Quantum transport in nanostructures-ii

Transcription

1 Quantum transport in nanostructures-ii Prof. Ilari Maasilta Nanoscience Center, Department of Physics, University of Jyväskylä YN 215,

2 Some topics in quantum transport of heat Energy transport mechanisms Electronic heat conduction Top-down nanofabrication Phonons, phononic heat conduction, quantum of thermal conduction Electron-phonon interaction Photonic heat radiation Applications in detectors

3 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 state 1. Particles of matter, i.e. atoms, ions, electrons. In solid state physics, atoms and ions do not move large distances but just vibrate 2. Insulators vs. conductors: In insulators also electrons are bound => no charge transport. Also no electric heat conductivity. In conductors there are free electrons that can carry charge and heat 3. Particles of fields, i.e. photons and phonons

4 Energy transport vs heat transport Energy vs. heat transport: Energy transport is a general concept, applies to all situations, including non-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: T 1 =T 2 Typically one measures thermal conductivity (Fourier s law): j q = κ T

5 Electronic heat transport At temperature T, electrons in a solid have a spread in energy of ~ k B T (the total energy is so called Fermi energy E F ~ K!) This is due to the quantum nature of electrons (no two electrons can occupy the same state, Pauli principle, QM matters even at RT) Both thermal and charge conductivities depend on differences of electron states => E F is not relevant (left-moving electrons and right moving electrons) At low temperatures, where most of scattering is from boundaries and impurities (not from vibrations), one can show that thermal conductivity κ is related to electrical conductivity simply by the Wiedemann-Franz law: κ = LσT 2 π L = 3 k e B 2

6 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 Ludvig Valentin Lorenz (January 18, 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 Hendrik Lorentz, who discovered it independently in How did Lorenz discover his number??? (No QM at the time)

7 Phonons But: We know that even an insulator tranports heat. How? Even though the atoms of the solid do not travel, they vibrate. These vibrations can be collective in such a way that the form a wave that extends over the whole sample. The smallest QM quantized (undivisible) unit of this wave is called a phonon. E = hω p

8 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 c s ~ m/s (speed of sound in air 300 m/s), remember the trick where you can hear the train coming by listening to the tracks? As phonons carry energy, they can conduct heat As they are quanta of thermal vibrations they can be created and annihilated (bosons) unlike electrons Number of phonons Speed of one phonon 1 κ p = l 3 Cc s scattering

9 Phonon thermal conductivity However, the scattering of phonons is very complicated Boundary scattering Phonon-phonon scattering Si crystal

10 Low-temperature limit (ballistic phonons) In the boundary scattering limit, phonons will not scatter inside the material, only at the edges. Thus they travel in straight lines, ballistically Interesting effects expected for lowdimensional phonon systems where d < λ At below 1K, λ 100 nm- 1 μm! hω = 2. 8k T dom B λ dom 2πc = ω s dom

11 Etching Plasma etching: Create reactive ions inside a plasma (ions+electrons). Accelerate ions to the substrate (physical+chemical etching) Chemical etching: use selectivity of chemistry (acids) to etch only certain materials or certain crystal directions We have both at NSC

12 2D phonons Anisotropic etching of Si in KOH Thermal conductance can increase with decreasing membrane thickness! (T. Kühn and I. J. Maasilta, Journal of Physics: Conf. Series 92 (2007) )

13 1D phonon transport Landauer transport works also for phonons! Each 1D phonon channel conducts exactly G th 2 2 kb 2 Gth = π T = ( pw / K ) T 3h This is the maximum conducting capacity Conductance can be lowered by introducing scatterers

14 Heat transport from electrons to phonons (=dissipation) In metals there are two subsystems, the free electrons and the phonons, and they can exchange energy via electron-phonon interaction P heat N island electrons R e-p local phonons R K substrate C e T e T p T s

15 What is electron-phonon interaction? The interaction arises, because when the lattice atoms move with the vibration mode (phonon), the effective electrostatic potential acting on an electron changes, thereby allowing scattering of the electron For simple metals (no transition metals), the simplest scalar potential 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 Δ = n / D( ε 2 F ) = ε, where D(ε F ) is the density of 3 F states, and the last equality is for a parabolic band. Spherical symmetry of the Fermi-surface and q < k F was also assumed.

