quantum transport in nanostructures - jyväskylän...

Fundamentals of nanoscience 12.2.2008

Quantum transport in nanostructures-II

Prof. Ilari MaasiltaNanoscience Center, Department of Physics, University of Jyväskylä

YN 215, [email protected]


Some topics in quantum transport of heat

• Energy transport mechanisms• Electronic heat conduction• Top-down nanofabrication• Phonons, phononic heat

conduction, quantum of thermalconduction

• Electron-phonon interaction• Photonic heat radiation• Applications in detectors


Energy transport mechanisms• Most QM particles will carry energy with them (but not

Cooper pairs!)• Energy is involved if the particle is an excitation of the

ground state1. Particles of ”matter”, i.e. atoms, ions, electrons. In solid

state physics, atoms and ions do not move largedistances but just vibrate

2. Insulators vs. conductors: In insulators also electronsare bound => no charge transport. Also no electricheat conductivity. In conductors there are freeelectrons that can carry charge and heat

3. Particles of ”fields”, i.e. photons and phonons


Energy transport vs heat transport

• Energy vs. heat transport: Energy transport is a general concept, applies to all situations, includingnon-equilibrium

• In addition, one can study emission and absorption of energy when bodies have well defined temperatures(near thermal equilibrium) => transport of heat

• Thermal equilibrium: T1=T2• Typically one measures thermal conductivity

(Fourier’s law):

Tjq ∇−= κ


Electronic heat transport• At temperature T, electrons in a solid

have a spread in energy of ~ kBT (the total energy is so called Fermi energy EF~ 50000 K!)

• This is due to the quantum nature of electrons (no two electrons can occupythe same state, Pauli principle, QM matters even at RT)

• Both thermal and charge conductivitiesdepend on differences of electron states=> EF is not relevant (left-movingelectrons and right moving electrons)

• At low temperatures, where most of scattering is from boundaries and impurities (not from vibrations), one canshow that thermal conductivity κ is related to electrical conductivity simplyby the Wiedemann-Franz law:

TLσκ =22

3⎟⎠⎞

⎜⎝⎛=

ekL Bπ


Wiedemann and Franz• This empirical law is named after Gustav Wiedemann

and Rudolph Franz, who in 1853 reported that K/σ has approximately the same value for different metals at the same temperature. The proportionality of K/σ with temperature was discovered by Ludvig Lorenz in 1872.

• Ludvig Valentin Lorenz (January 18, 1829 - June 9, 1891) was a Danish mathematician and physicist.

• Not to be confused with Hendrik Antoon Lorentz (July 18, 1853, Arnhem – February 4, 1928, Haarlem) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect.

• The Lorentz-Lorenz formula is named after the Danish mathematician and scientist Ludvig Lorenz, who published it in 1869, and the Dutch physicist HendrikLorentz, who discovered it independently in 1870.

How did Lorenz discover his ”number”??? (No QM at the time)

http://en.wikipedia.org/wiki/Empirical

http://en.wikipedia.org/wiki/Gustav_Heinrich_Wiedemann

http://en.wikipedia.org/w/index.php?title=Rudolph_Franz&action=edit

http://en.wikipedia.org/wiki/1853


http://en.wikipedia.org/wiki/Ludvig_Lorenz

http://en.wikipedia.org/wiki/Ludvig_Lorenz


http://en.wikipedia.org/wiki/January_18


http://en.wikipedia.org/wiki/June_9


http://en.wikipedia.org/wiki/Danish_people

http://en.wikipedia.org/wiki/Physicist

http://en.wikipedia.org/wiki/July_18







http://en.wikipedia.org/wiki/Physics

http://en.wikipedia.org/wiki/Nobel_Prize_in_Physics




http://en.wikipedia.org/wiki/Lorentz-Lorenz

http://en.wikipedia.org/wiki/Hendrik_Lorentz




Phonons• But: We know that even

an insulator tranportsheat. How?

• Even though the atoms of the solid do not travel, they vibrate. Thesevibrations can becollective in such a waythat the form a wave thatextends over the wholesample. The smallest QM ”quantized” (undivisible) unit of this wave is calleda phonon.

pE ωh=


Phonon velocity and occupation• Phonons form energy

bands in the solid, and the low energy (acoustic) modes are always linear in wavevector. Slope is the velocity of the wave = speed of sound cs ~ 1000 –10000 m/s (speed of soundin air 300 m/s), rememberthe trick where you canhear the train coming bylistening to the tracks?

