Quantum Interference Enhancement of the Spin-Dependent Thermoelectric Response

We investigate the influence of quantum interference (QI) and broken spin-symmetry on the thermoelectric response of node-possessing junctions, finding a dramatic enhancement of the spin-thermopower (Ss), figure-of-merit (ZsT), and maximum thermodynamic efficiency (ηsmax) caused by destructive QI. Using many-body and single-particle methods, we calculate the response of 1,3-benzenedithiol and cross-conjugated molecule-based junctions subject to an applied magnetic field, finding nearly universal behavior over a range of junction parameters with Ss, ZsT, and reaching peak values of , 1.51, and 28% of Carnot efficiency, respectively. We also find that the quantum-enhanced spin-response is spectrally broad, and the field required to achieve peak efficiency scales with temperature. The influence of off-resonant thermal channels (e.g., phonon heat transport) on this effect is also investigated.

Calculations were performed via exact diagonalization of the effective Hamiltonian derived from first principles.Although the transmission function is not symmetric (the HOMO and LUMO resonances differ significantly due to orbital symmetry, charging, and image charge effects), the transmission near the nodal energy µ 0 ≈ 1.94eV is nearly quadratic (fits are shown in the dashed lines).Consequently, the thermopower near µ 0 is nearly identical in all cases, despite the wide range of coupling strength and asymmetry values.In these calculations the constant A = 5 × 10 −5 and the temperature was set to 300K.
terms are included leading to the following renormalized parameters: An on-site repulsion of U nn = 9.69eV , an effective inter-molecular hopping integral t = 2.70eV , an on-site energy of ε n = −4.06eV, effective π quadrupole moment of Q = −0.64eÅ2 and a dielectric screening of ε = 1.56.S3 Molecular geometries were obtained by optimizing the isolated molecules using Q-Chem 3.0 S4 with density functional theory employing the B3LYP functional and 6-311G** basis.
The molecules were then chemisorbed (terminal hydrogens removed) to the FCC hollow binding site of a Au(111) surface with the Au-S bond lengths of 2.10 Å and 2.48 Å in the BDT and CC junctions, respectively.S5,S6 Image charge effects reduce the fundamental gap of each molecule significantly.
As indicated in Figs.S1 and S2, variations in the tunnel-coupling symmetry and total strength of several orders of magnitude play a negligible role on the form of the transport near the nodal energy, i.e.T (E) ∝ (E − µ 0 ) 2 , where µ 0 ≈ 1.94eV and ≈ 1.333eV in the BDT and CC junctions, respectively.Comparing these results to the Hückel calculations in Fig. 1 of the manuscript shows that electron-electrons also don't influence the form of T and therefore S near the node.These conclusions are supported by alternative theoretical methods S6-S12 and direct experimental observations S13-S16 of similar systems.
The near-universality of the thermopower near a node detuned from any molecular resonances can also be understood by realizing that the dominant contribution to the transport in the mid-gap region are the tails of the HOMO and LUMO resonances.In junctions with a node, the junction's Green's function in that case gives a transmission which tends to zero quadratically in that case.S13,S15,S17 The Kondo temperature The Kondo effect involves correlations between the lead and molecular electrons, leading to screening of unpaired spins within the junction at temperatures below the Kondo temperature T K .The Kondo temperature in a SMJ may be estimated from the formula S18-S20 where µ ≈ µ 0 , Γ = α Γα is the total effective tunnel-coupling strength, U = ε LUMO −ε HOMO is the HOMO-LUMO gap and µ is the chemical potential.Γα is not equal to α Γ α since the many-body matrix elements are not equal to unity.We extract Γ from the transmission function where Γ is the full-width at half maximum (FWHM).
We can estimate the Kondo temperature for the 1,3-BDT junction, using the nodal We have assumed the same FWHM for both the HOMO and LUMO resonance in the 1,3-BDT junction since the LUMO resonance deviates strongly from a Lorentzian in this case.For the CC junction, µ ≈ 1.33eV, U ≈ 6.9eV, µ − ε HOMO =1.33+1.82eV,µ + ε HOMO =5.08-1.33eV,and Γ ≈ 0.1882 + 0.1255eV, giving T K ≈ 2.98 × 10 −4 K.These calculations support our claim that the T K is below the mK range.
Molecules containing magnetic elements are more favorable for the observation of the Kondo effect.S21,S22 Peak splitting and chemical potential for Z el (s) T The total and spin figures-of-merit, which may be expressed as S24,S25 has a peak value which differs from S s since G s and κ are functions of electron energy and temperature.Although analytic solutions for the spin-splitting and chemical potential needed for peak performance, i.e. ∆ peak and µ peak , are possible for simple quadratic nodes they are complex expressions.Here, we extract the peak values numerically from the calculated response of a BDT junction, where we vary temperature, ∆, and µ using Hückel+NEGF theory.
