Ballistic magnetic thermal transport coupled to phonons
BBallistic magnetic thermal transport coupled to phonons
P. Zavitsanos and X. Zotos , , Department of Physics, University of Crete, 70013 Heraklion, Greece Foundation for Research and Technology - Hellas, 71110 Heraklion, Greece and Leibniz Institute for Solid State and Materials Research IFW Dresden, 01171 Dresden, Germany (Dated: February 18, 2021)Motivated by thermal conductivity experiments in spin chain compounds, we propose a phe-nomenological model to account for a ballistic magnetic transport coupled to a diffusive phononicone, along the line of the seminal two-temperature diffusive transport Sanders-Walton model. Al-though the expression for the effective thermal conductivity is identical to that of Sanders-Walton,the interpretation is entirely different, as the ”magnetic conductivity” is replaced by an ”effectivetransfer conductivity” between the magnetic and phononic component. This model also reveals thefascinating possibility of visualizing the ballistic character of magnetic transport, for appropriatelychosen material parameters, as a two peak counter-propagating feature in the phononic tempera-ture. It is also appropriate for the analysis of any thermal transport experiment involving a diffusivecomponent coupled to a ballistic one.
The thermal transport by magnetic excitations hasbeen extensively studied over the last few years [1] andit was established as a novel thermal conduction mech-anism besides the well known electronic and phononicones. In particular, it was studied in one dimensionalquantum magnets [2] where it was shown that, in com-pounds accurately described by the Heisenberg spin-1/2chain model, there is a ballistic magnetic component ofthermal transport [3] interacting with the phononic one.Besides steady state studies of the thermal conductiv-ity, the flash method[4] provides information on the dy-namic (in time) propagation of heat and in particular onthe interaction between the magnetic and phononic com-ponents [5]. These studies could provide a playgroundfor confronting experiments to recent theoretical devel-opments on the far-out of equilibrium dynamics in inte-grable spin Hamiltonians [6].However, the magnetic excitations in a quantum mag-net are interacting with the phonons and disentanglingtheir contribution in the total thermal conductivity isan important issue. A minimal phenomenological frame-work of thermal conduction in a diffusive magnon plusphonon system was proposed by Sanders and Walton(SW) [7] and it has become the standard model for an-alyzing magnetic thermal transport experiments [8]. Inthe SW model the magnetic subsystem is assumed diffu-sive so it is interesting to revisit this model in the caseof a ballistic magnetic component more appropriate forthe spin-1/2 Heisenberg chain compounds and not only.In the following we will try to highlight the differencesin the expected effective thermal conductivity and ther-mal pulse propagation between the two models, namelythe two-temperature SW diffusion model and the presentadvection-diffusion one.In SW the starting relation is the equilibration in timeof the phonon ( T p ) and magnetic ( T m ) temperature dif- ference, ∂ ∆ T∂t = − ∆ Tτ , ∆ T = T p − T m (1)where τ is a characteristic phenomenological relaxationtime. This basic relation is also satisfied by a system ofindividual contributions, ∂T p ∂t = c m C T m − T p τ∂T + ∂t = c p C T p − T + τ∂T − ∂t = c p C T p − T − τ , (2)from right (left) moving magnetic carriers with temper-ature T ± ( T m = ( T + + T − ) /