16 Theory for pure samples In the lowest order, one has two significant scattering processes: absorption and emission of a phonon of wavevector q and energy hω(q) see diagrams These satisfy the quasimomentum and energy conservation laws E( k) = E( k) = E( k E( k q) + hω( q) + q) hω( q) q k k-q (+K) q Umklapp-scattering Not significant at low T k k+q

17 Result Simple result follows for T < T D where is a material dependent parameter, value for Cu ~ 1 x 10 8 W/K 5 m 3

18 What does this mean? Can produce the hot-electron effect with miniscule amounts of power in (P in =P) If V= 100 μm x 1 μm x 100 nm, and T p = 0.1 K, get for 1 pw of power T e = 0.25 K => for 1 μm x 20 nm x 10 nm T e = 2.2 K!! => T e = K for W (10 zeptow) The e-p interaction can be used as a sensitive bolometer (if one knows how to measure T e sensitively)

19 A superconducting Nb bridge Resistance (Ω) Temperature (K) A sub-mm radiation detector M. Nevala, K. Kinnunen, I. Maasilta, unpublished

20 Weakening? of e-p interaction in samples on thin SiN membranes SiN X Si A Al Cu Nb/Al t=30 nm-750 nm SiN membrane, 200 nm Cu wire with 2 SN junctions SINIS thermometer

21 Results for 2D phonons J. T. Karvonen, I. J. Maasilta, Phys. Rev. Lett. 99, (2007). All Cu wires have the same size, 30 nm membrane sample is cooler at low powers but hotter at high powers => phonon dimensionality effect (has not been observed before) Temperature (K) T e of M T e of M2 2 (a) M1 B1 0.6 T e of M4 T e of B1 and B T e of B (b) M4 B (c) M2 B2 Heating power density [pw / (μm) 3 ] Heating power density [pw / (μ m) 3 ] d(log p)/ d(log T e )

22 Radiation by photons? In addition to phonons, an electronic system can radiate energy to EM fields=photons. Black-body radiation law (3D) (Stefan-Bolzmann law): This is why mushroom pickers who get lost are found At 4 K power has decreased by a factor 3x π P / A = T h c hω dom = 2. 8k B T At RT, Infrared frequencies, λ= 17 μm 4

23 The most famous black-body source? 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 Missions are to end September K ± 200 μk

24 NSC Nanoscience Center Simple system At the limit where electromagnetic environment is connected via transmission line (d > λ th ) to the resistor, a simple circuit model is appropriate to describe the heat flow Photonic thermal conductance has the form (1D black-body radiation) R e R γ Theory: D.R. Schmidt R.J. Schoelkopf and A.N. Cleland, PRL 93, (2004) Experiment: M. Meschke, W. Guichard and J. P. Pekola, Nature 444, 187 (2006). Photons and phonons have the same thermal conductance quantum!

25 NSC Nanoscience Center Estimate for theoretical heating power due to evanescent modes (d << λ) is calculated for parallel plate model according to Polder and Van Hove with different separation distances 200 nm and 2 μm and different temperatures. P. J. Koppinen, J. T. Karvonen, L. J. Taskinen, and I. J. Maasilta AIP Conf. Proc. 850, (2006) 1556

26 NSC Nanoscience Center Experiment We consider a situation where two mesoscopic Cu wires are separated by ~2 μm. The phonon thermal pathway is removed by using suspended structures. Have observed heating in second wire! Koppinen, Maasilta, unpublished

27 Applications: ultrasensitive detectors Bolometric principle: measure the temperature change in a small system caused by the photon From sub-mm to gamma-rays

28 Satellite based molecular spectroscopy of the atmosphere in sub-mm The ODIN satellite, collaboration between Sweden (leader), France Canada and Finland

29 Passive sub-mm camera

30 X-ray calorimetry/motivation XEUS is a follow-on to ESA's Cornerstone X-Ray Spectroscopy Mission (XMM-NEWTON). XEUS will be a permanent spaceborne 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

31 Motivation II For materials analysis, improvement over 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 commercial SEM at NIST NSC in collaboration with NIST and Lund University to develop X-ray detectors for femtosecond structural dynamics NIST data

32 Summary of part II Quantum laws apply also to heat transport in the nanoscale Quantization laws similar to electronic transport Quantum heat transport can be utilized for novel devices such as ultrasensitive radiation detectors