Download Laboratory 1

EEW 522 Quantum Transport (Fall 2012)
Final Project: Phonon Transport
(Due: Thursday, December 20)
I. References and recommended reading
[0] B. J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988). Measurement of quantized electrical
conductance 2e2/h = 77.6 S.
[1] K. Schwab et al., Nature 404, 74 (2000). Measurement of quantum thermal conductance
π 2 k B2T 3h = (9.456  10-13 W/K2) T.
[2] J.-S. Wang et al., Eur. Phys. J. B 62, 381 (2008) + Mathematica codes.
Mandatory reading: Ch. 1, Ch. 2.1, Ch. 2.2.6, & Ch. 2.2.7. Optional recommended reading for the
materials we have discussed in terms of electrons: Ch. 3.1 (for the transition from ballistic to diffusive
regimes), Chs. 3.2.1, 3.2.2, 3.2.3, 3.2.5 (for the “advanced” derivation of the NEGF formalism), Ch. 4
(for electron-phonon coupling).
[3] W. Zhang et al., Numerical Heat Transfer: Part B (Fundamentals), 51, 333 (2007).
[4] NanoHub simulation tool "Atomistic Green's Function Method 1-D Atomic Chain Simulation,"
(http://nanohub.org/resources/greentherm).
II. Computational implementation
 3  ω  2  n 
0
 
1. Plot the window function for the phonon transport  2 
 at different temperatures.
k
T

(

ω
) 
π
  B  

Compare its behavior with that for the electronic transport [cf. HW #1-3(a)].
2. By modifying the subroutine written for HW #1-3(c), construct a program that calculates the thermal
conductance K according to:
K 
Q
,
ΔT
where
Q
1
d (ω)ωT ph (ω)n1 (T1 )  n2 (T2 ) .
h
Use the unit of quantum thermal conductance.
3. After analyzing the Mathematica codes by referring to Refs. [2] and [3], transcribe it to your own
program using the programming language of your choice.
EEW 522 Quantum Transport (Fall 2012)

Input should be (1) atomic mass m in atomic mass unit and (2) spring constant k in N/m unit, i.e.
you should construct a dynamical matrix in the form of Eq. (2) for the given m and k.

Number of atoms in electrode 1, channel, and electrode 2 should be flexible. Note that the case
given in the Mathematica codes corresponds to 2, 2, 1 atoms for the “left contact”, “device”, and
“right contact”, respectively.

Transcribe and explain the implemented mathematical formulae and algorithms for
- eigen / generalizedeigen
- surface1green / surface2green / surface3green.
- leftlead / flip
- caroli
III. Computer experiments.
4. First, experiment the infinite homogeneous chain case (i.e.
kL01=VLC=kR01, kL11=kL00=KC).
Check the
convergence of the results with respect to the number of atoms within the device region. Varying m and
k, calculate transmissions, conductances, and density of states (DOS). Compare the results with those
from "greentherm" tool of Ref. [4]. Discuss the results.
5. Next, reproduce the results of the reference Mathemtaica code for the inhomogeneous chain case.
6. Finally, explore various inhomogeneous chain cases. Compare the results with those from
"greentherm" tool of Ref. [4]. Using DOS, explain the observed variations of transmission (and
conductance). Explain the corresponding physical implications.