2) and phonons at tem-perarure T p . Here, c ± are the corresponding magnetic( c m = c + + c − ) and c p phonon specific heats, C = c p + c m the total specific heat.These relations are for space independent temperatureprofiles. We can extend them to a space dependent en-ergy diffusion equation for the phonon subsystem and twoadvection equations for the ballistic magnetic system, ∂(cid:15) p ∂t = D ∂ (cid:15) p ∂x + c p c m C T m − T p τ∂(cid:15) ± ∂t ± v ∂(cid:15) ± ∂x = c p c ± C T p − T ± τ . (3) D is the phonon diffusion constant, v the characteristicvelocity of magnetic excitations, (cid:15) p the phonon energydensity, (cid:15) ± the magnetic ones and δ(cid:15) p, ± = c p, ± δT p, ± . Tohave a concrete model in mind, for the low energy spinongas in the 1D spin-1/2 Heisenberg chain with energy dis-persion (cid:15) spinon (cid:39) v | p | , (cid:15) ± = π T ± v , c ± = π T ± v and v thespinon velocity. Here and in the following, we consider a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b small deviations from thermal equilibrium and take thespecific heats independent of temperature.Furthermore, the total energy current density Q = Q p + Q m is given by the phonon Q p and magnetic Q m energy currents, Q p = − κ p ∂T p ∂xQ m = v(cid:15) + − v(cid:15) − , (4)with κ p = c p D the phonon thermal conductivity. Re-verting back to temperature dependent equations, ∂T p ∂t = D ∂ T p ∂x + c m C T m − T p τ∂T ± ∂t ± v ∂T ± ∂x = c p C T p − T ± τQ = − κ p ∂T p ∂x + v c m T + − T − ) . (5)We will first consider the effective thermal conductivityof a system − L/ < x < L/ Q | x = ± L/ = Q p at its bordersand no magnetic current ( T + − T − ) | x = ± L/ = 0. Thesteady state equations become, v ∂T m ∂x = − c p C ∆ T m τ , ∆ T m = T + − T − D ∂ T p ∂x + c m C T m − T p τ = 0 , (6) Q = − κ p ∂T p ∂x + vc m ∆ T m . (7)Solving (7) for ∆ T m we obtain,∆ T m = 1 vc m ( Q + κ p ∂T p ∂x ) ∂T m ∂x = − κ m ( Q + κ p ∂T p ∂x ) , (8)where ˜ κ m = ( c m /c p ) Cv τ is an effective transfer thermalconductivity . We solve this equation by, (i) assumingthat c p , c m are temperature independent (which strictlyspeaking is not the case) and (ii) taking the boundarycondition T p ( x = 0) = T m ( x = 0) = T . We find, T m = T − κ p ˜ κ m ( T p − T ) − κ m Qx (9)and by substituting in (6), ∂ ( T p − T ) ∂x − A ( T p − T ) − A κ t Qx = 0 A = c p c m Cτ · κ t κ p ˜ κ m , κ t = κ p + ˜ κ m . (10) The solution of (10) with the boundary condition ∂ ( T p − T ) /∂x = − Q/κ p gives the phonon temparature profile, T p = T − xκ t Q − ˜ κ m κ t κ p sinh AxA cosh
AL/ Q, (11)and (9) the magnetic temperature one.The effective thermal conductivity obtained from κ eff = − QL/ ∆ T p [7] is given by, κ eff = κ t (cid:16) κ m κ p tanh( AL/ AL/ (cid:17) − ,κ eff ∼ κ p , AL → κ eff ∼ κ p + ˜ κ m = κ t , AL → ∞ . (12)The above relations are identical to those of the SWtwo-temperature model with the replacement of the mag-netic conductivity by the effective magnetic transfer one˜ κ m = ( c m /c p ) Cv τ .Next, we will discuss the time dependent evolutionof phonon and magnetic temperature profiles (5) thatcan be probed by the flash method [4, 5]. Consider-ing an open system, 0 < x < L , we set as zero en-ergy current boundary conditions, ∂T p /∂x | x =0 ,L = 0 and T + = T − | x =0 ,L . We seek solutions of the form, T p = a + ∞ (cid:88) n =1 a n cos q n x, q n = πnLT ± = b + ∞ (cid:88) n =1 b n cos q n x ± c n sin q n x. (13)By substituting (13) in (5), we obtain the time depen-dence of a n , b n , c n ,˙ a + c m Cτ ( a − b ) = 0˙ b + c p Cτ ( b − a ) = 0 (14)with solution, a ( t ) = a (0) − c m C ( a − b ) | t =0 (1 − e − t/τ ) b ( t ) = b (0) − c p C ( b − a ) | t =0 (1 − e − t/τ ) . (15)For finite wavevector q n ,˙ a n + ( Dq n ) a n + ¯ c m ( a n − b n ) = 0˙ b n + ( vq n ) c n + ¯ c p ( b n − a n ) = 0˙ c n − ( vq n ) b n + ¯ c p c n = 0 , (16)where ¯ c p = c p / ( Cτ ) , ¯ c m = c m / ( Cτ ) are O (1 /τ ). Solu-tions of the form e λt are obtained by solving the charac-teristic 3rd order polynomial equation,˜ λ + ( Dq n + ¯ c m − ¯ c p )˜ λ + (( vq n ) − ¯ c p ¯ c m )˜ λ +( Dq n + ¯ c m − ¯ c p )( vq n ) = 0 λ = ˜ λ − ¯ c p . (17)The roots of this polynomial, although known [9], arenot physically transparent. To proceed, we will makethe physical assumption that the relaxation time τ is theshortest scale in the problem. Thus we expect two rootsof the order 1 /τ presenting the relaxation of the magneticexcitations and one root of the order of the diffusion con-stant. Once the roots determined, the constants α i , β i , γ i of the time evolutions, a n ( t ) = (cid:80) i =1 α i e λ i t , b n ( t ) = (cid:80) i =1 β i e λ i t , c n ( t ) = (cid:80) i =1 γ i e λ i t are evaluated from(17) and the initial conditions a n ( t = 0) , b n ( t = 0) , c n ( t =0).As an example, in the recent experiment [5] the rel-evant quantities for SrCuO are, κ (cid:39)
50 W/Km, κ p (cid:39) c p (cid:39) . · J/Km , c m (cid:39) · J/Km ( c p ∼ c m ), D = κ p /c p ∼ · − m /s, v (cid:39) · m/s, τ ∼ O (10 − s), which gives ˜ κ m ∼ O (10 W/Km).The largest uncertainty in these parameters is in the re-laxation time τ . For a typical sample of length L ∼ D/L ∼ − , ( v/L ) ∼ · s − and1 /τ ∼ s − . For these parameters, AL ∼ O (10 ) sothat κ eff (cid:39) κ p + ˜ κ m .Furthermore, for these experimental values, keepingthe dominant terms in (17) (e.g. dropping the 1st term,taking Dq n τ → c m << c p ) we obtain , λ (cid:39) − τ + ( vq n ) ¯ c m λ (cid:39) − c p Cτ − ( vq n ) ¯ c m . (18)Next, taking λ ∼ O ( (cid:15) ), (cid:15)τ <<
1, substituting in (17)˜ λ = (cid:15) + ¯ c p and keeping 1st order terms in (cid:15) , we find, λ (cid:39) − ( D + ¯ c m ¯ c p · v ¯ c p ) q n . (19)This relation implies a total diffusion constant composedof a phononic component D (cid:39) · − m /s enhanced bythe ballistic magnetic component ¯ c m ¯ c p · v ¯ c p = c m c p · v Cτc p ∼ O (10 − m /s). Consistently, multiplying (19) by c p werecover (12) with the second term corresponding to theeffective magnetic transfer ˜ κ m . In this limit, α (cid:39) β (cid:39)− β (cid:39) a n ( t = 0) and α (cid:39) β (cid:39) γ i (cid:39)
0. Thus, a flashmethod experiment that probes the long time behaviorof the temperature profile, when analyzed in terms of adiffusion equation, gives an effective diffusion constantwith a phononic and magnetic contibution.Whether we have three real roots or one real and twocomplex conjugate ones, indicating oscillatory behavior,depends on the sign of the discriminant in the roots λ , ,∆ = (¯ c m ) − vq n ) . (20)Assuming the experimental values quoted above and tun-ing the relaxation time, we find oscillatory behavior (complex roots) of the magnetic component relaxationfor a window of τ less than about ∼ − sec giving, λ , ∼ − τ ± i ( vq n ) . (21)Thus, the typical scenario emerging is that, within a time τ there is equilibration of the magnetic temperature pro-file to the phononic one, eventually with an oscillatorybehavior, followed by diffusive propagation of the com-bined system with an effective diffusion constant.Note that, in a flash experiment the heat is depositedon the phonon system and then it relaxes to the magneticsystem, propagating coupled thereafter. And, at any in-stant, it is the phonon temperature that it is probed. Itwould be particularly interesting if for some material pa-rameters we can visualize the ballistic propagation of themagnetic temperature profile. It is clear that at any timethe magnetic profile separates in two wavefronts propa-gating left-right while relaxing to the phonon bath. Thequestion is whether this two-bump profile can be man-ifested on the phonon temperature profile. This wouldbe a telltale sign of ballistic propagation. To realize sucha scenario within the diffusion time, it is favorable tohave a magnetic system with large specific heat that willrapidly propagate, thus large velocity and relaxation timecomparable to the diffusion