Phonon Scattering Processes Affecting Thermal Conductance at Solid-solid Interfaces in Nanomaterial Systems Patrick Hopkins University of Virginia Department.

Phonon Scattering Processes Affecting Thermal Conductance at Solid-solid Interfaces in Nanomaterial Systems Patrick Hopkins University of Virginia Department of Mechanical and Aerospace Engineering March 10, 2008

Equivalent power density [W/m2]
Moore’s Law Rocket nozzle 107 W/m2 Nuclear reactor 106 W/m2 Hot plate 105 W/m2 Equivalent power density [W/m2] The most commonly known effects of the nanorevolution is driven by Moore’s Law. In 1975, Gordon Moore, the cofounder of Intel predicted a doubling of transistors on and Integrated Circuit (IC) every 24 months. As IC’s are getting smaller and faster, these transistors must also get smaller and faster, leading to many thermal problems that device engineer’s must take into account in order to maintain the goal set by Moore’s Law. The thermal problems that are encountered on the nanometer length scales are the fundamental drive behind Microscale and Nanoscale Heat Transfer (not just a clever name) and this project aims to address some of the thermal problems that can arise during miniaturization. 45 nm 100 nm 500 nm Transistor size

Field effect transistors
Heat generated Rejected heat Thermal management is highly dependent on the boundary between materials

Thermoelectrics ZT = figure of merit S = Seebeck coefficient
σ = electrical conductivity k = thermal conductivity T = temperature

Thermal boundary resistance
Thermal boundary resistance creates a temperature drop, DT, across an interface between two different materials when a heat flux is applied. First observed by Kapitza for a solid and liquid helium interface in 1941. A typical resistance of m2K/W is equivalent to ~ mm Si ~ nm SiO2 T Mismatch in materials causes a resistance to heat flow across an interface.

Two types of interface resistance
Thermal Boundary Resistance Thermal Contact Resistance Due to difference in the acoustic properties: Phonon reflection at the interface Electron-phonon interaction Present even in the case of perfect contact with no roughness Microscopic quantity Important for bulk surfaces Macroscopic quantity Due to imperfect contact or voids in microstructure A Thot B Voids, imperfect contact Tcold Thot Tcold A B hBD= thermal boundary conductance 1/hBD = thermal boundary resistance Distance Thot Tcold DT Thot DT Tcold Distance

Major research objectives
the role of interface disorder on interfacial heat transfer the effects of different phonon scattering mechanisms on interfacial heat transfer

Outline of presentation
Theory of phonon interfacial transport Measurement of interfacial transport with the transient thermoreflectance (TTR) technique Influence of atomic mixing on interfacial phonon transport Influence of high temperatures on interfacial phonon transport Summary

Thermal conduction in bulk materials
Microscopic picture T L Z l = mean free path [m] phonon-phonon scattering length in homogeneous material k = thermal conductivity [Wm-1K-1] = thermal flux [Wm-2] What happens if l is on the order of L?

Thermal conduction in nanomaterials
Z T Microscopic picture of nanocomposite T Z Ln < ln q=hBDDT keffective of nanocomposite does not depend on phonon scattering in the individual materials but on phonon scattering at the interfaces hBD = thermal boundary conductance [Wm-2K-1] Change in material properties gives rise to hBD

Theory of hBD Phonon flux transmitted across interface
1 2 I q Phonon distribution Calculate the phonon flux across the interface, # of phonons of frequency approach interface at a certain angle, energy, and speed. Only a certain percentage will be transmitted Assuming diffuse scattering, the phonon looses memory of the incident angle, so the flux calcualtions can be simplified and hbd can be easily solved. Projects phonon transport perpendicular to interface Spectral phonon density of states [s m-3] Phonon energy [J] Phonon speed [m s-1] Phonon interfacial transmission

Diffuse Mismatch Model (DMM) Is this assumption valid?
Diffuse scattering Diffuse Mismatch Model (DMM) E. T. Swartz and R. O. Pohl, 1989, "Thermal boundary resistance,” Reviews of Modern Physics, 61, diffuse scattering – phonon “looses memory” when scattered T > 50 K and realistic interfaces Scattering completely diffuse Elastically isotropic materials Single phonon elastic scattering Averaged properties in different crystallographic directions Is this assumption valid?

