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 Bi2Te3 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., Bi2Te3).

• Work has appeared showing that TI surface states in ultra-thin films (<10 nm) can lead to enhanced thermoelectric properties.

ZT ~ 2P. Ghaemi et al., Phys. Rev. Lett. 105, 166603 (2010).

ZT ~ 7F. Zahid and R. Lake, Appl. Phys. Lett. 97, 212102 (2010).

Next step is to reproduce and perhaps expand these results.

Overview of thermoelectric effect

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Ie = G ⋅ΔV + GS ⋅ΔTIQ = −GTS ⋅ΔV −κ 0 ⋅ΔT

Electric current:Heat current:

IeIQ

T1 T2ΔT = T1 – T2

ΔV = V1 – V2V1 V2

External parameters

G: Electrical conductanceS: Seebeck coefficientκ0: Thermal conductance (electronic contribution)

Material properties

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ZT = S2GTκ

Thermoelectricefficiency

Overview of thermoelectric effect

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ΔV = Ie G −S ⋅ΔTIQ = Π ⋅ Ie −κ e ⋅ΔT

Electric current:Heat current:

G: Electrical conductanceS: Seebeck coefficientκ0: Thermal conductance (electronic contribution)

IeIQ

T1 T2ΔT = T1 – T2

ΔV = V1 – V2V1 V2

Material propertiesSeebeck (S) : factor relating ΔT to ΔV (zero current).

Peltier (Π) : factor relating Ie to IQ (zero T-gradient).

Electronic transport in the Landauer picture

• Electrons flow when there is a difference in carrier occupation (f1 and f2).

• Carriers travel through the device region both elastically and ballistically (i.e. quantum transport).

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f1 = 1e(E−μ1 ) kBT1 +1

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μ1

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f2 = 1e(E−μ 2 ) kBT2 +1

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μ2 = μ1 − qVI

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μ1

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μ2

e- e-

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qV

Device/structureReservoir in thermodynamic

equilibirum

Reservoir in thermodynamic

equilibirum

Electronic transport in the Landauer picture

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I = 2qh

T ε( ) M ε( )∫ f1 ε( ) − f2 ε( )[ ] dε • Non-equilibrium transport

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G = 2q2

hT ε( ) M ε( )∫ −∂f0

∂ε ⎡ ⎣ ⎢

⎤ ⎦ ⎥dε • Near equilibrium (linear response)

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G' ε( ) = 2q2

hT ε( ) M ε( ) −∂f0

∂ε ⎡ ⎣ ⎢

⎤ ⎦ ⎥

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G = G' ε( ) dε∫• Differential conductance (energy-

dependent G)

• Average transmission times the number of conducting channels

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G = 2q2

hT ˜ M (Ballistic)

Scattering

Band structure

Diffusive transport in the Landauer picture

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σ =LA

G = LA

2q2

hT ε( ) M ε( )∫ −∂f0

∂ε ⎡ ⎣ ⎢

⎤ ⎦ ⎥dε

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σ ' ε( ) = 2q2

hλ ε( )

M ε( )A

−∂f0

∂ε ⎡ ⎣ ⎢

⎤ ⎦ ⎥

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σ =2q2

hλ

˜ M A

• Average mean-free-path times the number of conducting channels per unit area.

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σ =2q2

hλ ε( )

M ε( )A∫ −∂f0

∂ε ⎡ ⎣ ⎢

⎤ ⎦ ⎥dε

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T ε( ) =λ ε( )

λ ε( ) + L

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T ε( ) ≈λ ε( )

L

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λ ε( ): 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 G0.

In 2D or 3D, the “band counting” method for applies to every transverse k-state.

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M ε( ) = 12π( )

2 M ε,k⊥( ) dk⊥BZ∫

Si Fermi surface M(ε,k)

Effect of dimensionality on M(ε)

Parabolic bands

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E r k

= h2k 2

2m∗

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M3D ε( ) = m∗

2π h2 ε −εc( )

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M2D ε( ) =2m∗ ε −εc( )

π h€

M1D ε( ) = Θ ε −εc( )1D:

2D:

3D: S. Kim, S. Datta and M. Lundstrom, J. Appl. Phys. 105, 034506 (2009).