time. T e m pe r a t u r e p r o f il e s x T p T m FIG. 1. Temperature profiles at t = 0 . s for L =1 mm, D p /L = 1 s − , c p /C = 0 . , c m /C = 0 . , v =20 m/s, τ = 10 − s . Initial temperature profiles, T p =(40 / √ π ) e − (40( x − / , T m ( t = 0) = 0. It is clear that the parameter space of magnetic mate-rials to search for such a behavior is very extended. Herewe will discuss just an arbitrary example with model pa-rameters indicated in Fig.1, where we observe such anevolution by numerical integration of (5). We first take asinitial phonon temperature profile a narrow Gaussian atthe center of the sample. Note, that for these parameters,the discriminant (20) is large and negative, indicative ofa strong oscillatory behavior. T p x t=0.01t=0.02t=0.03t=0.04t=0.05t=0.06t=0.08 FIG. 2. Phonon temperature profiles for the same param-eters as in Fig.1. Initial temperature profiles T p ( t = 0) =(40 / √ π ) e − (40 x ) , T m ( t = 0) = 0. T e m pe r a t u r e p r o f il e s t T p (x=1)T m (x=1) FIG. 3. Temperature profiles at x/L = 1 for the sameparameters as in Fig.1. Initial temperature profiles T p =(40 / √ π ) e − (40 x )) , T m ( t = 0) = 0. Next, for an initial heat pulse at the left-side of thesample deposited at the phonon subsystem as in a flashexperiment, we observe in Fig.2 the propagation of a heatwave, very unlike to a diffusive one, due to the hybridiza-tion of the phonon and magnetic components. It resultsto a characteristic non-monotonic behavior at the rightside of the sample as shown in Fig.3. Certainly, furtherstudy is required to establish the range of parameterswhere such a singular temperature evolution can be ob-served. In conclusion, to appropriatelly describe the ballis-tic thermal transport of magnetic excitations coupledto phonons we have introduced a phenomenologicaladvection-diffusion model. This model shows qualita-tively different behavior to the SW two-temperature dif-fusion model. In particular, it implies a greatly enhancedeffective diffusion constant for the material parametersof a recent dynamic heat experiment due to the coupledpropagation of magnetic and phononic excitations.Most important, this model predicts a specific two-bump counter-propagating temperature profiles for cer-tain material parameter range. It would by particularlyinteresting to find materials with appropriate parame-ters in order to observe such a behavior in a dynamicheat propagation experiment, e.g. direct evidence of bal-listic transport by spinons in spin-1/2 Heisenberg chaincompounds. Last but not least, althought the proposedmodel was motivated by the ballistic magnetic transportcoupled to the diffusive phonon one in spin chains, it canbe applied to any two-component system with coupledadvection-diffusion transport.
ACKNOWLEDGMENTS
This work was supported by the Deutsche Forschungs-gemeinschaft through Grant HE3439/13. X.Z. acknowl-edges fruitful discussions with Profs. P. van Loosdrechtand C. Hess on existing and future dynamic heat exper-iments. [1] C. Hess, Phys. Rep. , 1 (2019).[2] N. Hlubek, P. Ribeiro, R. Saint-Martin, A. Revcolevschi,G. Roth, G. Behr, B. B¨uchner, C. Hess, Phys. Rev. B ,020405 (2010).[3] X. Zotos, F. Naef and P. Prelovsek, Physical Review B ,11029 (1997).[4] W. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott,J. Appl. Phys. , 1679 (1961).[5] M. Montagnese, M. Otter, X. Zotos, D.A. Fishman, N.Hlubek, O. Mityashkin, C. Hess, R. Saint-Martin, S.Singh, A. Revcolevschi, P.H.M. van Loosdrecht, Phys.Rev. Lett. , 147206 (2013).[6] O.A. Castro-Alvaredo, B. Doyon and T. Yoshimura Phys.Rev. X , 041065 (2016); B. Bertini, M. Collura, J. DeNardis and M. Fagotti, Phys. Rev. Lett. , 207201(2016).[7] D.J. Sanders and D. Walton, Phys. Rev. B , 1489(1977).[8] A. V. Sologubenko, K. Giann`o, H. R. Ott, A. Vietkine andA. Revcolevschi, Phys. Rev. B64