Maximum hBD with elastic scattering
Phonon radiation limit (PRL) Same assumptions as DMM DOS side 1 (softer) in DMM DOS side 2 (harder) in PRL DMM PRL

DMM and PRL calculations
DMM and PRL calculations for Al/al2O3 and Pt/al2O3, interfaces that are studied in this work. Note the PRL is always higher than the DMM and the PT/al2O3 calculations predict a more constant hBD than the Al/Al2O3 over the range of (insets) since this temperature range is above the debye temperature of Pt therefore, based on our previous discussions, the constant change in <n> of the Pt would give rise to a constant hBD.

Theory of phonon interfacial transport Measurement of hBD with the TTR technique Influence of atomic mixing on hBD Influence of high temperatures (T > D) on hBD Summary

Transient ThermoReflectance (TTR)
Verdi V10 = 532 nm W Mira 900 tp ~ 190 fs @ 76 MHz l = nm 16 nJ/pulse RegA 9000 tp ~ 190 fs single shot kHz 4 mJ/pulse Verdi V5 = 532 nm W Probe Beam Delay ~ 1500 ps l/2 plate Beam Splitter Sample Dovetail Prism Lenses Detector Polarizer Pump Beam The TTR technique Pump probe method Increase the length that the probe pulse has to travel to reach the sample surface. To, the pump and probe are arriving at the same time By increasing the length of the probe path, the probe arrives at the sample after the pump, and the change in reflectance is related to the cooling of the high temperature electrons Our setup has a delay of 1500 ps, which is sufficient to resolve TBC in certain situations Variable ND Filter Acousto-Optic Modulator Lock-in Amplifier Automated Data Acquisition System

Free Electrons Absorb Laser Radiation Ballistic Electron Transport PROBE HEATING “PUMP” Electrons Transfer Energy to the Lattice Electron-Phonon Coupling (~2 ps) Thermal Diffusion by Hot Electrons Thermal Equilibrium Thermal Diffusion within Thin Film Thermal Diffusion (~100 ps) FILM Thermal Diffusion Resolving the TBC on 2 ns time scale is dependent on film thickness and thermal diffusivity of the film material Thermal Conductance Across the Film/Substrate Interface Thermal Boundary Conductance (~2 ns) SUBSTRATE Thermal Diffusion within Substrate Substrate Thermal Diffusion (~100 ps – 100 ns) Focus of current analysis

TTR data Free Electrons Absorb Laser Radiation
Thermal Conductance across the Film/Substrate Interface Thermal Boundary Conductance (~1-10 ns) 50 nm Cr/Si Resolving the TBC on 2 ns time scale is dependent on film thickness and thermal diffusivity of the film material

DMM compared to experimental data
Stevens, Smith, Norris, JHT, 2005 Lyeo, Cahill, PRB, 2006 Stoner, Maris, PRB, 1993 New data Stevens, Smith and Norris extended the work by Stoner and Maris by measuring hBD at interfaces with a wide variety of debye temperature ratios. The previous work by stoner and maris show the underprediction of the DMM to experimental data at room temperature. The recent results that our lab put out a few years ago show that better matched interfaces (where inelastic scattering may not play as great of a role), the DMM actually UNDERPREDICTS the data. Goal: investigate the over- and under-predictive trends of the DMM based on the single phonon elastic scattering assumption

Theory of phonon interfacial transport Measurement of hBD with the TTR technique Influence of atomic mixing on hBD Influence of high temperatures (T > qD) on hBD Summary

DMM assumptions DMM Assumption Realistic Interface Slight changes in deposition conditions can give rise to different elemental compositions around solid interfaces

AES depth profiles Cr-1: no backsputter Elemental Fraction
Cr/Si mixing layer 9.5 nm Cr-1: no backsputter Si change 9.7 %/nm Elemental Fraction Cr/Si mixing layer 14.8 nm Cr-2: backsputter Si change 16.4 %/nm Label curves and mixing layer thickness and label si slope Depth under Surface [nm]

