, d† satisfy the usual fermionic commutationrelations.

(c, c†

)are associated with the SPG network

and(d, d†

)with the source and drain. ǫn represents

the on-site potential at the n-th site. tnm denotes thenearest-neighbor hopping (NNH) integral in the SPG inthe absence of light. On-site potential ǫ0 and hoppingamplitude t0 are assumed to be the same for both thesource and drain. The coupling strength between thesource and SPG is τS , and that between the drain andSPG is τD. The source and drain are connected to theSPG at the p-th and q-th sites, respectively.

B. Incorporation of light irradiation

When a system is irradiated with light, the system be-comes a periodically driven one. Under this situation,the problem becomes quite complicated and challengingas well. But in the minimal coupling regime, such a time-dependent problem can be simplified using Floquet-Blochansatz13,24–27. Following the Floquet approximation, theeffect of light incorporation can be taken care of througha vector potential A(τ). In the tight-binding framework,the vector potential manifests itself in the hopping inte-gral through Peierls substitution e

cℏ

∫A(τ)·dl, where the

symbols e, c, and ℏ carry their usual meaning. Withoutlosing any generality, we can write the vector potential inthe form A(τ) = (Ax sin(Ωτ), Ay sin(Ωτ + φ), 0), whichrepresents an arbitrarily polarized field in the X-Y plane.Ax and Ay are the field amplitudes, and φ is the phase.Depending upon the choices Ax, Ay, and φ, we can getdifferent polarized lights, such as circularly, linearly, orelliptically polarized lights. After rigorous mathematicalcalculation, the effective hopping integral in the presence

3

of irradiation gets the form

tnm → tpqnm = tnm ×1

T

∫T

0

eiΩτ(p−q)eiA(τ)·dnmdτ (3)

where dnm is the vector joining the nearest-neighbor sitesin the SPG. tnm is the NNH strength in the absence oflight and is assumed to be isotropic that is tnm = t. pand q correspond to the band index of Floquet bands. Weassume the driving field to be uniform with frequency Ωand time-period T. Here the vector potential is expressedin units of ea/cℏ (a being the lattice constant, is takento be 1Ao).Finally, with the modified hopping integral, the SPG

Hamiltonian (Eq. 2(a)) can be written as

HSPG =∑

n

ǫnc†ncn +

∑

pq

∑

〈nm〉

tpqnmc†ncm + h.c.

− pℏωδpq

.

The last term (−pℏωδpq) originates due to the Fouriertransformation of −iℏ∂t. The mathematical steps arenot shown here to save space. (For a detailed derivationof the modified NNH term and the Hamiltonian, see Refs.13,26)

C. Two-terminal transmission probability

We employ the Green’s function formalism to calculatethe transmission probability of an electron from sourceto drain through the SPG network. Here, we neglectthe Coulomb interaction term and also restrict ourselveswithin the regime of coherent transport. The effectiveGreen’s function can be written as

Gr = (E −HSPG − ΣS − ΣD)−1 (4)

where ΣS and ΣD represent the self-energies of the sourceand drain, respectively. So, the two-terminal electronictransmission probability can be written in terms of re-

tarded (Gr) and advanced(Ga (= Gr)

†)

Green’s func-

tions as

T = Tr [ΓSGrΓDG

a] (5)

where ΓS and ΓD are the coupling matrices that describethe rate at which particles scatter between the leads andthe fractal network.