• As phonons carry energy, they can conduct heat

• As they are quanta of thermal vibrations they canbe created and annihilated(bosons) unlike electrons lCcsp 3

1=κ

scattering

Speed of one phononNumber of phonons


Phonon thermal conductivity

• However, the scattering of phonons is verycomplicated…

Si crystal

Phonon-phonon scatteringBoundary scattering


Low-temperature limit (ballisticphonons)

• In the boundaryscattering limit, phononswill not scatter inside the material, only at the edges. Thus they travel in straight lines, ballistically

• Interesting effectsexpected for low-dimensional phononsystems where d < λ

• At below 1K, λ ∼ 100 nm-1 μm !

TkBdom 8.2=ωh

dom

sdom

cωπλ 2

=


Etching• Plasma etching: Create

reactive ions inside a plasma (ions+electrons). Accelerate ions to the substrate(physical+chemicaletching)

• Chemical etching: useselectivity of chemistry(acids) to etch onlycertain materials orcertain crystal directions

• We have both at NSC


2D phonons

• Anisotropic etching of Si in KOH

• Thermal conductancecan increase withdecreasing membranethickness ! (T. Kühnand I. J. Maasilta, Journal of Physics: Conf. Series 92 (2007) 012082)


1D phonon transport • Landauer transport works

also for phonons!• Each 1D phonon channel

conducts exactly Gth

• This is the maximumconducting capacity

• Conductance can belowered by introducingscatterers

TKpWThkG B

th )/9456.0(3

222

==π


Heat transport from electrons to phonons (=dissipation)

• In metals there aretwo subsystems, the free electrons and the phonons, and theycan exchange energyvia electron-phononinteraction

substrate Ts

N island

electrons

local phonons

TC eePheat

RK

Re-p

Tp


What is electron-phononinteraction?

• The interaction arises, because when the lattice atoms move withthe vibration mode (phonon), the effective electrostatic potentialacting on an electron changes, thereby allowing scattering of the electron

• For simple metals (no transition metals), the simplest scalarpotential theory is sufficient, where the e-p deformation potential Δ is simply determined by the gross properties of the Fermi surface and is given by where D(εF) is the density of states, and the last equality is for a parabolic band. Sphericalsymmetry of the Fermi-surface and q < kF was also assumed.

,32)(/ FFDn εε ==Δ


Theory for pure samples• In the lowest order, one has

two significant scatteringprocesses: absorption and emission of a phonon of wavevector q and energysee diagrams

• These satisfy the quasimomentum and energyconservation laws

)(qωh

q q

k-q (+K) k+q

)()()()()()(

qqkEkEqqkEkE

ω

k k

ωh

h

−+=+−=

Umklapp-scatteringNot significant at low T


Result

• Simple result follows for T < TD

where

is a material dependent parameter,value for Cu ~ 1 x 108 W/K5m3

24.886


What does this mean?• Can produce the hot-electron effect with

miniscule amounts of power in (Pin=P)• If V= 100 μm x 1 μm x 100 nm, and Tp= 0.1 K,

get for 1 pW of power Te= 0.25 K• => for 1 μm x 20 nm x 10 nm Te= 2.2 K !!• => Te= 0.101 K for 10-20 W (10 zeptoW)• The e-p interaction can be used as a sensitive

bolometer (if one knows how to measure Tesensitively)


A superconducting Nb bridge

4 5 6 7 8 9 10 11 12-505

10152025303540

Res

ista

nce

(Ω)

Temperature (K)

M. Nevala, K. Kinnunen, I. Maasilta,unpublished

A sub-mm radiation detector


Weakening ? of e-p interaction in sampleson thin SiN membranes

t=30 nm-750 nm SiN membrane, 200 nm Cu wire with 2 SN junctionsSINIS thermometer

Si

SiNX

A

CuAl Nb/Al


Results for 2D phononsJ. T. Karvonen, I. J. Maasilta, Phys. Rev. Lett. 99, 145503 (2007).

