Some pictures are taken from the UvA-VU Master Course: Advanced Solid State Physics by Anne de Visser (University of Amsterdam), Solid State Course

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1 Some pitures are take rom the UA-VU Master Course: Adaed Solid State Physis by Ae de Visser (Uiersity o Amsterdam), Solid State Course by Mark arrel (Ciiati Uiersity), rom Ibah ad Lüth, rom Ashrot ad Mermi ad rom seeral soures o the web.

2 the study o trasport pheomea i physis is related with the ehage o mass, eergy, ad mometum studied systems. luid mehais, heat traser, ad mass traser i solid state physis, the motio ad iteratio o eletros, holes ad phoos are studied uder "trasport pheomea".

3 Cotet: 1. Itrodutio. Geeral. Fik Law. Boltzma Eq. Relaatio time.. Eletroi trasport i odutors. Eletro-phoo satterig. 3. Eletro-imperetio satterig 4. Eletrial odutiity. Bloh-Grueise 5. Mageti satterig 6. Thermal odutiity 7. Thermoeletri pheomea 8. Eletrial odutiity i mageti ields. 9. Aomalous Hall eet 10. Magetoresistae : AMR, CMR 11. Magetoresistae : GMR TMR. 1. Strogly orrelated eletro systems

4 1. C. Kittel, H. Kroemer, Thermal Physis, W.H. Freema Co. New York. C. Kittel, Itrodutio to Solid State Physis (7-8 ed., Wiley, 1996) 3. N. W. Ashrot, N. D. Mermi, Solid State Physis, U. Mizutai, Itrodutio to the Eletro Theory o Metals, Cambridge Uiersity Press Ch. Ess, S. Hukliger, Low-Temperature Physis, Spriger-Verlag Berli Heidelberg M. Coldea, Magetorezisteta. Eete si Apliatii, Presa Uiersitara Clujeaa, E. Dagotto, Naosale Phase Separatio ad Colossal Magetoresistae, Spriger- Verlag Berli Heidelberg M deteresa, New mageti materials ad their utios 007, Cluj-Napoa, Romaia. Summer Shool ( 9. UA-VU Master Course: Adaed Solid State Physis 10. H. Ibah ad H. Lüth: Solid State Physis 3rd editio (Spriger-Verlag, Berli, 003) ISBN X 11.. M. ZIMAN, ELECTRONS AND PHONONS,The Theory o Trasport Pheomea i Solids, UNIVERSITY OF CAMBRIDGE, OXFORD AT THE CLARENDON PRESS 1960 Some pitures are take rom the UA-VU Master Course: Adaed Solid State Physis by Ae de Visser (Uiersity o Amsterdam), Solid State Course by Mark arrel (Ciiati Uiersity), rom Ibah ad Lüth, rom Ashrot ad Mermi ad rom seeral soures o the web.

5 Itrodutio. Trasport Proesses Noequilibrium steady state System T 1 T Etropy (reseroir 1+reseroir + System) INCREASES Driig oretemperature gradiet Trasport o Eergy Flu (oeiiet) (driig ore) Liear pheomeologial law (i the ore is ot to large) e.g. Ohm s law or the odutio o eletriity

6 A lu desity o A et quatity o A trasported aross area i uit time Net trasport the trasport i oe diretio the trasport i oposite diretio

7 grad ρ A D Summary o pheomeologial trasport laws A rom Ess

8 Partile diusio rom Kittel, Thermal Physis T ostat µ 1 µ partile low Etropy (reseroir 1+reseroir + System) INCREASES Presume: the dieree o hemial potetial is aused by a dieree i partile oetratio - the umber o partiels passig through a uit area i uit time Fik s law: D grad D partile diusio ostat - diusiity

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10 l the mea ree path At positio z the partiles ome ito a loal equilibrium oditio µ(z) ad (z) z θ z+l z-l z 1 d dz [ ( z l ] z ) ( z + lz ) z zlz We epress z l z i terms o l l z l osθ osθ z The aerage is take oer the surae o a hemisphere The elemet o surae area is π siθ dθ l z l π π os θ siθ dθ 1 l 0 z π 3

11 So that z 1 d 1 l D l 3 dz 3 Partile diusio is the model or other trasport probelms Partile diusio trasport o partiles Thermal odutiity trasport o eergy by partiles Visosity trasport o mometum by partiles Eletrial odutiity trasport o harge by partiles The liear trasport oeiiets that desribes the proesses are proportioal to the partiles diusiity D

12 let ρ A the oetratio o the physial quatity A. -the lu desity o A i the z diretio is: z A ρ A z z is the mea drit eloity o the partiles i the z diretio (drit eloity is zero i thermal equilibrium) I A (e.g. eergy, mometum ) depeds o the eloity o a moleule: z A A ρ A z A is a ator with magitude o the order o uity it depeds o the eloity depedee o A ad may be alulated (e.g. by usig the Boltzma trasport equatio) By aalogy with Fik law A D grad ρ A

13 Thermal odutiity Fourier law s A K grad T Desribes the eergy lu desity i terms o the thermal odutiity K ad the temperature gradiet Τ 1 Τ System This orm assumes that there is a et trasport o eergy, but ot partiles Aother term must be added i additioal eergy is trasported by meas o partile low (as whe eletros low uder the iluee o a eletri ield.)

14 The eergy lu desity i the z diretio is: z u ρ u z ρ u is the eergy desity By aalogy with the diusio equatio: Diusio o eergy D dρ u d D( ρ u T )( dt d ) ρu T Is the heat apaity per uit olume, C V.