D. Thermoelectric quantities

Thermoelectric materials convert heat into electric en-ergy and vice versa. The heat-to-electric energy conver-sion efficiency is expressed by a dimensionless quantity,known as the figure of merit (FOM) which is denoted byZT . The expression of FOM is given by

ZT =GS2T

k(= ke + kph)(6)

where G is the electronic conductance, S is the Seebeckcoefficient (thermo power), and T is the temperature. krepresents the total thermal conductance which is a sumof electronic conductance (ke) and phononic conductance(kph). Each of these quantities, apart from kph, can becalculated from Landauer prescription19,33 as

G =2e2

hL0 (7a)

S = −1

eT

L1

L0(7b)

ke =2

hT

(L2 −

L21

L0

). (7c)

In the above expressions, the Landauer integrals Ln aredefined as

Ln = −

∫T (E)(E − Ef )

n ∂fFD

∂EdE (8)

where Ef is the fermi energy, T (E) is the transmissionprobability of the system, and fFD(E) represents theFermi-Dirac distribution function. Typically ZT > 1 isregarded to be a good thermoeletric material. However,for large-scale energy-conversion systems ZT ∼ 2 − 3 isoften prescribed37.

E. Phonon thermal conductance

The phononic contribution to the thermal conductancekph is often neglected as an approximation in Eq. 6. Thisis due to the fact that at nanoscale regime, the systemcontains less number of lattice sites, and therefore, at lowor even moderate temperatures, the contribution is rela-tively small. But for precise estimation of ZT , one needsto include kph. When the temperature difference betweenthe two contact electrodes is infinitesimally small, thephonon thermal conductance is evaluated from the ex-pression34–36

kph =ℏ

2π

∫ ωc

0

Tph∂fBE

∂Tωdω. (9)

Here, ω is the phonon frequency ωc is the cut-off fre-quency. Here we consider only elastic scattering. fBE

is the Bose-Einstein distribution function. Tph is thephonon transmission coefficient across the SPG, and itis computed using the well known Green’s function pre-scription through the relation

Tph = Tr[ΓphS GphΓ

phD (Gph)

†]

(10)

ΓphS/D = i

[ΣS/D − Σ†

S/D

]is the thermal broadening and

ΣS/D is the self-energy matrix for the source/drain elec-trode. The Green’s function for the SPG can be writtenas

Gph =[Mω2 −K− ΣS − ΣD

](11)

4

where M is a diagonal matrix representing the mass ma-trix of the SPG. A diagonal element Mnn of this matrixdenotes the mass of the atom at the n-th position in theSPG. K is the matrix of spring constants in the SPG. Theelement Knn represents the restoring force of the n-thatom due to its neighboring atoms, whereas the elementKnm describes the effective spring constant between n-th atom and its m-th neighboring atom. The self-energy

matrices ΣS and ΣD have the same dimension as M andK and can be computed by evaluating the self-energy

term ΣS/D = −KS/D exp[2i sin−1

(ωωc

)], where KS/D

is the spring constant at the electrode-SPG contact in-terface.

The spring constants are calculated from the secondderivative of Harrison’s interatomic potential38. For the1D electrodes, the spring constant is given by K =3dc11/16, while for the SPG K = 3d (c11 + 2c12) /16.Here d denotes the interatomic spacing and c11 and c12are the elastic constants. The difference in the expres-sions of the spring constants arises because, in a 1D elec-trode, there is no transverse interaction, but in a 2Dsystem like SPG, one needs to consider it39. With theknowledge of the mass and spring constant, the cut-offfrequency for the 1D electrode is determined from therelation ωc = 2

√K/M . For a detailed description of the

procedure to calculate the phonon thermal conductance,see Refs.34–36.

III. RESULTS

Before discussing the results, let us first mention theparameters used in the present work. The on-site energiesin the fractal network as well as in the source and drainelectrodes are set at zero. All the energies reported hereare measured in units of electron-volt (eV). The NNHhopping integrals are considered in the wide-band limit,where it is set for the electrodes as t0 = 2 eV, and inthe SPG as t = 1 eV. The coupling strengths of theSPG to the source and drain electrodes, characterizedby the parameters τS and τD, are also fixed at 1 eV. Forany other choice of parameter values, the physical picturewill be qualitatively the same, which we confirm throughour exhaustive calculation. The rest of the parametervalues, which are not common for the entire analysis, arementioned in the appropriate places of our analysis.