• All Cu wires have the same size, 30 nm membranesample is cooler at low powers but hotter at high powers=> phonon dimensionality effect (has not been observedbefore)

2468

2468

0.1 1 10 100 1000

2468

M1 B1

M2 B2

d(lo

g p)

/ d(lo

g T e)

M4 B4

(c)

(b)

Heating power density [pW / (μ m)3]

(a)

0.1 1 10 100 10000.1

1 Te of M1 Te of M2 Te of M4 Te of B1 and B2 Te of B4

0.6

0.4

0.2

Tem

pera

ture

(K)

Heating power density [pW / (μm)3]

0.8


Radiation by photons?• In addition to phonons, an

electronic system canradiate energy to EM fields=photons.

• Black-body radiation law(3D) (Stefan-Bolzmannlaw):

• This is why mushroompickers who get lost arefound

• At 4 K power hasdecreased by a factor3x107

423

2

60/ T

cAP

h

π=

TkBdom 8.2=ωh

At RT, Infrared frequencies, λ= 17 μm


The most famous black-bodysource? The Universe as a baby

• WMAP (Wilkinson Microwave Anisotropy Probe)• WMAP was launched on June 30, 2001 aboard a

Delta II rocket.WMAP completed its prime 2 years of mission operations in its L2 orbit by September 2003. Missions are to end September 2009.

2.725 K ± 200 μK


NSCNanoscience Center

Simple system• At the limit where electromagnetic environment is connected via transmission line(d > λth) to the resistor, a simple circuit model is appropriate to describe the heatflow

Photonic thermal conductance has the form

(1D black-body radiation)

Theory: D.R. Schmidt R.J. Schoelkopf and A.N. Cleland, PRL 93, 045901 (2004)

Experiment: M. Meschke, W. Guichard and J. P. Pekola, Nature 444, 187 (2006).

Re Rγ

Photons and phonons have the same thermal conductance quantum!



Estimate for theoretical heating power due to evanescent modes (d << λ) is calculated for parallel plate model according to Polder and Van Hove withdifferent separation distances 200 nm and 2 μm and different temperatures.

P. J. Koppinen, J. T. Karvonen, L. J. Taskinen, and I. J. MaasiltaAIP Conf. Proc. 850, (2006) 1556



ExperimentWe consider a situation wheretwo mesoscopic Cu wires areseparated by ~2 μm. The phononthermal pathway is removed byusing suspended structures.

Have observed heating in second wire!Koppinen, Maasilta, unpublished


Applications: ultrasensitivedetectors

• Bolometric principle: measure the temperature change in a small systemcaused by the photon

• From sub-mm to gamma-rays


Satellite based molecular spectroscopy of the atmospherein sub-mm

The ODIN satellite, collaboration between Sweden (leader), FranceCanada and Finland


Passive sub-mm camera


X-ray calorimetry/motivation

• XEUS is a follow-on to ESA's Cornerstone X-Ray Spectroscopy Mission (XMM-NEWTON).

• XEUS will be a permanent space-borne X-ray observatory with a sensitivity comparable to the most advanced planned future facilities such as JWST, ALMA and Herschel.

• XEUS will be around 200 times more sensitive than XMM-NEWTON. The scientific goals include the study of the:

• First massive black holes. • First galaxy groups and their

evolution into the massive clusters observed today.

• Evolution of heavy element abundances.

• Intergalactic medium using absorption line spectroscopy.

STJs+Transition edge Sensor array at 100 mK

http://xmm.vilspa.esa.es/

http://www.stsci.edu/jwst/

http://www.mma.nrao.edu/

http://www.rssd.esa.int/Herschel/


Motivation II

For materials analysis, improvementover existing commercial technology~ factor of 100 in energy resolution,(world record at NASA 2.0 eV at 6 keV)

Only issue complexity of refrigeration,has been integrated into a commercialSEM at NIST

NSC in collaboration with NIST and Lund University to develop X-ray detectors forfemtosecond structural dynamics NIST data


Summary of part II

• Quantum laws apply also to heat transport in the nanoscale

• Quantization laws similar to electronictransport

• Quantum heat transport can be utilized for novel devices such as ultrasensitiveradiation detectors

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