Modeling thermoelectric properties of TI materials: a Landauer approach Jesse Maassen and Mark Lundstrom Network for Computational Nanotechnology, Electrical and Computer Engineering, Purdue University, West Lafayette, IN USA DARPA-TI meeting, April 25, 2012

Overview Motivation. Summary of the thermoelectric effect. Thermoelectric modeling within the Landauer approach. Example: effect of TI surface states on the thermoelectric properties of Bi 2 Te 3 films.

Motivation In recent years, much research has focused energy-related science and technology, in particular thermoelectrics. Some of the best known thermoelectric materials happen to be topological insulators (e.g., Bi 2 Te 3 ). Work has appeared showing that TI surface states in ultra-thin films (<10 nm) can lead to enhanced thermoelectric properties. ZT ~ 2 P. Ghaemi et al., Phys. Rev. Lett. 105, (2010). ZT ~ 7 F. Zahid and R. Lake, Appl. Phys. Lett. 97, (2010). Next step is to reproduce and perhaps expand these results.

Overview of thermoelectric effect Electric current: Heat current: IeIe IQIQ T1T1 T2T2 ΔT = T 1 – T 2 ΔV = V 1 – V 2 V1V1 V2V2 External parameters G: Electrical conductance S: Seebeck coefficient κ 0 : Thermal conductance (electronic contribution) Material properties Thermoelectric efficiency

Overview of thermoelectric effect Electric current: Heat current: G: Electrical conductance S: Seebeck coefficient κ 0 : Thermal conductance (electronic contribution) IeIe IQIQ T1T1 T2T2 ΔT = T 1 – T 2 ΔV = V 1 – V 2 V1V1 V2V2 Material properties Seebeck (S) : factor relating ΔT to ΔV (zero current). Peltier (Π) : factor relating I e to I Q (zero T-gradient).

Electronic transport in the Landauer picture Electrons flow when there is a difference in carrier occupation (f 1 and f 2 ). Carriers travel through the device region both elastically and ballistically (i.e. quantum transport). I e-e- e-e- Device/structure Reservoir in thermodynamic equilibirum Reservoir in thermodynamic equilibirum

Electronic transport in the Landauer picture Non-equilibrium transport Near equilibrium (linear response) Differential conductance (energy- dependent G) Average transmission times the number of conducting channels (Ballistic)

Scattering Band structure Diffusive transport in the Landauer picture Average mean-free-path times the number of conducting channels per unit area. : Mean-free-path for backscattering

What is M( ε )? Courtesy of Changwook Jeong M(ε) is the number of conducting channels. One band = One mode for conduction (“band counting” method). Roughly corresponds to number of half- wavelengths that fit in cross-section. Each mode contributes a conductance of G 0. In 2D or 3D, the “band counting” method for applies to every transverse k-state. Si Fermi surface M(ε,k)M(ε,k)

Effect of dimensionality on M( ε ) Parabolic bands 1D: 2D: 3D: S. Kim, S. Datta and M. Lundstrom, J. Appl. Phys. 105, (2009).

Thermoelectric transport coefficients Physically intuitive form (assuming constant λ 0 ): Conductivity Seebeck Electronic thermal conductivity (zero field) Electronic thermal conductivity (zero current) Lattice thermal Conductivity (phonon)

Lattice thermal transport within Landauer Lattice / phonon transport is the same as electron transport within the Landauer approach. In principle, one can utilize the Landauer model to perform a complete assessment of thermoelectric performance (electron + phonon). Figures: Bi 2 Te 3 phonon modes (top) and lattice thermal conductivity (bottom). [Courtesy of Changwook Jeong] T (K) κ ph (Wm -1 K -1 ) THz (s -1 ) M ph (10 18 m -2 ) [Courtesy of Changwook Jeong]

Example: TI states in Bi 2 Te 3 films Estimate impact of TI surface states on the thermoelectric characteristics of variable thickness Bi 2 Te 3 films. Electronic states of film: sum of bulk Bi 2 Te 3 states (varying with t film ) and TI surface states (independent of t film ). Bulk states calculated from first principles. TI surface states approximated by analytical expression. Neglect TI/bulk and TI/TI hybridization.

Bulk states Band structure Scattering Good comparison with experiment using constant MFP. Exp. data: Proc. Phys. Soc. 71, 633 (1958). Deeper in VB Deeper in CB

TI surface states [L.Fu, Phys. Rev. Lett. 103, (2009)] Analytical model: v k = 2.55 eV Åλ = 250 eV Å 3 Shape of the Fermi surface confirmed experimentally [Y. L. Chen et al., Science 325, 178 (2009)]. Iso-energy of TI state Dispersion of TI state Alignment of TI surface state relative to bulk Bi 2 Te 3 taken from exp. study. [Y. L. Chen et al., Science 325, 178 (2009)]. Distribution of modes is linear in energy. Distribution of modes (TI state) λ is taken to be 100 nm [F. Xiu et al., Nature Nano. 6, 216 (2011)].

Conductivity (TI + bulk states) Sheet conductivity Conductivity > 10x σ Bulk at t film = 10 nm. Significant difference between film and bulk σ at t film =100 nm. Surface conduction largest in bulk band gap. Large fraction of surface conduction for n-type (exp. E 0.05 eV above CB*). * Y. L. Chen et al., Science 325, 178 (2009).

Seebeck coefficient (TI + bulk states) S weighted by conductance Max. Seebeck reduced 100nm and 10nm. Effect of TI surface state observed at 1µm. How do results change with λ surf ? When λ surf decreases 10x, S increases < 2x. Decreasing λ surf one order of magnitude is equivalent to increasing t film by the same factor. t film = 10 nm t film = 100 nm

Power factor (TI + bulk states) Significant reduction in power factor with the presence of TI surface states. Aside from conductivity, all thermoelectric characteristics are degraded with the surface states. Hinder surface conduction by enhancing scattering or destroying the surface states. Surface roughness or adding magnetic impurities may enhance thermoelectric performance.

Conclusions Landauer approach is a powerful formalism for calculating the thermoelectric coefficients of materials, particularly when combined with full band descriptions of electronic dispersion. This method naturally spans from ballistic to diffusive transport regimes and considers bulk and nano-scale systems. Within our example, TI surface states were shown to degrade the thermoelectric performance of Bi 2 Te 3 films (when the thickness is large enough to form a gap in the TI states). Hindering surface conduction may enhance thermoelectric performance, e.g. introducing surface roughness and/or magnetic impurities.