Due to the time-periodicity of the driving field, a peri-odically driven D-dimensional lattice is equivalent to anundriven D + 1-dimensional lattice13,26. For such a pe-riodically driven system, the initial Bloch band breaksinto Floquet-Bloch (FB) bands, where the coupling be-tween FB bands depends directly on the driving fre-quency regime. This time-independent D+1 dimensionallattice can be visualized as if the SPG is connected to itsseveral virtual copies arranged vertically to the latticeplane. In the high-frequency limit, the Floquet bandsdecoupled from each other, and only the zeroth-order

Floquet band (p = q = 0) has the dominant contribu-tion in Eq. 3, while other higher-order terms in p and qessentially have a vanishingly small contribution. Due tothis decoupling process, the coupling between the par-ent SPG lattice and its virtual copies becomes vanish-ingly small. This scenario is no longer valid in the low-frequency regime, where the virtual copies are directlycoupled to the parent SPG lattice. Therefore, in the low-frequency limit, several virtual copies of the SPG latticecome into the picture. Consequently, the effective sizeof the system increases. This could decrease the phase-relaxation length, and at finite temperature, it might bereduced further. Thus, it will be quite hard to get favor-able transport properties in the low-frequency regime.Because of the above facts, we restrict the present anal-

ysis in the high-frequency limit ℏω ≫ 4t. The lightfrequency for this limiting case should be at least ∼

1015Hz, which is in the near-ultraviolet/extreme ultravi-olet regime. The corresponding electric field ∼ 104V/m,while the magnetic field is ∼ 10−5T. Since the magneticfield due to the light irradiation is vanishingly small, itseffect can safely be ignored. The intensity of the lightirradiation is ∼ 105W/m2, and is certainly within theexperimental reach. Since much higher light intensitieshave been used in several other recent works40,41, westrongly believe that our chosen intensity will no longerdamage the physical system.

A. Detection of mobility edge

We begin our discussion by analyzing the two-terminaltransmission coefficient along with the energy eigenval-ues of an SPG network in the absence and presence oflight, as shown in Fig. 2. The transmission spectrum(red color) is superimposed on the spectrum of energyeigenvalues, where we draw a vertical line of unit magni-tude (cyan color) in each of these eigenvalues. To checkthe localization behavior in the asymptotic limit, we con-sider a bigger SPG (8th generation) that contains a fairlylarge number of lattice sites. Figure 2(a) shows that theeigenenergies are highly degenerate. Few sub-bands areformed, providing finite gaps. Along with the bands,some isolated energy eigenvalues are also visible. Here,the transmission coefficient becomes zero or vanishinglysmall for the entire allowed energy window, which em-phasizes a complete localization of all the states. Thislocalized behavior for the irradiation-free SPG is knownin the literature.The situation becomes quite interesting and important

as well when the system is irradiated. Figure 2(b) showsthat degeneracies get removed, and thus wider bands areformed. The most striking feature is observed in thetransmission spectrum, where almost all the energy lev-els are associated with finite transmission probabilities.Thus, it gives a clear indication of a localization to de-localization transition in the presence of light irradia-tion. Now, a careful inspection reveals that near the

5

energy E ∼ 0, there is a fine strip of eigenvalues that arecompletely localized since the transmission coefficient isidentically zero within this fine strip. To the immediateleft/right of the strip, the transmission coefficient is fi-nite which clearly manifests that the states are extended.Thus, we have a sharp edge that separates the extended

-2 -1 0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

E

(a)

-2 -1 0 1 2

0.0

0.2

0.4

0.6

0.8

1.0

E

(b)

FIG. 2: (Color online). Transmission probability T (E) (redcolor) as a function of energy along with the energy eigenval-ues (cyan color) for an 8th generation SPG network, where(a) and (b) correspond to the results in the absence and pres-ence of light respectively. At each eigenvalue, we draw a singlevertical line of unit magnitude for better visibility of the en-ergy eigen spectrum. The light parameters are Ax = 0 andAy = 2.

and localized energy eigenstates which validates the exis-tence of a mobility edge in the presence of light in SPG.Interestingly, we find that such a mobility edge appearsat multiple energies like E ∼ −1, 1, and 2, etc. The re-gion across a mobility edge is marked by a black dottedellipse in Fig. 2(b) for better visualization.Another interesting feature we observe here is that in

the presence of light, the allowed energy window getsreduced (∼ −2 - 2.5) than in the absence of light (∼ −2- 4). We shall talk about this feature at length in thenext subsection. Overall, what we accumulate is thatlocalization to delocalization transition is obtained in thepresence of light along with multiple mobility edges.

