Scaling analysis of negative differential thermal resistance Dahai He Department of Physics, Xiamen University, China Workshop on Nanoscale Heat Transport @IPM, Tehran April, 205
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Part I Outline. Introduction 2. Analysis 3. Examples 4. Conclusion 5
Negative differential electrical resistance in tunneling diode Leo Esaki (925- ) Nobel Prize in Physics(973) Illustrative figure for the work by L. Esaki, Phys. Rev. 09, 603 (958) http://en.wikipedia.org/wiki/file:negative_differential_resistance.png 6
NDTR in thermal diode T L T R simulation analytical calculation 0.04 0.02 j 0.00 0.008 0.006 0.004 0.002 0.000-0.6-0.4-0.2 0.0 0.2 0.4 0.6-0.002-0.004 B. Li, L. Wang, G. Casati, PRL 93, 8430 (2004) B. Hu, DH, L.Yang, Y. Zhang, PRE 74, 0600 (2006) Negative differential thermal resistance (NDTR) 7
NDTR in thermal transistor B. Li, L. Wang, G. Casati, APL 88, 4350 (2006) 8
Mechanism of NDTR K int T L T R H L H R K 2 ( ) int 2 H = HL + x x0 + HR 2 pi 2 whre HLR, = + ( xi+ xi) + ULR, ( xi) 2 2 Difficulty of analysis : I. Nonequilibrium stationary state II. Nonlinearity III. Qusi-particle entity of phonons 9
Mechanism of NDTR K int T L T R H L K 2 ( ) int 2 H = HL + x x0 + HR H R 2 pi 2 whre HLR, = + ( xi+ xi) + ULR, ( xi) 2 2 j = σ T k B max where ( ) ωmin ω σ= αωdω 2π Transport process in the nonlinear response regime DH, S. Buyukdagli, B.Hu, Phys. Rev. B 80, 04302 (2009) 0
T + T - 8.5 8.0 7.5 7.0 6.5 NDTR region j (0-4 ) 6.0 5.5 5.0 4.5 4.0 3.5 0.025 0.050 0.075 0.00 0.25 0.50 0.75 0.200 0.225 T Negetive differential thermal resistance Negetive in differential electrical the Frenkel- Kontorova (FK) model resistance in tunneling diode DH, B. Ai, H. Chan, and B. Hu, Phys. Rev. E 8, 043 (200) Illustrative figure for the work by L. Esaki, Phys. Rev. 09, 603 (958) http://en.wikipedia.org/wiki/file:negative_differential_resistance.png
Size effect of NDTR N=32 N=64 N=28 N=52 0. J J = Nj 0.05 0.0 0.5 0.20 0.25 0.30 0.35 T Shrinkage of the NDTR regime for increasing N: FK model DH, B. Ai, H. Chan, and B. Hu, Phys. Rev. E 8, 043 (200) 2
Size effect of NDTR N=32 N=64 N=28 N=256 N=52 N=024 N=2048 continuum J J = Nj 0 00 T Heat current approaches to saturation as N increases: ϕ 4 model DH, B. Ai, H. Chan, and B. Hu, Phys. Rev. E 8, 043 (200) 3
Q: what is the necessary conditions for the occurring of NDTR?. Spatially asymmetric structure? ( ) 2. Nonlinearity? ( ) (Φ 4 vs. FPU-β?) 3. Temperature? 4. System size? Q2: Is it possible to give a prediction of the occurring of NDTR? 4
A general theoretical analysis for NDTR jt ( +, T ) = jt (, T) 2 jt (, T) jt (, T) dj( T, T ) = dt + d T T T T T T = T + + T T + T T [ 0,2T ] For a particular constraint F( T, T ) = 0 Along this curve, negative differential thermal resistance (NDTR) corresponds to jt (, T) T F( T, T) = 0 < 0 H.-K. Chan, DH, B.Hu, Phys. Rev. E 89, 05226 (204) 5
A general theoretical analysis for NDTR NDTR: n ( T, T) n ( T, T) < [ + n ( T, T)] 2 3 where n ( T, T) n ( T, T) 2 dlnt d T = ln F( T, T) 0 ln κe( T, T) lnt a particular way of varying the temperature difference T Dependence of the effective thermal conductivity κ e on T ln κe( T, T) n3 ( T, T) ln T (, ) Nj κe T T T T Dependence of the effective thermal conductivity κ e on ΔT Example: T T dt = 0, T = + const, n ( T, T ) = [0,] 2 2T 6
NDTR: nt (, Tn ) ( T) < 2 3 n (T, T) NDTR 2 PDTR PDTR 0-3 -2-0 2 3 PDTR - n 2 (T) PDTR -2-3 NDTR Note: one can always ensure the inequality is satisfied by choosing a suitable value of n ( T, T) (, ) + H.-K. Chan, DH, B.Hu, Phys. Rev. E 89, 05226 (204) 7
κ e 0 2 0 0 0 0 - φ 4 model FPU-β model 0-2 0-3 Fixing the value of T + or T - at 0 0. 0-0 0 0 0 2 T + t ( ) Nj κ ( ) e T CT+ t T γ 8
Example Φ 4 model 2 pi H = + ( x x ) + x 2 2 4 i 2 4 i+ i i Constraint: fix T - N=32 N=64 N=28 N=256 N=52 N=024 N=2048 γ κeff ( T) = CT ( + t), where γ [0.95,.24], t 0 NDTR: nn 2< κ eff 0. γ >, T > T* 0.0 0 _ T where: T* = 2( T + t) γ 9
N=32 N=64 N=28 N=256 N=52 N=024 N=2048 continuum.25 NDTR: γ >, T > T* = 2( T + t) γ J.20.5 * NDTR: N < N 300.0 γ.05.00 0 00 0.95 T N* The vertical lines indicates the 0.90 theoretical 0 value 250 500 of ΔT* 750 000 250 500 750 2000 2250 N 20