Slope of Si in Beginning of Mixing Layer [%/nm]
Results from AES data Sample ID Cr Film Thickness [nm] Mixing Layer [nm] Slope of Si in Beginning of Mixing Layer [%/nm] Cr-1 38 ± 2.1 9.5 ± 0.6 9.7 ± 0.7 Cr-2 37 ± 0.4 14.8 ± 1.0 16.4 ± 0.7 Cr-3 35 ± 0.5 11.5 ± 0.7 16.6 ± 1.0 Cr-4 35 ± 2.8 10.8 ± 0.8 7.4 ± 1.0 Cr-5 39 ± 0.5 5.8 ± 0.5 24.1 ± 1.0 Cr-6 45 ± 0.5 7.0 ± 0.4 28.1 ± 1.2 uncertianty

TTR testing TTR data of two different samples with severely different boundaries as seen from AES profiles Fitted with thermal model presented earlier Fit at 100 ps which ensures there is not diffusion in film but still gives ~1.3 ns to resolve interface diffusion, both of which can be resolved as was discussed earlier, so can apply lumped capacitance P. E. Hopkins and P. M. Norris, Applied Physics Letters 89, (2006).

DMM predicts a constant hBD = 855 MWm-2K-1
hBD results Decreasing hBD with increasing mixing layer thickness DMM predicts a constant hBD = 855 MWm-2K-1 Hopkins, Norris, Stevens, Beechem, and Graham, to appear in the Journal of Heat Transfer, 2008

Virtual crystal DMM The disordered region is replaced by a homogenized virtual crystal of thickness Dint having effective properties based on the disordered medium with MFP= lint. multiple scattering events from interatomic mixing T. E. Beechem, S. Graham, P. E. Hopkins, and P. M. Norris, Applied Physics Letters 90, (2007)

Virtual crystal DMM multiple scattering events from interatomic mixing
In well-matched material systems such as Cr on Si, Rpp is very small and on the same order as Rep, so this additional resistance must be considered and added in parallel with Rpp. G = electron-phonon coupling factor Majumdar and Reddy, APL, 2006

Virtual crystal DMM multiple scattering events from interatomic mixing
Majumdar and Reddy, APL, 2006

VCDMM DMM predicts hBD that is almost 8 times larger than that measured on the samples and no dependence on mixing layer thickness or composition. The VCDMM calculations are within 18% of the measured values and show the same trend with mixing layer thickness as the measurements. Hopkins, Norris, Stevens, Beechem, and Graham, to appear in the Journal of Heat Transfer, 2008

Summary Investigate the role of interface disorder on interfacial heat transfer Examined the effects of interfacial properties on hBD in the acoustically matched Cr/Si system with TTR DMM predicts hBD 855 MWm-2K-1 at room temperature Measured data varies from MWm-2K-1, depending on deposition conditions Multiple phonon elastic scattering could cause this over- prediction of the DMM DMM only takes into account single scattering events DMM assumes a perfect interface, but interface disorder will increase the scattering thus decreasing the hBD VCDMM is introduced and predicts same values and trends for Cr/Si at room temperature as experimental data

Summary Stevens, Smith, Norris, JHT, 2005 Lyeo, Cahill, PRB, 2006 Stoner, Maris, PRB, 1993 New data The presence of an interfacial mixing region causing multiple elastic scattering events which are not accounted for and may be the cause of the overestimation of the DMM in well matched material systems with Debye temperature ratios close to one. Stevens, Smith and Norris extended the work by Stoner and Maris by measuring hBD at interfaces with a wide variety of debye temperature ratios. The previous work by stoner and maris show the underprediction of the DMM to experimental data at room temperature. The recent results that our lab put out a few years ago show that better matched interfaces (where inelastic scattering may not play as great of a role), the DMM actually UNDERPREDICTS the data. Goal: investigate the over- and under-predictive trends of the DMM based on the single phonon elastic scattering assumption