B. Spectral analysis

Now, we discuss the spectral behavior of an SPG net-work, which are extremely crucial to understand the elec-tronic transport phenomena. Here, we consider a 5th

generation SPG to analyze the results. In Fig. 3, weshow the spectra of energy eigenvalues both in the pres-ence and absence of irradiation. In the absence of light,the spectrum is highly degenerate and gapped, as shownin Fig. 3(a). These are the basic characteristics of any

-2 -1 0 1 2 3 4

125

100

75

50

25

0

Eigenvalues

Energ

yle

vels

(a)

-0.5 -0.25 0.0 0.25 0.5 0.75 1.0

125

100

75

50

25

0

Eigenvalues

En

erg

yle

ve

ls

(b)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

125

100

75

50

25

0

Eigenvalues

En

erg

yle

ve

ls

(c)

-2 -1 0 1 2 3

125

100

75

50

25

0

Eigenvalues

Energ

yle

vels

(d)

FIG. 3: (Color online). Eigenvalue spectra in the absenceand presence of light with different irradiation parameters.(a) Absence of light, (b) Ax = Ay = 2 and φ = π/2, (c)Ax = Ay = 2 and φ = π/4, and (d) Ax = 0 and Ay = 2.

fractal geometries and are well-known in literature.2,42.In the case of a circularly polarized (CP) light, the na-ture of the spectrum is identically the same like whatwe find in the irradiation-free case, but the allowed en-ergy window gets shortened as depicted in Fig. 3(b). Theidentical energy spectrum and reduced energy window forthe case of CP light can be explained as follows. For theSPG network, there are two hopping directions, one isthe horizontal hopping, and the other is the angular one.In the presence of an arbitrarily polarized light, the hop-ping strengths in both directions are modified accordingto Eq. 3 by the Bessel function of the first kind13,43, buttheir strengths become the same for the CP light.

Therefore, the hopping strength is isotropic for the CPlight, like the case in the absence of light but with a re-duced hopping terms. This isotropic nature of the hop-ping integrals makes the spectrum identical with that ofthe irradiation-free case, and the reduced hopping termis responsible for the decrease in the allowed energy win-dow. For an elliptically polarized light (the horizontaland angular hopping strengths are different now), thedegeneracy is broken, and the energy levels are regularlyarranged (Fig. 3(c)). Here the allowed energy window isalso different than the previous two cases. For the lin-early polarized light as shown in Fig. 3(d), the brokendegeneracy levels with modified allowed energy windowsare observed due to the modified NNH integrals in boththe directions.

To have a better understanding of the effect of lightirradiation, in Fig. 4 we examine the spectrum of energyeigenvalues in terms of the light parameters. Figure 4(a),shows the eigenvalues as a function of phase of the vec-

6

tor potential φ for Ax = Ay = 2. The allowed energywindow oscillates with a period φ = π/2. Therefore, it

0 π/2 π 3π/2 2π-2

-1

0

1

2

ϕ

Eig

en

va

lue

s

(a)

0 2 4 6 8 10

-4

-2

0

2

4

Ax (= Ay)E

ige

nva

lue

s

(b)

FIG. 4: (Color online). Eigenvalues as a function of (a) phaseφ with Ax = Ay = 2 and (b) field amplitude Ax(= Ay) withφ = π/2.

is possible to control the energy window externally byadjusting the light parameter φ, which is useful in en-gineering transport phenomena that can be understoodfrom our forthcoming analysis.Figure 4(b) describes the behavior of energy eigenval-

ues with field amplitudes for CP light that is by varyingAx (= Ay) keeping phase fixed at φ = π

