**6.2 Results and Discussions**

**6.2.1 Seebeck Coefficient and Figure of Merit**

Figure 6.1: *The variation of the Seebeck coefficient, S as a function of temperature,*
*T (scaled by superconducting order parameter,* ∆0*)* (*a*) *for pristine graphene,* (*b*)
*for Au decorated graphene. The values of the spin-orbit coupling are shown in the*
*figure.*

Initially we show the results of Seebeck coefficient for a pristine graphene
(*λ**R* =^{0, λ}*I* = 0). The variation of the Seebeck coefficient,*S* as function of the tem-
perature (in units of superconducting gap, ∆0) for a pristine graphene is shown
in Fig.(6.1a). This is included for comparison with that of Fig.(6.1b). The Seebeck
coefficient, *S* is dimensionless (since *e* = ^{1 and} ^{k}*B* = 1). It is understood that
the Seebeck coefficient increases initially with temperature and after attaining
a certain value it decreases very slowly. In this regard we should remind our-
selves that the temperatures should be in a range such the superconductivity
is not destroyed. From BCS theory it can be shown that the relationship be-
tween superconducting temperature and the superconducting order parameter
is *T*_{c} ∼ (0*.*5− _{0.6})∆0. Now to get an idea about the role of spin-orbit couplings

Figure 6.2: *The variation of the Seebeck coefficient, S as a function of temperature,*
*T/*∆0*for a larger RSOC parameter by one order greater magnitude compared to that*
*of the Au decorated graphene.*

on value of the thermopower, in Fig.(6.1b) we present the variation of the spin
resolved Seebeck coefficient, *S* as function of the temperature (in units of su-
perconducting gap, ∆0) for an Au decorated graphene. From the first principal
calculations, in the Au decorated graphene the values of the following parameters
are, *λ**I* = ^{0.007t}1 and *λ**R* = ^{0.0165t}1 [87] and one does not notice any significant
change in the thermopower profile. Thus, by some means if we are able to en-

Figure 6.3: *The variation of the spin resolved Seebeck coefficient, S as function*
*of λ**R* *and λ**I* *for* (*a*) *up spin,* (*b*) *down spin. Reddish yellow regions indicate the*
*parameters values needed for the achieving maximum, S.*

hance the SOCs by one order of magnitude compared to the value present in the Au decorated graphene, there could be noticeable effects of SOC. Thus in Fig.(6.2) we have shown the thermopower profile with one order of greater magnitude of RSOC strengths where, indeed noticeable changes are obtained. For this reason in the later discussions we shall use these values of the SOCs strength. The in- teresting fact is that, in case of a normal junction (that is not graphene based) in the presence of RSOC, there is no spin resolved thermopower, which graphene

based junction devices, in presence of a bit high RSOC strengths show the spin resolved thermopower. The reason behind the spin resolved Seebeck coefficient is the same as that of the spin resolved conductance which has been explained in previous chapter. It is clearly understood that with the inclusion of the SOC parameters the Seebeck coefficient increases. Further the up spin shows larger values of thermopower compared to that of the down spin.

Figure 6.4: (*a*)*The variation of the charge Seebeck coefficient as function of λ*_{R}*and*
*λ*_{I}*,*(*b*)*The variation of the spin Seebeck coefficient as function of λ*_{R} *and λ*_{I}*.*

To get an idea how the spin resolved Seebeck coefficient vary with both of the
spin-orbit couplings, and also to get an operating regime in the parameter space,
we have shown the spin resolved Seebeck coefficient as a function of *λ*_{R} and *λ*_{I}
in Fig.(6.3a) and Fig.(6.3b) with temperature, *T* = ^{0.5}∆0. The color plots yield
the information of the Seebeck coefficient for different values of the RSOC and the
ISOC parameters. For certain values of the RSOC parameter (*>*0*.*1*t*1), irrespective
of the ISOC strength, both spins show higher values of thermopower, So we can
infer that for both the spins RSOC enhances the thermopower.

Further we have shown results of the charge and spin Seebeck coefficients
in Fig.(6.4a) and Fig.(6.4b). The charge Seebeck coefficient shows higher values
for larger strengths of RSOC, and for certain values of the SOC parameters, the
spin Seebeck coefficient vanishes. This map gives an idea of the magnitude of
the Seebeck coefficient corresponding to a variety of choices of *λ*_{R} and *λ*_{I}. As the
strengths of SOCs correspond to presence of different adatoms, a careful choice
of the periodic table may provide useful information on tunable thermopower of
these junction devices.

Now we show the results on the ’Figure of Merit’ (FM) which defines the effi-
ciency of this system as a thermopower device. The variation of the Figure of Merit,
*ZT* as function of the temperature (in units of superconducting gap,∆0) is shown
in Fig.(6.5). It shows that, initially with temperature the efficiency of the system
as a thermopower device increases and, after attaining a certain temperature it

decreases. The variations of the spin dependent FM, *Z*_{σ}*T* as the function of the

Figure 6.5: *The variation of Figure of Merit, ZT as a function of temperature, T*
*(scaled by superconducting order parameter,* ∆0*) for* (*a*) *λ*_{R} = ^{0} ^{and λ}*I* = ^{0,} (*b*)
*λ**R* =0*.*165*t*1*and λ**I* = 0*.*007*t*1*.*

spin-orbit couplings are presented in Fig.(6.6). This map gives an idea of the FM
for different spins corresponding to different choices of *λ**R* and *λ**I*. Interestingly,
the down spin shows more efficiency compared to that of the up spin. Further we

Figure 6.6: *The variation of the ’Figure of Merit’, ZT as function of λ*_{R}*and λ*_{I} *for*(*a*)
*up spin and*(*b*)*down spin.*

have shown the results for charge and spin FM in Fig.(6.7a) and Fig.(6.7b) where it is observed that for higher values of ISOC the charge FM becomes larger. The spin FM becomes zero for the lower values of the RSOC parameters irrespective of the ISOC strengths. Such regions, along with others, are shown by black patches in Fig.(6.7b). Thus these maps aid in deciding on the values of the parameters that may be used for maximizing the gain of these KMNIS junction devices.

Figure 6.7: (*a*)*The variation of the charge ’Figure of Merit’, Z*_{ch}*T as function of λ*_{R}
*and λ*_{I}*,*(*b*)*The variation of the spin ’Figure of Merit’, Z*_{sp}*T as function of λ*_{R}*and λ*_{I}*,*