Single phonon elastic scattering
hBD from DMM limited by f1 f=T/qD f There are 3 major assumptions associated with phonon transport when employing the DMM. Completely diffuse scattering, i.e. a phonon has no recollection where it came from. This enabled us to simplify our equation for phonon flux in the previous slide. Both materials on either side of the interface are completely isotropic, this means that the phonon speeds are the same in all crystallographic directions. Phonons are elastically scattered, i.e. a phonon can only scatter with another phonon with the same frequency. This is the specific assumption that we will be testing in this study. How does this come into play in the DMM? It simplified the transmission coefficient. Also, since the DMM is dependent on the change in <n> with temperature. So, if D<n>/DT is constant with temperature, then the hBD predicted by the DMM should not change, assuming elastic scattering. Let me divulge on this… Linear in classical regime (T>qD) *Kittel, 1996, Fig. 5-1

Elastic scattering – hBD is a function of f/T in lower qD material qDAl=428 K DMM Predictions f Again, <n> is shown on the left as a function of temperature, and on the right is D<n>/DT, again, the xaxis is normalized at the Debye temperature. You can see that above 1, Dn/Dt is constant, indicating that the change in phonon population with temperature is constant. This is a result of the linear change in phonon population with temperature in the classical limit (i.e. T>debye). So now to examine specific material systems that we will be studying, let’s look at D<n>/DT of Pt and Al2O3. Since Pt has a lower Debye cutoff frequency, the DMM would be calculated assuming that the phonon are transmitted into the Al2O3 from the Pt. Therefore, at temperatures around and above the Debye temperature, the change in phonon occupancy in Pt is constant. If we assume only elastic scattering, then hBD will also be constant in this regime. However, if inelastic scattering processes are occurring, then we would expect the hBD to have some influence from the change in phonon states of the Al2O3. Therefore, as temperature increases above the Debye temperature of Pt, and if inelastic scattering is occurring between the Pt and al2O3, then we would see an increase in hBD above the debye temperature of Pt instead of a flat line. qDPb=105 K T/D

Molecular dynamics simulations
Lennard-Jones Potential with Different Atomic Sizes Kr/Ar Superlattice Nanowire Stevens, Zhigilei, and Norris, IJHMT, 2007 Chen, Li, Yang, Wu, Lukes and Majumdar, Physica B, 2004 Computational results indicate a linear increase in conductance (decrease in resistance) with temperature.

Mismatched samples at low temperatures
Lyeo and Cahill, PRB, 2006 Stoner and Maris, PRB, 1993

hBD results at temperatures above qD of the softer material
Pt/Al2O3 Pt/AlN P. E. Hopkins, R. J. Stevens, and P. M. Norris, To appear in the Journal of Heat Transfer, HT (2008).

Analysis DMM JOINT FREQUENCY DMM
Linear trend in MDS in classical regime (T>>qD) MDS calculates hBD making no assumption of elastic scattering in interfacial phonon transport Several samples show linear hBD trends around classical regime DMM f/T JOINT FREQUENCY DMM Substrate (diamond) Film (Pb) P. E. Hopkins and P. M. Norris, Nanoscale and Microscale Thermophysical Engineering 11, 247 (2007)

DMM vs. JFDMM DMM JFDMM

Summary Investigate the effects of different phonon scattering mechanisms on interfacial heat transfer Measured hBD at different metal-dielectric interfaces with a range of acoustic similarity Observed linear trend in hBD around qD Evidence of inelastic scattering – not predicted with DMM JFDMM takes into account substrate phonons – and provides better agreement with experimental data

Summary Stevens, Smith, Norris, JHT, 2005 Lyeo, Cahill, PRB, 2006 Stoner, Maris, PRB, 1993 New data The presence of inelastic scattering events, which add an additional channel of interfacial energy transport may be the cause of the underestimation of the DMM in mismatched material systems with distinctly different Debye temperatures. Stevens, Smith and Norris extended the work by Stoner and Maris by measuring hBD at interfaces with a wide variety of debye temperature ratios. The previous work by stoner and maris show the underprediction of the DMM to experimental data at room temperature. The recent results that our lab put out a few years ago show that better matched interfaces (where inelastic scattering may not play as great of a role), the DMM actually UNDERPREDICTS the data. Goal: investigate the over- and under-predictive trends of the DMM based on the single phonon elastic scattering assumption

Conclusions Investigate the role of interface disorder on interfacial heat transfer Determined that interfacial mixing can play a role in phonon transport by inducing multiple phonon scattering events Accurately described with VCDMM taking into account e-p resistance Investigate the effects of different phonon scattering mechanisms on interfacial heat transfer Inelastic scattering contributes to hBD at temperatures close to qD of the softer material where substrate phonon population is still quantum mechanically increasing Developed JFDMM to take into account some portion of inelastic scattering

How does the knowledge of phonon scattering affect nanoapplications?
Impact How does the knowledge of phonon scattering affect nanoapplications?