2 . The obser-vation is quite remarkable in the sense that the allowedenergy window can significantly be modified by tuningthe field amplitude. For instance, in the absence of light,the eigenvalues are distributed within the interval -2 to4. Once we irradiate the SPG with CP light, the al-lowed window gets shortened with increasing the fieldamplitude, and around Ax = Ay = 2.4, all the eigen-values coincide exactly at zero energy. This particularfeature could be useful in switching applications due tothe following reason. Suppose we are measuring the elec-trical conductance, setting the Fermi energy apart fromzero. In the absence of light, the conductance is finite.Now, in the presence of CP light, when the field ampli-tude is around 2.4, there will be no such state at thatenergy, and consequently, the conductance will be zero.Beyond Ax = Ay = 2.4, we note another two values ofthe field amplitude where all the eigenvalues again co-incide to zero. A band inversion is also observed forAx > 2.4. This is due to the modified hopping term,which stays positive up to Ax > 2, and beyond that,it becomes negative. As the phase of the hopping inte-gral reveres, the band inversion takes place. Earlier inFig. 3(b), we claimed that in the presence of CP light,the fractal nature of the energy spectrum remains intactwith a modified bandwidth. This is true for any otherfield amplitude, including the band inversion cases whereonly the structure of the energy spectrum gets inverted.This claim also holds for the field amplitudes where allthe eigenvalues coincide at zero energy which we confirmthrough our exhaustive analysis. Here we would like tonote that the energy eigenvalues coincide to zero as weset ǫn = 0. For the situation, when ǫn is finite, all theeigenvalues will coincide to that particular value withoutaltering the characteristic features.In Figs. 3, and 4, we find that the eigenenergies are

significantly modified in the presence of light. Thus, itwill be interesting to study how the transmission coeffi-

cient T behaves with the light parameters since T playsa crucial role in transport phenomena. In Fig. 5, we plot

0 π/2 π 3π/2 2π0.0

0.2

0.4

0.6

0.8

1.0

ϕ

m

ax

FIG. 5: (Color online). The maximum transmission coeffi-cient Tmax as a function of φ with Ax = Ay = 2.

the maximum of the transmission coefficient, Tmax as afunction of φ. Tmax is evaluated by taking the maximumof T within the allowed energy window for a given lightparameter. Here Tmax shows oscillatory behavior similarto the eigenvalue spectrum with a period φ = π/2. Inter-estingly, Tmax varies in a wide range (0.2 to unity), andtherefore, the conducting behavior of the SPG networkcan be adjusted by regulating the phase factor.So far, several interesting features have emerged from

the study of the energy spectrum and transmission char-acteristics, such as the localization to delocalization phe-nomena, the existence of multiple mobility edges, break-ing of the level degeneracy, the possibility to engineer thebandwidth, etc. Now, we try to exploit the delocaliza-tion phenomena and the existence of the multiple mo-bility edges (Fig. 2) in TE application for this quantumnetwork.

C. Thermoelectric properties

Unless mentioned otherwise, all the TE quantities areevaluated at room temperature T = 300K. We considera 5th generation SPG for demonstration and the lightparameters are Ax = 0 and Ay = 2. In Fig. 6(a), thetransmission spectrum (red color) is superimposed on thespectrum of energy eigenvalues (cyan color). At eacheigenvalues, we draw a vertical line of unit magnitude forbetter visualization.The transmission spectrum shows almost similar be-

havior with energy that was obtained for the 8th gener-ation (Fig. 2) with the identical light parameters owingto the self-similar structure of SPG. Across E ∼ −1, amobility edge is found, similar to what we get for the 8thgeneration SPG, and we vary the Fermi energy windowacross this energy. Now, we compute the electrical con-ductance, thermopower, and thermal conductance dueto electron as a function of the Fermi energy as shown inFigs. 6(b), (c), and (d) respectively. The G-EF spectrumbasically follows the T -E curve as G is evaluated using

7

Eqs. 7(a) and 8. The spikes in the transmission spec-trum smeared out due to the temperature broadening.We observe a dip around the mobility edge in Fig. 6(a).On the other hand, the thermopower becomes maximumaround the mobility edge (Fig. 6(b)), acquiring a value∼ 380µV/K. The behavior of the electronic thermal con-ductance kph is similar to that of G and is of the orderof a few hundreds of pW/K.

