**Thermoelectric effects of Kane-Mele NIS junction**

**Kane-Mele NIS junction**

The recent development in feasible fabrication techniques for graphene, has allowed for exploratory studies of this system. From the application point of view, graphene is a potential candidate due to its high mobility. Further in graphene the carrier density can be controlled by external gate voltage, which makes it a good candidate for fabrication of devices. In order to apply graphene to electric devices, it is an important topic to study the characteristics of charge and heat transport.

Recently, the thermal and thermoelectric properties of graphene structure have gained much attention because of the large Seebeck coefficient and high thermal conductivity obtained in graphene sheets [76–78]. Previously, due to the experimental limitations in accessing nano-scale devices, the charge and spin dependent thermoelectric properties were often ignored [162,163]. But recently improved techniques in low temperature measurements devices provides an op- portunity of experimental observation of thermoelectric physics. Very recently Zuev et al [77] and Wei et al. [133] have performed theoretical and experimental investigation of the thermoelectric effects of graphene sheets.

In a parallel front, the quantum transport through the junction devices are gaining increased attention in the field of modern research for developing the nano-devices at atomic/molecular level. The junction devices have interesting applications in the fields of thermoelectric, thermometric, solid state cooling etc.

In the past a good number of studies on junction has been performed in these fields [7,17,18] and the junction devices are very useful in wide range of experi- ments and applications [8–10]. The recent development in the field of thermoelec- tric physics in small scale junction devices provides a new direction for fabricating self-cooling devices, thermopower devices etc.

We have already discussed that two types of spin-orbit couplings (SOC) are proposed in graphene by Kane and Mele. Though the strengths of the spin- orbit couplings are very small, it is possible to induce enhanced SOC strengths in graphene via different techniques, such as, such as via adatoms [87], using proximity effect of a three dimensional topological insulators [50,88], by function- alization with methyl groups [89] etc. Moreover the tunability of RSOC strength via an external gate voltage provides an additional impetus in the field of spin- tronics. It is worth to mention that SOCs are very significant and hence cannot be skipped in the context of transport. Since both the charge and thermal cur- rent are very sensitive to the strength of SOCs and owing to the tunability of the SOC parameters, a Kane-Mele NIS junction can be a good candidate for a tunable

thermoelectric device.

Motivated by the above, we have performed an extensive study of the ther- moelectric effect of a Kane-Mele normal-insulator-superconductor (KMNIS) nano- junction by employing the modified Blonder-Tinkham-Klapwĳk (BTK) theory. Phys- ically the scenario corresponds to adatom decorated graphene NIS junction to ac- count for finite strengths of the spin-orbit couplings. We have computed the spin resolved thermopower, Figure of Merit, thermal current, coefficient of performance and explored how the spin-orbit couplings (induced by adatoms or otherwise) as- sume roles in shaping up the thermoelectric properties of such a junction.

For better readability, let us comment on the organization of our work. To make notations clear, we briefly describe the system in section(6.1). Formulae of Seebeck coefficient and thermal current are depicted in subsection(6.1.1) and in subsection(6.1.2) respectively. The results on the thermopower and thermoelec- tric cooling of this junction device are discussed in section(6.2). In section(6.3) we summarize our results and highlight the tunability of the junction device with regard to its thermoelectric properties.

**6.1** **Kane-Mele NIS junction**

In Fig.(2.5) the schematic diagram of the junction setup has been shown where
the left electrode is a normal (*x* ≤0) and the right electrode (*x* ≥ *d*) is a supercon-
ducting material with the insulating layer is extending from *x* = 0 to *x* = *d*. As
earlier, the insulating layer is modeled by an external gate voltage such that the
Fermi energy is ramped by*V*0. It is considered that the*x* ≥*d* region of the junction
system has been produced by proximity effect by an external superconductor.

The effective Hamiltonian for Kane-Mele system is given in Eqn.(5.1) and the simplification of Kane-Mele Hamiltonian is elaborately discussed in Chapter 2.

The Fermi energy variation across the junction system is assumed as,

*E**F*(*x*)= ^{E}*F*^{N}Θ(−*x*)+^{E}*F*^{I}Θ(*d*−*x*)+^{E}*F*^{S}Θ(*x* −*d*) 6.1
where *E*_{F}^{N} and *E*_{F}^{S} are the Fermi energies of the normal and the superconducting
leads. *E*_{F}^{I} is the Fermi energy of the insulating barrier which is defined by, *E*_{F}^{I} =
*E*_{F}^{N}+^{V}0.