Acknowledgments Pamela Norris, my doctoral advisor and head of the Microscale Heat Transfer laboratory at UVA Funding from the National Science Foundation (NSF) Graduate Research Fellowship Program (GRFP) Funding from the Virginia Space Grant Consortium (VSGC) Collaborators: Leslie Phinney (Sandia), Robert Stevens (RIT), Samuel Graham (GaTech), Thomas Beechem (GaTech) Rob Kelly (UVA), Avik Ghosh (UVA), Mikiyas Tsegaye (UVA), David Cahill (UIUC), John Hostetler (Trumpf Photonics), Mike Klopf (Jefferson Lab), Vickie Connors (NASA Langley) Microscale Heat Transfer Crew – Rich Salaway, Jennifer Simmons, John Duda, Justin Smoyer

Free Electrons Absorb Laser Radiation Focus of current analysis Ballistic Electron Transport SUBSTRATE FILM HEATING “PUMP” PROBE Thermal Diffusion Electrons Transfer Energy to the Lattice Electron-Phonon Coupling (~2 ps) Thermal Diffusion by Hot Electrons Thermal Equilibrium Thermal Diffusion within Thin Film Thermal Diffusion (~100 ps) Resolving the TBC on 2 ns time scale is dependent on film thickness and thermal diffusivity of the film material Thermal Conductance across the Film/Substrate Interface Thermal Boundary Conductance (~2 ns) Thermal Diffusion within Substrate Substrate Thermal Diffusion (~100 ps – 100 ns) Focus of previous analysis

Electron-phonon (e-p) nonequlibrium
Electron-phonon coupling factor Energy stored in e- system Energy conducted through e- system Energy deposited into e- system Energy transferred from e- system to l system Energy stored in l system Energy gained by l system from e- system PARABOLIC TWO-STEP MODEL (PTS) *Anisimov, 1974 For the electron phonon studies that I am going to look at, the rate of electron-phonon equilibration is quantitatively described the electron phonon coupling factor in the two temperature model. This model assumes a thermalized electron system so that an electron temperature can be defined. This set of nonlinear pdes represents an energy balance between the two carrier baths, the electrons and the phonons. Notice there is not thermal conductivity term in the lattice energy balance, which is due to the relative populations of the electrons and phonons in metals. Through an appropriate reflectance model, the TTR experimental data can be related to the temperature profiles calculated from this equation, and G, the electron phonon coupling factor can be deduced. To further the conceptual discussion this equation temporally and spatially describes the absorption and transport of heat in thin film. The processes that are mathematically described by the TTM are pictorially represented by the squares in this slide. Consider the blue square as a metal film at ambient. The incident laser is absorbed by the electrons in the metal to a distance of the optical penetration depth into the film from the film/ambient interface. Through ballistic transport, the electrons are stretched out in space. Eventually (10’s of fs) these electrons relax into a fermi distribution. At this point, e-p collisions start to dominate the thermal transport. Once the e&p have equilibrated, a temperature gradient exists in the film which creates diffusion which is driven the gradient and predicted by the thermal conductivity of the metal. Eventually, the thin metal film will reach an equilibrium temperature. This begins the time when energy transport out of the film system and into the substrate system dominates energy transport. time z

Relate temperature to reflectance
DR/R = aDTe + bDTl – only valid for DTe < 150 K Test at fluences up to 15 J m-2 DTe in Au of up to ~ 4000 K ITT 2.4 eV > 1.55 eV TTR energy Intraband reflectance model Valid for all electron temperatures Christensen, PRB, 1976 Smith and Norris, APL, 2001