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1E

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E

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E

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E

Thermoelectric transport coefficients

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σ = σ ' ε( )dε∫

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S = − kB

q ⎛ ⎝ ⎜

⎞ ⎠ ⎟

ε − EF

kBT

⎡ ⎣ ⎢

⎤ ⎦ ⎥∫ σ ' ε( )

σdε

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κ0 = T kB

q ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

ε − EF

kBT

⎡ ⎣ ⎢

⎤ ⎦ ⎥2

∫ σ ' ε( ) dε

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κe = κ 0 − S2σ T

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σ ' ε( ) = 2q2

hM ε( )

Aλ ε( ) −∂f0

∂ε ⎛ ⎝ ⎜

⎞ ⎠ ⎟

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σ =G0 λ 0

˜ M A

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S = − kB

q ⎛ ⎝ ⎜

⎞ ⎠ ⎟

ε − EF

kBT

⎡ ⎣ ⎢

⎤ ⎦ ⎥

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κ0 = 2TLkB2

hλ 0

ε − EF

kBT

⎛ ⎝ ⎜

⎞ ⎠ ⎟2 ˜ M

A

Physically intuitive form (assuming constant λ0):

Conductivity

Seebeck

Electronic thermal conductivity (zero field)

Electronic thermal conductivity (zero current)

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ZT = S2σ Tκ e + κ l

Lattice thermalConductivity

(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: Bi2Te3 phonon modes (top) and lattice thermal conductivity (bottom). [Courtesy of Changwook Jeong]

T (K)

κ ph (W

m-1

K-1)

THz (s-1)

Mph

(1018

m-2

)

[Courtesy of Changwook Jeong]

Example: TI states in Bi2Te3 films

Estimate impact of TI surface states on the thermoelectric characteristics of variable thickness Bi2Te3 films.

• Electronic states of film: sum of bulk Bi2Te3 states (varying with tfilm) and TI surface states (independent of tfilm).

• Bulk states calculated from first principles.

• TI surface states approximated by analytical expression.

• Neglect TI/bulk and TI/TI hybridization.

Bulk statesBa

nd st

ruct

ure

Scatt

erin

g

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, 266801 (2009)]

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E±

r k ( ) = k 2

2m∗ ± vk2k 2 + λ6k 6 cos2 3θ( )

Analytical model:

vk = 2.55 eV Å λ = 250 eV Å3

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m∗ → ∞

Shape of the Fermi surface confirmed experimentally[Y. L. Chen et al., Science 325, 178 (2009)].

Iso-energy of TI state

Dispersion of TI stateAlignment of TI surface state relative to bulk Bi2Te3 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)

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σ =σ Bulkt film + GTI

s( )t film

Sheet conductivity

• Conductivity > 10x σBulk at tfilm = 10 nm.

• Significant difference between film and bulk σ at tfilm =100 nm.

• Surface conduction largest in bulk band gap.

• Large fraction of surface conduction for n-type (exp. EF @ 0.05 eV above CB*).

* Y. L. Chen et al., Science 325, 178 (2009).

Seebeck coefficient (TI + bulk states)

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S =SBulkσ Bulkt film + STI GTI

s( )σ Bulkt film + GTI

s

S weighted by conductance

• Max. Seebeck reduced ~35% @ 100nm and ~70% @ 10nm.

• Effect of TI surface state observed at 1µm.• How do results change with λsurf?

• When λsurf decreases 10x, S increases < 2x.

• Decreasing λsurf one order of magnitude is equivalent to increasing tfilm by the same factor.

tfilm = 10 nm tfilm = 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.

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ZT = S2σ Tκ e + κ l

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 Bi2Te3 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.