Example 2 Φ 4 model Constraint: 2 pi H = + ( x x ) + x 2 2 4 i 2 4 i+ i i T+ + T m0 T = m0 m = T + T Tmax = 2 2 /2 n m T γ =, n = T + t/ T 2 m NDTR: ( γ ) m > 0, T > T* = m0 + t ( γ ) m m + t/ m 0 > 2 γ + t/ m0 2
.00 0.75 m 0.50 Presence of a NDTR regime 0.25 Absence of a NDTR regime 0.00 0 2 4 6 8 0 m o Phase diagram of m and m o for the ϕ 4 model 22
0.08 m o = 2 0.06 j 0.04 0.02 0.00 0 5 0 5 20 25 T Values of m 0.45 0.55 0.65 0.75 m0 + t The vertical lines indicate the theoretical values of T* = ( γ ) m 23
FPU-β model Example 3 2 pi H = + ( x x ) + ( x x ) 2 2 4 i 2 4 i+ i i+ i Constraint: n T = m0 + m T m T γ =, n = T + t/ T 2 NDTR: ( γ ) m > 0, T > T* = m0 + t ( γ ) m m t, m + t/ m 0 0 > < γ 2 γ + t/ m0 24
(b) 00 m 50 0-50 Absence of a NDTR regime Absence of a NDTR regime Possible presence of a NDTR regime 0-5 -0 80 90 00-00 0 20 40 60 80 00 -t / γ m o Phase diagram of m and m o for the FPU-β model 25
0.25 0.20 m o = 40 0.5 j 0.0 0.05 T* T* 0.00 0.0 0.5.0.5 2.0 T Values of m -20-50 m Note that 0 T*/ Tmax = 0.987 Tmax = γ /2 m NDTR regime is too narrow to be observed! 26
FPU-β model A very specific example Consider a change of temperatures of heat baths ( T, T ) = (5, 49) ( T, T ) = (4.005, 38.995) + + T = 2 T = 2.0 Theory: n T dt = 44.56, nn 5.35 < T d T 2 Numerics: j = 0.2664 j = 0.2457 27
Conclusion We develop a system-independent scaling analysis, and obtain the general condition for the occurrence of NDTR. Based on the condition, one can judge whether NDTR exist; If NDTR exist, ΔT* and N* can be predicted. The occurrence of NDTR can be manipulated for any nonlinear model by suitably choosing the way of varying T + and T -. 28
Thank You! 29
Part II Thermal expansion and its impacts on thermal transport in the FPU-α-β model X. Cao, DH, H. Zhao, and B. Hu, AIP Advances 5, 053203 (205) 30
Motivation I Recent controversy on the effect of asymmetric interaction potential on normal thermal conduction. (Hong Zhao s and Shunda Chen s talk) 3
Motivation II: application aspects With the rapid development of nanotechnique, thermal expansion plays an important role for thermal measurement, designing nanodevices with intriguing electronic, mechanical and thermal properties. Thermal Expansion Thermometry Appl. Phys. Lett. 66, 694 32
Motivation III: theoretical aspects Most of previous analytical studies used the perturbation approach, such as lattice-dynamics calculations, and nonequilibrium Green s function theory, which is incapable of dealing with strong anharmonicity for which some concerned intriguing properties occur. 33
Potential Profile of FPU-ab model 60 0 V(x) -60 α=0 α= α= 2 α= 3 α= 4 α= 5 α= 6 x -4-2 0 2 4 6 8 0 34
Quantify the asymmetry 20 0 (a) 0 - (b) -20 0-2 V(x) -40 Σ 0-3 -60-80 Σ= S -S 2 /(S +S 2 ) x S x S 2 0-4 -00-2 0 2 4 6 8 x x c 0-5 -6-5 -4-3 -2-0 α 35
Temperature profile T.9 α=0.8.7.6.5.4.3.2. α= α= 2 α= 3 α= 4 0 0 20 30 40 50 60 i 36
Thermal conductance 0.5 G γ ~ 0 γ < 0 γ > 0 0.0-6 -5-4 -3-2 - 0 α The nonmonotonic behavior of G can be divided by three domains, corresponding to negative, positive and vanishing coefficient of thermal expansion γ, respectively. X. Cao, DH, H. Zhao, and B. Hu, AIP Advances 5, 053203 (205) 37
Self-consistent phonon theory (SCPT) Incorporating the nonlinearity into normal modes by renormalizing the harmonic frequency spectrum, which is realized by performing thermal average with respect to a trial Hamiltonian 2 eff pi f H = + ( ui+ ui) 2m 2 Where the effective harmonic potential coefficient f(t) can be obtained from the self-consistent equations: 2 V x x 2 ( ) V( x) = 0, = x 2 0 0 f T. Dauxois, et al, Phys. Rev. E 47, 684(993) DH, S. Buyukdagli, and B. Hu, Phys. Rev. E 78, 0603 (2008) 38
Coefficient of thermal expansion 0 γ ~ 0 γ < 0 l γ > 0 0. α=- α=-3 α=-6 0 2 4 6 8 0 T 39
Effect of nonlinearity on thermal expansion 6 5 MD SCPT 4 3 l 2 0 α -6-5 -4-3 -2-0 X. Cao, DH, H. Zhao, and B. Hu, AIP Advances 5, 053203 (205) 40
Conclusion Three domains of thermal conductance with respect to α are identified, which is related to thermal expansion effect. Self-consistent phonon theory is developed to study the effect of thermal expansion, which agrees well with the numerical simulations. 4
Thank You! 42