Different e-p equilibration curves for different fluences
Measure G in Au with TTR 20 nm Au/glass 20 nm Au/glass Different e-p equilibration curves for different fluences But G should be a material property???? Hopkins and Norris, App. Surf. Sci., 2007

Simplifies transmission coefficient There are 3 major assumptions associated with phonon transport when employing the DMM. Completely diffuse scattering, i.e. a phonon has no recollection where it came from. This enabled us to simplify our equation for phonon flux in the previous slide. Both materials on either side of the interface are completely isotropic, this means that the phonon speeds are the same in all crystallographic directions. Phonons are elastically scattered, i.e. a phonon can only scatter with another phonon with the same frequency. This is the specific assumption that we will be testing in this study. How does this come into play in the DMM? It simplified the transmission coefficient. Also, since the DMM is dependent on the change in <n> with temperature. So, if D<n>/DT is constant with temperature, then the hBD predicted by the DMM should not change, assuming elastic scattering. Let me divulge on this…

hBD results for Al/Al2O3 Stoner and Maris, PRB, 1993
P. E. Hopkins, et al., International Journal of Thermophysics 28, 947 (2007)

Thermal model Change in temperature across a 50 nm Cr film on Si substrate interface Model sensitivity to thermal boundary conductance

Resolving TBC with TTR Al/Al2O3 interfaces Resolving TBC with TTR
kf = 237 Wm-1K-1 hBD = 2.0 x 108 Wm-2K-1 In order to resolve TBC, the time constant for the film should be significantly smaller than the time constant associated wit the interface or else it will be difficult to extract the TBC from thermal diffusion in the film. LABEL CURVES HBD INSTEAD OF TBC ti tf

Thermal Model Lumped capacitance T film substrate Al/Al2O3 interfaces
kf = 237 Wm-1K-1 hBD = 2.0 x 108 Wm-2K-1 Bi<<1 d =75 nm< 120 nm Bi = 1 Assume lumped capacitance the biot number must be less than 1 and for minimal error must be less than 0.1 Film thickness, diffusion is less than 100 ps but for finite interfacial regions, time constant is linear with thickness, so 1.5 ns stage is more than enough Bi>>1 x

Sample fabrication Sample ID Backsputter Etch Heat Treat Prior
to Deposition Deposition Notes (on Si substrates) Cr-1 none 50 nm 300 K Cr-2 5 min Cr-3 K Cr-4 K Cr-5 50 nm 573 K Cr-6 10 nm of Cr at 300 K; heating to 770 K; 40 nm of Cr at 300 K 6 chrome sample were made on Si substrates

Interface characterization
Auger electron spectroscopy (AES) Electron bombardment Ionization Relaxation and Auger emission Monitor energy e- [3 keV] Vacuum Energy Higher levels A high intensity electron bombards the sample surface and causes a core electron to eject from the core This leaves the electron in a highly excited state that needs to rapidly relax back to a lower state An electron falls from a higher core level to fill the lower core level, and the energy liberated in the process is immediately transferred to a second electron The second electron then uses the energy to over come the binding energy ,and the remainder of the energy permits the electron to emit from the surface. Depending on the element and incident electron energy, the emitted auger electron will have different energies, by monitoring the energy of the emitted auger electron, the type of element can be determined Core level

AES depth profiling detector e- gun O2 Ar+ gun C Cr dN/dE Si
Argon gun sputters away film material for a certain amount of time then AES data can be taken with the electron gun Look after sputtering Energy [eV]

AES depth profile Depth proflie for Cr-1
C and O2 at surface which is blow away after initial sputter, not interface region is of interest so the C and O2 surface amounts in this profile are not representative Mixing region about 10 nm defined mixing region at 10% of Si and 10% if Cr

Energetic Electrons Doped Semiconductors (T = Troom) E
k E Energetic Electrons Doped Semiconductors (T = Troom) Conduction Band Intraband Intraband excitation – E<Eg Ef Interband Interband excitation – E>Eg Eg The active layer in the previous example is typically made of an n-doped semiconductor to increase electron mobility. Doping the semiconductor with an element that has more valence electrons, extra valence electrons become unbounded an act as free electrons, similar to metals. At room temperature due to fermi smearing, almost all the electrons in the donor band are in the conduction band. Upon electron excitation, if E<Eg then only the conduction electrons will contribute to transport since the valence electrons will not be able to excite into the conduction band, however, at higher energies (E>Eg), the valence electrons can be excited into the conduction band and participate in transport processes until recombination. So in the non equilibrium electron-lattice realm, the electrons transfer their energy to the lattice by colliding with the lattice and emitting a phonon. Valence Band