-2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

E

(a)

-1.5 -1.2 -0.9 -0.6 -0.3 0.00.0

0.2

0.4

0.6

0.8

1.0

Ef

G(2e

2/h

)(b)

-1.5 -1.2 -0.9 -0.6 -0.3 0.0

-400

-200

0

200

400

Ef

S(μ

V/K

)

(c)

-1.5 -1.2 -0.9 -0.6 -0.3 0.00

100

200

300

400

500

Ef

ke

(pW

/K)

(d)

FIG. 6: (Color online). (a) Transmission coefficient T (redcolor) as a function of energy E along with the energy eigen-values (cyan color). At each eigenvalue, we draw a singlevertical line of unit magnitude for better visibility of the en-ergy eigen spectrum. (b) Electrical conductance G, (c) ther-mopower S, and (d) thermal conductivity due to electrons keas a function of the Fermi energy EF . Here a 5th generationSPG is considered and the light parameters are Ax = 0 andAy = 2.

The suppression of G and ke around the mobilityedge is simply due to the sharp fall in the transmissionspectrum. The asymmetry in the transmission functionaround the mobility edge is responsible for the enhance-ment of S. The thermopower is evaluated using Eq. 7(b)and the corresponding thermal integral L1 (Eq. 8), wherethe transmission spectrum T (E) is weighted by the terms(E −Ef ) and (∂fFD/∂E). The latter term provides thebroadening and is antisymmetric around EF . Thus, ifT (E) is symmetric around EF , then S will be zero irre-spective of the value of the transmission. Therefore, toget higher thermopower, we need an asymmetric trans-mission function that can be easily obtained across themobility edge.So, what we gather from Fig. 6 is that both G and

kph decrease, and S increases around the mobility edge.Though G is directly proportional to ZT , the decreasein G does not suppress the FOM considerably since kecomes in the denominator in the expression of FOM(Eq. 6). As ZT ∝ S2, a small increase in the ther-mopower enhances the FOM significantly. Therefore, allthe TE results indicate a high ZT around the mobilityedge. However, before we make any definite conclusionregarding the efficiency, it is important to study the ther-mal conductance due to phonon, which we discuss now.

Phonon thermal conductance: Before discussing theresults due to phonons let us briefly mention the valuesthose are considered for our calculation. We describethe system to calculate kph as a spring-mass system.The spring constants are usually calculated from thesecond derivation of the harmonic Harrison potential38.For the 1D electrodes, the spring constant is consideredas 16.87N/m, while in the SPG, the spring constant is23.93N/m. The considered values of the sping constantsare corresponding to typical semiconductors such as Geand Si36. The cut-off frequency of vibration depends onthe material properties of the electrodes and the SPGsince we assume that two different atoms are adjacentto each other in the present work. In our chosen setup,by averaging the spring constants of the electrodes andSPG, and the masses, the cut-off frequency comes out tobe ωc = 31.3Trad/s. Since the frequency of the light ir-radiation

(∼ 1015 Hz

)is about three orders of magnitude

higher than the phonon vibrational frequency, we assumethat the irradiation will not affect the lattice vibrationsignificantly and hence the effect of irradiation on latticevibration can safely be ignored.Analogous to electronic transport, the localization phe-

nomena are also expected to occur in the phonon trans-mission due to the fractal nature of SPG. In Fig. 7(a),the phonon transmission coefficient Tph is plotted as afunction of the phonon frequency ω for a 5th generationSPG. Here the acoustic vibrational modes are greatlysuppressed within the frequency range ∼ 16−27 Trad/s.For the higher generation SPGs, we get more localized