We include analyses for both ZT and Z s T , shown in Fig. S4c and Fig. S3b, respectively.
As discussion in the main text, according to theory ZT should be peaked when ∆ = 0 and The influence of interactions on the spin-thermoelectric response In this section, we repeat the calculations used to generate Fig. 2 and 3 of the manuscript using an effective Hückel calculation This method utilizes the optimized geometries, renormalized Hamiltonian, etc. but sets U nm = 0.As shown in Fig. S4, S (s) , Z (s) and η max (s) are all identical to the many-body case, as expected since the transmission function's form is nearly universal.
As in the many-body case, max(S s ) = 2S 0 when µ = 0 and ∆ = ∆ max Ss , where In contrast, the influence of κ alt , shown in Fig. S5 is different when Hückel or many-body theory is used.Comparing the left-and right-hand panels of the figure reveals that for the same κ alt , the peak values of Z s T are reduced (proportionally) less than those of ZT .For the systems considered here, this occurs because Z s T peaks when ∆ > 0 which increases κ el and reduces κ alt /κ el .For instance, when κ alt = 10 −6 κ 0 and ∆ is fixed (top panels), the peak value of Z s T is reduced by 14.8% while the peak value of ZT is reduced by 44.9%.
When ∆ is tuned, the reductions generally decrease and the spectral width of the enhancement increases.For the same κ alt value with tuned ∆ (bottom panels), the reduction of the peak value of Z s T becomes 11.5% while ZT remains unchanged, although in both cases the enhancement has broadened significantly in µ.As κ alt is increased so does the  Owing to the influence of ∆ on κ, Z s T is less sensitive to additional thermal channels than ZT .When κ alt = 10 −6 κ 0 , Z s T is reduced by 11.5% when ∆ is tuned and 14.8% when ∆ is fixed while ZT is reduced by 44.9% in both cases.
The optimal ∆ is a function of the total transport, so the importance of tuning ∆ increases with κ alt .When κ alt = 10 −5 κ 0 , Z s T is reduced by 34.2% and 61.7% when ∆ is tuned or fixed, respectively, while ZT is reduced by 65.2% and 87.5%, respectively.
importance of tuning ∆.With κ alt = 10 −5 κ 0 , Z s T is reduced by 61.7% when ∆ is fixed but only 34.2% when ∆ is tuned.Similarly, ZT is reduced by 87% when ∆ is fixed and 65% when ∆ is tuned for the same κ alt .Although the values are different for the Hückel calculations, the trends and relative influence of κ alt on the spin-response are very similar to the many-body case presented in the main text.
Relationship between the η max (s) and Z (s) T spectra and the importance of interactions The calculated ZT and η max spectra and Z s T and η max s spectra of a 1,3-BDT junction are shown in panels (a) and (b) of Fig. S6, respectively, as a function of electrode chemical , indicating that ZT is a good measure of maximum thermodynamic performance in these system.Calculations utilize many-body theory with π-EFT and are for junctions operating at 300K.potential µ.Calculations were performed using MDE many-body theory with π-EFT with the nodal energy is set to zero for convenience.Solid lines correspond to ∆ = 0 in the ZT and η max plots and ∆ = 2π/ √ 3kT in the Z s T and η max s plots.In both the fixed and tuned cases, variation in Z (s) T match variations in η (s ) max , supporting our claim that the figure-of-merit is an accurate measure of the maximum thermodynamic device performance in these systems.S23 For example, with κ alt = 10 −6 κ 0 and ∆ fixed, ZT is reduced by 73.5% and η max by 67.4% while Z s T is reduced by 37.4% and η max s by 27.3%.
As usual, the reduction in these quantities can be mitigated when ∆ is tuned.For κ alt = 10 −6 κ 0 this gives ZT and η max reductions of 60.6% and 53.5%, respectively, with significantly  Unlike the nodal response, interaction play a role in determining the influence of κ alt .
This occurs because the magnitude of the thermal conductance is roughly proportional to the transmission which does depend strongly on the chemical structure, energy levels, etc.The response for the same junction found using Hückel+NEGF (i.e.MDE many-body theory with U nm = 0) is shown in Fig. S7.The reduction values in this case are given in Tab.S2, where smaller reductions are exhibited since the transmission predicted by Hückel+NEGF theory is significantly larger than that predicted by many-body theory.

Figure S1 :
Figure S1: The calculated transmission function (top panel) and thermopower (bottom panel) of a 1,3-BDT junction with different coupling strengths and coupling symmetries.Calculations were performed via exact diagonalization of the effective Hamiltonian derived from first principles.Although the transmission function is not symmetric (the HOMO and LUMO resonances differ significantly due to orbital symmetry, charging, and image charge effects), the transmission near the nodal energy µ 0 ≈ 1.94eV is nearly quadratic (fits are shown in the dashed lines).Consequently, the thermopower near µ 0 is nearly identical in all cases, despite the wide range of coupling strength and asymmetry values.In these calculations the constant A = 5 × 10 −5 and the temperature was set to 300K.