Energetic Electrons Metals (T = Troom) E Noble Metal
Electron transport – Noble metals s Ef ITT d Transition Metal Electron transport – Transition metals Things get a little more complicated in metals since their band structure is not as simplified. Typical in metals is the crossing of the conduction band and the Fermi energy. This allows for intraband transitions to occur at low energies and gives rise to an interband transition threshhold in which the metal losses it’s “free electron” qualities. This ITT is distinct in noble metals since there is a large separation between the p band and d band. However, in transition metals, it is common to have overlapping of he d-band and s-band close to the fermi energy, therefore, electron transport can involve energetic electrons undergoing both intra and interband transitions, even at low energies. s Ef ITT d k

Band Structures GaAs Gold Nickel s c s v d d
* Swaminathan and Macrander, 1991 Band Structures GaAs Gold Nickel s c s v d d As an example of these band structures that were just discussed, GaAs, Gold, and Ni band structures are presented. The nickel demonstrates band splitting due to its ferromagnetic state. Notice the dband to fermi transition in gold as opposed to nickel, so even at low energies, interband transitions can occur. *Christensen, 1976 *Weiling and Callaway, 1982

Reflectance at ITT Gold Silver *Sun, et al., 1994 *Fatti et al., 2000
Fatti and sun studied ep in Au and silver over a range of energies straddling the ITT. Two important things to note about this data. When probing at the first ITT, the proportionality constant relating Dr/DT is large compared to energies removed from the ITT. The data taken by fatti and sun are reduced around the ITT due to the huge increase in DR at the same Dtemp compared to other energies. Also, notice at energies removed from the ITT, a sign change of the reflectance occurs, which sun relates to a presence of a non thermal electron distribution that changes with incident laser fluence. *Sun, et al., 1994 *Fatti et al., 2000

G Measurement in Gold G @ energies lower than ITT
energies around ITT s Ef *Hohlfeld et al., 2000 *Smith and Norris, 2001 d

Transient Thermoreflectance Data

Thermal Model k = 91 Wm-1K-1 G = 3.6 x 1017 Wm-3K-1

G Measurements R = aDTe + bDTl 30 nm Ni/Si 1.3 eV
G = 5.8 x 1017 Wm-3K-1 1.55 eV G = 3.7 x 1017 Wm-3K-1

Analysis G measurement at 1.3 eV interband transition yields higher results than measurements taken at other energies (~6.0 x 1017 Wm-3K-1) This study: ~3.7 x 1017 Wm-3K-1 at 1.55 eV Wellershoff et al., 1998: ~3.6 x 1017 Wm-3K-1 at 3.11 eV Previous Au measurements of G were same at ITT and other energies 1.3 eV in Ni is not at ITT but at a higher interband transition lower d-band  Fermi level Ni/Si Ni/Glass

Reflectance Model De2 De1 R = aDTe + bDTl &

How do we define temperature?
Temperature is an equilibrium property How can we define temperature when there is a nonequilibrium (TTM)? Consider case of homogeneous heating in Au t > tee Temporally Both e- and phonon systems are in local thermal equilibrium When e- scatters and emits phonon, e- system redistributes into a new temperature distribution (lower T). What about e- substrate scattering work? Homogeneous heating so lumped capacitance hBDe-=1E8 Wm-2K-1 d=50 nm ke=317 Bi = .02 very conservative

How do we define temperature?
However, how can we determine temperature spatially if there is a thermal gradient? Can temperature, which is an equilibrium concept, still be invoked in a nonequilibrium process such as heat conduction? (Cahill, JAP, 2003) Consider 1D conduction Kinetic Theory speed Mean free path #Collisions/time Box is 4l long Cannot resolve local temperature Can resolve local temperature at Te=Tl=300, Q~1E13, ~1E4 m/s l~1 nm *Since equilibrium is achieved through multiple collisions d~15 nm

