Thermoelectric properties of nanograting layers

A. Tavkhelidze

Ilia State University, Cholokashvili Ave. 3-5, Tbilisi 0162, Georgia

A.Tavkhelidze, Large enhancement of the thermoelectric figure of merit in a ridged quantum well, Nanotechnology 20 (2009) 405401.

Introduction

Geometry dependentquantum effects:

Periodic curved surfaces Nanotubes Cylindrical surfaces with non-constant diameterStrain-driven

nanostructuresGraphene Quantum billiards

Density of state (DOS) of nanograting layer

)(ρ0 EDOS in plain layer DOS in nanograting layer GEE /)(ρ)ρ( 0

G >1 is a geometry factor.

According to Fermi's golden rule, the electron scattering rate is proportional to Consequently,

)(E0 G

The geometry factor calculation requires solving the time-independent Schrödinger equation in NG geometry. Mathematically, there is no difference between DOS reduction and electromagnetic (TM) mode depression. The Helmholtz equation and Dirichlet boundary conditions are used in both cases.

J. H. Kim, M. Barth, U. Kuhl, H.-J. Stockmann and J. P. Bird, Phys. Rev. B 68, 045315 (2003).K.-F. Berggren, I. I. Yakimenko and J. Hakanen, New J. Phys. 12, 073005 (2010).

Geometry factor calculation

Literature related to Casimir effect, review: T. Emig, Casimir Forces and Geometry in Nanosystems, Nonlinear Dynamics of Nanosystems, ed. by G. Radons, B. Rumpf, H. G. Schuste (Wiley-VCH Verlag GmbH & Co. KGaA, 2010)

Software for mode calculation in ridged waveguides: FIMMMWAVE, photon design software (A fully vectorial 2D Mode Solver), ttp://www.photond.com/products/fimmwave.htm.CONCERTO, software for electromagnetic design, Vector Fields, http://www.vectorfields.com.

The approximate analytical expression known as Weyl’s formula allows the calculation of TM modes by using a ratio of layer surface area and volume.

H. P. Baltes and E. R. Hilf, Spectra of Finite Systems (Wissenschaftsverlag, Mannheim 1976).B. Eckhardt, Phys. Rep. 163, 205-297 (1988).

Perturbation method was used to obtain approximate formula G=(2H-a)/2a within the range of 3<.G<10 and for the case H, w>>a.

A.Tavkhelidze, V. Svanidze and I. Noselidze, Fermi gas energetics in low-dimensional metals of special geometry, J. Vac. Sci. Technol. B, v. 25(4), p.1270, (2007).

is number of TM modes from 0 to k.

for w=a and Hk >2.5.

Nanograting layer

Energy diagrams metal

Energy diagrams semiconductor

Chemical potential of NG layer

Sample preparation

Si substrateAu, Nb, Cr films wer quench depositedAt T=300 K and T=80 K.

Films deposited at T=300 K had polycrystalline structure.

Films deposited at T=80 K had amorphous structure.

A.Tavkhelidze et al., Observation of Quantum Interference Effect in Solids , J. Vac. Sci. Technol. B 24(3), p. 1413 (2006).

Maximum work function reductions of 0.5 eV in Au, 0.4 eV in Cr, 0.35 eV in Nb and 0.2 eV in SiO2 films were observed.

Kelvin probe was used to measure difference in work function between nanograting and plain areas.

AFM image of Au Nanogratilg layer

PEEM images of ridged Au film surface

Transport coefficients

)/( le2 SZ Materials having high S have low

Increasing leads to an increase ine (Wiedemann–Franz law)

We present large enhancement in S without changing e

)d()(eff TaTa

)G(G T

Calculate Z and compare with Zo where, Zo corresponds to )G(G T

J0

in Boltzmann transport equations and calculate S asWe insert

J0 SSS

Charge and heat transport

GDepletion depth depends on Y, and geometry factor gradient

appears in the Y-direction.

and modify the electron distribution function and cause electron motion from the hot side to the cold side

TG

TeJ 1211 / LL TeJ Q 2221 / LL and

jiLWithin the parabolic bands approximation are integrals

)(y)τ()ρ()(Ω2

0)( v EEEEfdEE

The NG does not change dispersion relation and consequently yv

Charge and heat transport

GEE /)(ρ)ρ( 0 )(τ)τ( 0 EGE For NG layer

2yv)τ()ρ( EE

and and consequently

product is G independent.

The NG influences integrals )(Ω )( Eby changing alone.

jiji0LL

C2/3

CB /2/ln NnNnTk

CBcon2/3

CBconB )1(2)1(ln NGnNGnTk

CBcon02/3

0B0 2)1/(1ln NnGGGGTk

)( 00 G

)1(COND GnN

Introduction of defines reference material as n+-type semiconductor with electron concentration of or NG having constant geometry

factor 0GG 0)/( TG

T θ0

T

G

GTGG

G

GkB

1

1

2

1

1

1lnθ 0

0

CBcon2/3 /2 Nn

Charge and heat transport

TeeJ // 110

1200

110 LLL

TeeJ Q // 210

2200

210 LLL

eSeS // 0110

110

120

LLL

0110

210

110

120

210

220 /)/( eeee

LLLLLL

0110 L

Geometry factor temperature dependence

)d()(eff TaTa

dTTdaGdTdGTG /)d(/)/(/ eff

2/1

Bbi

DAD

AS 2

)(

2)d(

e

Tk

NNN

N

eT

eE gbipbinbi

2/1

11Bg

co2

S

)]1(1[)1(

)2(2)d(

GG

TkE

neT

n

Acon / Nn

BBB1

np 23)1(2/35//)( kGkkTTETG )/(2 VA

2/3 NN

1)1()1(

2)2( B1

11

BgnpG

GTk

GG

G

G

G

GTkEE da /eff

CBcon2/3 /2 Nn

Seebeck coefficient of NL with p+–n+ junctions

T

G

Ge

TkSS

1

1B0

**)(F

*)(F

2/3

2/5

1/2r

3/2rB0

r

r

e

kS

r is a scattering parameter

TkB/*

200/ SSZZ

Fabrication – UV Interference lithography

H. S. Jang et al. Current Applied Physics , 10, 2010, pp. 1436–1441

C. P. Funcetola, H. Korre, and K. K.Berggren. Low –cost interference lithography. J.Vac. Sci. Technol. B 27, (2009)

Low cost <1000 $ interference lithography from MIT

Optical microscope images of nanogratings formed in photo resist

Simple Lloyd interferometer based on 405 nm laser diode.

Preliminary experiments and results

PSI, Laboratory for Micro- and Nanotechnology

Fabrication – X-ray Interference lithography

Fabrication – multilayer epitaxy

G. W. Pickrell et al., JOURNAL OF APPLIED PHYSICS V 96, 4050, 2004

Cross-sectional, transmission electron microscopy micrograph of the sample grown using an interfacial superlattice with a growth rate of 1.0 ML/ sec. The diffraction grating can be seen at the bottom of the figure and the planarized DBR layers can be seen near the top of the micrograph.

Multiple NG layers

Conclusions

1. Nanograting on the surface of thin quantum well layer reduces density of quantum states and increase chemical potential (Fermi level).

2. Work function reduction has been observed in NG films made from Au, Cr, Nb.

3. NG reduces electron scattering rates and increases electron mobility.

4. When p-n junctions are grown on the top of NG additional builds up under influence of . This leads to dramatic increase in ZT.

5. Large areas of NG having pitch of 20 nm can be fabricated using interference lithography without masks.

6. Multiple NG layers can be fabricated by epitaxial grown on NG base substrate.

T