The expression for the charge current through the KMNIS junction using BTK theory can be found to have the following form,

*I*_{NS}_{σ}(*E*^{N}_{F}*, T*_{N}*, E*_{F}^{S}*, T*_{S})= ^{eA}*r**v*_{F}^{N}
Z Z

*τ*_{σ}(*E, θ*_{N1}) 6.2
[*f*^{N}(*E*_{f}^{N}*, T*^{N})−*f*^{S}(*E*_{F}^{S}*, T*_{S})]*N*(*E*)*dE*cos*θ**N*1*dθ*_{N1}

where *N*(*E*) denotes the density of states,*v*^{N}_{F} is the Fermi velocity,*A*_{r} is the area
of contact, and*f*^{N},*f*^{S} are the Fermi distribution functions for the normal and the
superconducting leads respectively. *τ*_{σ}(*E, θ*_{N1}) is the transfer probability. where
*τ*_{σ}(*E, θ*_{N1})= ^{1}−|*b*_{σ}(*E, θ*_{N1})|^{2}+^{cos}_{cos}^{θ}_{θ}^{N2}_{N1}|*a*_{σ}(*E, θ*_{N1})|^{2},*a*_{σ}is amplitude of Andreev reflection,
*b*_{σ} is amplitude of normal reflection. The derivation for finding the amplitudes of
normal and Andreev reflections are presented in chapter 2. As the normal state
resistance, *R*_{N} is defined by, *R*_{N} = _{2e}2*N*^{1}_{0}*v*^{N}_{F}*A*_{r} (2 comes due to the spin degeneracy,
*N*_{0} denotes the density of state at Fermi level), the electrical charge current takes
the following form,

*I**NS*_{σ}(*E*_{F}^{N}*, T**N**, E*_{F}^{S}*, T**S*)= ^{1}
2*eR**N**N*0

Z Z

*τ**σ*(*E, θ**N*1) 6.3
[*f*^{N}(*E*^{N}_{f} *, T*^{N})−*f*^{S}(*E*_{F}^{S}*, T**S*)]*N*(*E*)*dE*cos*θ**N*1*dθ**N*1

where the energy dependent quantity,

*N*(*E*)= |*E*_{F}^{N}+^{E}|*W*
*π*~^{V}*F*^{N}

6.4

is the number of transverse modes in graphene sheet of width*W* [160].

**6.1.1** **Seebeck coefficient**

We consider the left and right electrodes serve as independent temperature
reservoirs where the left electrode is fixed at temperature,*T*^{N} =^{T} −*δT/*2 and the
right electrode is fixed at temperature*T*^{S} =^{T}+*δT/*2. The population of electrons
in the left and the right lead is described by the Fermi-Dirac distribution function,
*f*^{N} and*f*^{S} respectively, where*E*_{F}^{N} =^{E}*F*^{S} at zero external bias.

The temperature difference between two dissimilar materials produces a volt-
age difference between the two substances and this phenomenon is known as
Seebeck effect. Seebeck coefficient is a measurement of the amount of potential
induced in the device for unit temperature difference and defined by,*S* =^{δV/δT}^{.}

Let us now consider an extra infinitesimal current induced by an additional
voltage, *δV* and the temperature difference, *δT* across the junction in an open
circuit. The current induced by *δT* and *δV* are given by, (*dI*)*T* = ^{I}(*E*_{F}^{N}*, T*^{N}*, E*_{F}^{S} =
*E*_{F}^{N}*, T*^{S} = ^{T}^{N} +^{δT}^{) and} (*dI*)*V* = ^{I}(*E*_{F}^{N}*, T*^{N}*, E*_{F}^{S} = ^{E}*F*^{N} +^{eδV, T}^{S} = ^{T}^{N}). Suppose that
the current cannot flow in an open circuit, thus (*dI*)*T* counter balances (*dI*)*V*. It
allows us to write,

*dI* = (*dI*)*T* +(*dI*)*V* =^{0} ^{6.5}
where the expression for the(*dI*)*T* and(*dI*)*V* can be obtained from Eqn.(6.3). Now
first order expansion of Fermi-Dirac distribution function in(*dI*)*T* and(*dI*)*V* while
the energy is shifted by Fermi energy, yields the expression for the spin dependent

Seebeck coefficient,

*S*_{σ} = ^{δV}
*δT* =

R R *dEdθ*_{N1}cos*θ*_{N1}*E*(*E*_{F}^{N} +^{E})*τ*_{σ}(*E, θ*_{N1})_{∂E}^{∂f}
*eT*R R

*dEdθ*_{N1}cos*θ*_{N1}(*E*_{F}^{N} +^{E})*τ*_{σ}(*E, θ*_{N1})_{∂E}^{∂f} ^{6.6}
Now the charge and spin Seebeck coefficients are usually defined by [164],

*S**ch* = ^{1}

2(*S**up*+^{S}*down*) ; *S**sp* = ^{1}

2|*S**up*−*S**down*| _{6.7}
which can be computed from Eqn.(6.6) for *σ* =^{up/down.}