Joint frequency DMM DMM JFDMM Weighting factor x is simply a percentage of the composition of each material in the unit volume (M=atomic mass) (N=number of oscillators per unit volume)

Analysis Inelastic scattering – DMM does not account for this
Data at solid-solid interfaces taken at temperatures around Debye temperature show linear trend DMM predicts flattening of predicted hBD around Debye temperature Accounting for substrate phonons in DMM improves prediction (JFDMM) PRL DMM JFDMM Inelastic phonon radiation limit (IPRL) Is there an upper limit to inelastic scattering?

Elastic and inelastic contributions
Pb/diamond Pb/diamond Pb/diamond In classical limit

Elastic and inelastic contributions
qDPb/qDdiamond~0.05 qDPt/qDAl2O3~0.23 Hopkins, Norris, and Stevens, Submitted to the Journal of Heat Transfer Relative contribution to hBD of inelastic scattering compared to elastic scattering increases with sample mismatch and with temperature

Thermal testing in novel nanostructures
Future directions Thermal testing in novel nanostructures CNT and nanocomposites

Steady state and 3w electrical resistance techniques
Future directions Steady state and 3w electrical resistance techniques B. W. Olson, S. Graham, and K. Chen, Review of Scientific Instruments 76, (2005). Hopkins and Phinney, MNHT

NonEquilibrium Green’s Function (NEGF) modeling
Future directions NonEquilibrium Green’s Function (NEGF) modeling Now let’s switch gears to phonon transport. Instead of schodinger’s equation to govern electron transport, we use newton’s law with open boundary conditions to describe the atomic vibrations. Therefore, picturing atoms on a spring, we can describe the force of each atoms given by the displacement and the spring constant. Now, if instead we put an end on this atomic chain with another atomic chain of different atoms, where between atom A and atom B the spring constant is then given by Kb, we have two waves at the “source” “channel” interface, one of which is incident and one of which is reflected. In this case, we build a semiinfinite source of atoms A and a “channel” or “nanostructure” of atoms B, and can describe the force between these two material at the interface by F0.

Future directions NEGF to calculate phonon conductivity in nanostructures from first principles Relies on basic quantum mechanics No assumptions based on scattering or transport Can be extended to any nanostuctures To directly compare the two approaches, we use a 1D DOS and only assume longitudnal cutoffs and velocities in the calculations for each mode. Still, NEGF gives better agreement then the semiclassical models. Si wire data from: D. Li, Y. Wu, P. Kim, L. Shi, P. Yang and A. Majumdar, 2003, "Thermal conductivity of individual silicon nanowires," Applied Physics Letters, 83, Hopkins et al., MHT

Future directions More TTR applications – electron-phonon scattering
Free Electrons Absorb Laser Radiation Electrons Transfer Energy to the Lattice Thermal Diffusion by Hot Electrons 20 nm Au/glass Resolving the TBC on 2 ns time scale is dependent on film thickness and thermal diffusivity of the film material Insulated boundary conditions always assumed P. E. Hopkins and P. M. Norris, Applied Surface Science 253, 6289 (2007)

Future directions More TTR applications – electron-phonon scattering z
Different e-p equilibration curves for different fluences But G should be a material property P. E. Hopkins, et al., Submitted to Phys Rev B.

Future directions Extend nanoscale thermophysics to realistic low dimensional nanostructures Electrically based resistance techniques to measure thermal transport and thermophysical properties of nanomaterials NEGF formalism for accurate modeling of real nanosystems TTR technique to measure electron-phonon coupling and interfacial thermal transport

Phonon Scattering Processes Affecting Thermal Conductance at Solid-solid Interfaces in Nanomaterial Systems Patrick Hopkins University of Virginia Department.

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Phonon Scattering Processes Affecting Thermal Conductance at Solid-solid Interfaces in Nanomaterial Systems Patrick Hopkins University of Virginia Department.

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Presentation on theme: "Phonon Scattering Processes Affecting Thermal Conductance at Solid-solid Interfaces in Nanomaterial Systems Patrick Hopkins University of Virginia Department."— Presentation transcript:

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