modeling thermoelectric properties of ti materials: a landauer approach
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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.
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