The efficiency of the device depends upon a quantity called as ’Figure of Merit’

(FM). To get a clear idea of the efficiency, one should compute the spin dependent FM which is given by,

*Z*_{σ}*T* = ^{S}^{σ}^{2}^{G}^{σ}

*K*_{σ} *T* 6.8

where *S*_{σ} is Seebeck coefficient, *G*_{σ} is electrical conductance, *K*_{σ} is thermal con-
ductance, and*T* is absolute temperature. *G*_{σ} can be calculated from the relation
*G*_{σ} = ^{dI}_{dV}^{NS σ} and is given by the form,

*G**σ* = ^{1}
2*eR**N**E*_{F}^{N}

Z Z

*τ**σ*(*E, θ**N*1)(−^{∂f}

*∂E*)(*E*+^{E}*F*^{N})*dE*cos*θ**N*1*dθ**N*1 6.9
The thermal conductance *K**σ*, can be calculated from the relationship *K**σ* = ^{dJ}_{dT}^{NS σ}
where*J**NS σ*is the thermal current flowing from the normal region to the supercon-
ducting region. In the next subsection we shall discuss how the thermal current
and the thermal conductance can be calculated.

In addition, the charge FM (*Z**ch**T*) and spin FM (*Z**sp**T*) are defined as [164–166],
*Z**ch**T* = ^{S}

2

*ch*(*G*_{up}+^{G}*down*)*T*
*K**up*+^{K}*down*

;*Z**sp**T* = ^{S}

2

*ch*|*G*_{up}−*G*_{down}|*T*
*K**up*+^{K}*down*

6.10

**6.1.2** **Thermoelectric cooling**

As said earlier, the left electrode, that is the normal lead serves as the cold
reservoir and the right one serves as hot reservoir. The junction device is con-
nected to an external bias voltage,*V*_{B} =(*E*_{F}^{N}−*E*_{F}^{S})*/e*, which drives the electrons to
flow from the normal lead to the superconducting lead. Thus the electron removes
the heat energy from the normal lead and transfers it to the superconducting lead
which further makes the cold reservoir (normal) cool. The thermal current from

normal to superconducting leads is given by,

*J**NS*_{σ} = ^{1}
2*e*^{2}*R*_{N}*E*_{F}^{N}

Z Z

(*E*−*eV**B*)(*E*+^{E}*F*^{N})*τ*_{σ}^{0}(*E, θ**N*1) 6.11
[*f*^{N}(*E*−*eV*_{B}*, T*^{N})−*f*^{S}(*E, T*^{S})]*dE*cos*θ*_{N1}*dθ*_{N1}

Similarly,

*J*_{SN}_{σ} = ^{1}
2*e*^{2}*R**N**E*_{F}^{N}

Z Z

*E*(*E*+^{E}*F*^{N})*τ*^{0}_{σ}(*E, θ**N*1) 6.12
[*f*^{N}(*E*−*eV**B**, T*^{N})−*f*^{S}(*E, T*^{S})]*dE*cos*θ**N*1*dθ**N*1

while energies are shifted by Fermi energy and*τ*^{0}_{σ} is given by the form,
*τ*_{σ}^{0}(*E, θ**N*1)=1− |*b**σ*(*E, θ**N*1)|^{2}− cos*θ**N*2

cos*θ*_{N1}|*a**σ*(*E, θ**N*1)|^{2} 6.13
This normal-insulator-superconductor (NIS) junction can be regarded as the
electronic cooling device only when *J**NS*_{σ} *>* 0, which implies that it is capable to
remove the heat from the cold reservoir, thereby making it cooler.

Hence the thermal conductance,*K*_{σ} = ^{dJ}_{dT}^{NS σ} is given by the form,
*K**σ* = ^{1}

2*e*^{2}*R**N**E*_{F}^{N}
Z Z

(*E*−*eV**B*)(*E*+^{E}*F*^{N})*τ*_{σ}^{0}(*E, θ**N*1) 6.14

"

*f*^{N}(*E*−*eV**B**, T*^{N})

*dT* − ^{f}

*S*(*E, T*^{S})
*dT*

#

*dE*cos*θ**N*1*dθ**N*1

The performance of this junction as a self-cooling device can be measured by
the coefficient of the performance (*COP*) where*COP* is defined as the ratio of the
heat removed from the cold reservoir to the electrical power needed for driving the
system. The*COP* for electronic thermal current, namely, *COP**σ* is given by,

*COP*_{σ} = ^{J}^{NS}^{σ}
*I**NS*_{σ}*V**B*

= ^{J}^{NS}^{σ}
*J**NS*_{σ} −*J**SN*_{σ}

6.15