0 8 16 24 320.0

0.2

0.4

0.6

0.8

1.0

ω (Trad/sec)

p

h

(a)

100 150 200 250 300

3.1

3.2

3.3

3.4

Temperature (K)

kp

h(p

W/K

)(b)

FIG. 7: (Color online). (a) Phonon transmission Tph as afunction of phonon angular frequency ω. (b) Phonon thermalconductance kph as a function of temperature T .

phonon modes which we confirm by studying the phonontransport up to the 8th generation SPG (not shown here).Such localization of phonon modes has also been stud-ied in asymmetric harmonic chains35 and observed in 2Dfractal heterostructures44,45. The corresponding phononthermal conductance kph as a function of temperature isshown in Fig. 7(b). Here kph is of the order of a fewpW/K, about two orders of magnitude lower than itselectronic counterpart. It increases systematically butvery slowly with temperature.With all the TE quantities, we finally calculate the

FOM. ZT as a function of the Fermi energy is presentedin Fig. 8. The maximum ZT is ∼ 5 around the mobilityedge. This is, of course, a highly favorable response anda direct consequence of the existence of the asymmetricfunction across the mobility edges for the irradiated SPG

8

-1.5 -1.2 -0.9 -0.6 -0.3 0.0

0

1

2

3

4

5

Ef

ZT

FIG. 8: (Color online). ZT as a function of the Fermi energyEF . The SPG configuration and all the other parameters areidentical as mentioned in Fig. 6.

network.Here it is important to note that all the results studied

in this communication are worked out for a set of typicalparameter values. Now, if we had considered a real ma-terial like say Ge or Si, then the on-site energies wouldbe non-zero and the NNH integrals would be of the orderof eV46 which we have considered in the present work.Any non-zero on-site energy simply provides a shift ofthe energy spectrum, keeping all the physical picturesunaltered.

IV. SUMMARY

In conclusion, we have given a new prescription toget mobility edge separating the localized and extendedstates by irradiating an SPG network, and we exploitthe existence of mobility edge in TE applications. Suchan attempt is completely new to the best of our knowl-edge. The system under consideration has been describedwithin a tight-binding framework, and the irradiationeffect has been incorporated using the Floquet-Blochansatz in the minimal coupling scheme. The electronicand phononic transmission coefficients have been evalu-

ated using the standard Green’s function formalism basedon Landauer-Buttiker approach. At first, a higher gen-eration SPG has been studied to observe the existenceof mobility edge by studying the electronic transmissionspectrum and energy eigenvalues in the presence of lightirradiation. After that effect of light irradiation on theenergy spectrum has been examined in detail. Finally, wehave discussed the TE performance of a driven SPG byanalyzing the different TE quantities such as the electri-cal conductance, thermopower, and thermal conductancedue to electrons and phonons. Our essential findings aresummarized as follows.• Multiple mobility edges have been observed in the pres-ence of light.• The highly degenerate eigenvalues of SPG spread outin the presence of light.• The bandwidth can be modified by tuning the light pa-rameters.• The fractal nature of an undriven SPG remains intactin the presence of circularly polarized light with a renor-malized bandwidth.• A band inversion has been observed in the presence ofa circularly polarized light by tuning the field amplitude.• Controlling the phase of the vector potential it ispossible to have a highly conducting SPG from a poorone. This particular feature can be exploited to engineerswitching devices.• The electrical conductance and thermal conductancedue to electrons are suppressed appreciably around themobility edge while the thermopower acquires high value.• The localization phenomenon has also been observed inthe phonon transmission spectrum.• The phonon thermal conductance is two orders of mag-nitude lower than its electronic counterpart.• We have found that FOM is large and greater thanunity by setting upon the Fermi energy around the mo-bility edge.

Our analysis can be utilized to investigate electronicand spin-dependent transport phenomena in similarkinds of fractal lattices and also to design fascinatingspintronic and electronic devices.

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