Figure S2 :
Figure S2: The calculated transmission function (top panel) and thermopower (bottom panel) of a CC junction with different coupling strengths and coupling symmetries.Calculations were performed via exact diagonalization of the effective Hamiltonian derived from first principles including thiol end-groups, image charge effects, etc.Like the 1,3-BDT junction these spectra are not symmetric.Near the nodal energy (µ 0 ≈ 1.33eV) the transmission is nearly quadratic (see fits shown in the dashed lines).Consequently, the thermopower in each case are indistinguishable when µ ∼ µ 0 .For the quadratic fit A = 5 × 10 −4 .The temperature was set to 300K.
Figure S3: The calculated spin-splitting ∆ peak and chemical potential µ peak needed to realize peak performance for (a) ZT and (b) Z s T as a function of temperature.The analytic fit for ZT 's peak values and numerical fit for Z s T 's peak values are shown in teh solid lines of each panel.Calculations were for the 1,3-BDT junction although the results apply to any quadratic node.
15) 1/4 kT .The calculated ∆ peak and µ peak values shown in panel (a) of the figure support these theoretical results and show our method can accurately extract peak value quantities from the data.Applying this procedure to Z s T , shown in panel (b) of the figure, we find the values reported in the main text.
Similarly, Z el T and η max decrease monotonically as a function of ∆ while Z el s T and η max s are peaked, as indicated in sub-figures (c) and (d), respectively.Z el s T reaches a maximum value of 1.51 while η max s reaches 27.96% of Carnot when ∆ = ∆ max ZsT .

Figure S4 :
Figure S4: The calculated (a) charge thermopower S, (b) spin-thermopower S s , (c) Z el T , η max and (d) spin Z el s T , η max s spectra of a 1,3-BDT junction's π-system shown as functions of electrode chemical potential µ and spin-splitting ∆ = 2gµ B B using Hückel+NEGF theory.Charge quantities decrease monotonically with increasing ∆.Spin quantities are peaked, reaching maxima when ∆ = 2π/ √ 3kT ≈ 3.63kT and ∆ ≈ 11.8kT for S s and Z s T (η max s ), respectively.In the lower panels quantities are shown with µ tuned to give the maximum response (µ = µ peak ) and with µ fixed to specific values.Calculations are for T = 300K such that S 0 = π/ √ 3(k/e) ≈ 156µV/K.

Figure S5 :
Figure S5:The influence of κ alt on the Z s T (left panels) and ZT (right panels) spectra of a 1,3-BDT junction calculated using Hückel+NEGF theory.Values are normalized to the node-possessing channel's peak value.Owing to the influence of ∆ on κ, Z s T is less sensitive to additional thermal channels than ZT .When κ alt = 10 −6 κ 0 , Z s T is reduced by 11.5% when ∆ is tuned and 14.8% when ∆ is fixed while ZT is reduced by 44.9% in both cases.The optimal ∆ is a function of the total transport, so the importance of tuning ∆ increases with κ alt .When κ alt = 10 −5 κ 0 , Z s T is reduced by 34.2% and 61.7% when ∆ is tuned or fixed, respectively, while ZT is reduced by 65.2% and 87.5%, respectively.κ 0 = π 2 k 2 T /3h ∼ 289pW/K at T = 300K.

Figure S6 :
Figure S6: The calculated (a) ZT and η max spectra and (b) Z s T and η max s spectra of a 1,3-BDT junction as a function of electrode chemical potential µ.The nodal energy is set to zero for convenience.Dashed lines and solid lines represent fixed spin-splittings, respectively, with ∆ = 0 for the total charge quantities and ∆ = 2π/ √ 3kT for the spin quantities.Variations in Z (s) T match variations η max (s), indicating that ZT is a good measure of maximum thermodynamic performance in these system.Calculations utilize many-body theory with π-EFT and are for junctions operating at 300K.

Figure S7 :
Figure S7: The calculated (a) ZT and η max spectra and (b) Z s T and η max s spectra of a 1,3-BDT junction using Hückel+NEGF theory.The nodal energy is set to zero for convenience.Dashed lines and solid lines represent fixed spin-splittings, respectively, with ∆ = 0 for the total charge quantities and ∆ = 2π/ √ 3kT for the spin quantities.Although the trends are still essentially the same as in the many-body case, the reductions are slightly different (cf Tab.S2).Calculations are for junctions operating at 300K.

Table S1 :
Influence of κ alt on Z (s) T and η max (s) from many-body theory.
broader responses.The spin-response sees similar changes with Z s T and η max s exhibiting reductions of only 21.1% and 14.5%, respectively.A full accounting for the reductions found S-10 can be found in Tab.S1