15 u D C V grad T K DC V 1 C 3 V l The thermal odutiity o a gas is idepedet o pressure util ery low pressure whe the mea ree path beomes limited by the dimesios o the apparatus rather the by itermoleular ollisio.

16 Q (heat low) Hot T h Cold T L Q T ka T L h dt ka d Thermal odutiity

17 Visosity Visozity is a measure o the diusio o mometum parallel to the low eloity ad traserse to the gradiet o the low eloity Visosity oeiiet: z X d η dz z z ( p ) rom Ess

18 The partile lu desity i the z diretio: z z Dd dz The traserse mometum desity: M Its lu desity i the z diretio: ( M This lu desity Dd( M ) dz A D grad ρ A ( ) ) z Kittel, Thermal physis Mass desity: ρ M z ( p ) ρ Dρd dz η d dz η 1 Dρ ρl 3 CGS- poise SI uit: Pa.s

19 The mea ree path: l 1/ πd Visozity: η M 3πd idepedet o gas pressure The idepedee ails at ery high pressures whe the moleules are always is otat, or at ery low pressures whe the mea ree path is loger tha the dimesio o the apparatus. Robert Boyle 1660 air D η ρ Kiemati isosity K η C / ρ

20 Geeralized Fores C. Kittel, Thermal Physis The traser o etropy rom oe part o a system to aother is a osequee o ay trasport proess. We a relate the rate o hage o etropy to lu desity o partiles ad o eergy For V ost. ds 1 T The etropy urret desity : du µ dn T s 1 T u µ T (*) Etropy desity Ŝ The et hage o etropy desity at a ied positio Ŝ t Ŝ t g S di s Eq. o otiuity Rate o produtio o etropy

21 I a traser proess U ad N are osered The equatios o otiuity: u dt dt di u di di Diergee o S 1 ( T ) ( T ) ( T ) S di u + u grad 1 µ di grad µ T (*) Ŝ 1 u µ t T t T t

22 gs u grad grad µ ( 1 T ) + ( T ) g S u F u + F Geeralized ores F u grad 1 T ( ) F grad µ ( T ) Coupled eets Thermodyamis o irreersible proesses: u L L 11 Fu + 1 Fu + L L 1 F F L Osager relatio: ij B L ji B I mageti ield

23 Aaed Treatmet: Boltzma Trasport Equatio We work i the 6 D (si-dimesioal spae o Cartesia oordiates r ad ). The lassial distributio utio: ( r,) drd umber o partiles i drd The eet o time displaemet dt o the distributio utio: ( t + dt,r + dr, + d) ( t,r, ) I the absee o ollisios

24 With ollisios: ( t + dt,r + dr, + d) ( t,r, ) dt( / t ) ollisios dt( With a series deelopmet: t ) + dr a d dt grad r + d grad dt( / t ) ollisios t + grad r + a grad ( / t ) ollisios Boltzma trasport equatio

25 Relaatio time aproimatio: This is based o the assumptio that a oequilibrium distributio gradually returs to its equilibriu alue withi a harateristi time, the relaatio time, by satterig o partiles with the eloity ( τ r, ) ito states, ad ie ersa. ' ( / t ) τ ollisios ( 0 ) / rom Kittel, Thermal physis ( t,r, ) Suppose that a oequilibrium distributio o eloities is set up by eteral ores whih are suddely remoed. The deay o the distributio towards equilibrium is the obtaied: ( t 0 ) ( 0 ) / τ

26 ) / t ( ) ( ) ( t t τ ep t ( ) r, τ τ Geerally r grad a grad t τ I the steady state: 0 t / by deiitio

27 Partile Diusio Cosider a isothermal system with a gradiet o partile oetratio The steady-state Boltzma trasport equatio i the relaatio time approimatio: d d ( 0 ) / τ First order approimatio Seod order approimatio d 1 0 τ 0 d d d0 d d 0 τ d1 d 0 τ d0 d + τ d 0 d The iteratio is eessary or the treatmet o oliear eets

28 Classial Distributio 0 ep[( µ ε ) / k B T ] d / d ( d / d )( dµ / d ) ( / kb T )( dµ / 0 0 µ 0 d ) The irst order solutio or the oequilibrium distributio beomes: 0 ( τ 0 / kbt )( dµ / d ) The partile lu desity i the diretio: D( ε ) dε The desity o orbitals per uit olume per uit eergy rage: D( ε ) 1 4π M 3 ε 1

29 Presume τ ostat, idepedet o eloity D( ε )dε 0 ially, ( τ / M )( dµ / d ) ( k B Tτ / M )( d / d ) beause µ k B Tlog + ost. Diusiity: D τ k B T / M 1 3 τ I we presume τ l 1 D l 3

30 Fermi-Dira distributio 0 1 ep[( ε µ ) / k B T + 1 d dµ δ( ε µ ) 0 d d δ( ε µ )dµ / 0 + d F ( ε ) δ( ε µ )dε F( µ ) The partile lu desity We kow D( ε )dε ( dµ / d ) τ D( µ ) 3 / ε F δ( ε µ )D( ε ) dε µ ( / m )( 3π ) 3 ( τ / 3m ) ε d / d F 1 3 F τ d / d D 1 3 F τ

31 Eletrial odutiity We multiply the partile lu desity by the partile harge q qdϕ / d qe dµ / d q ( τ q / m )( dµ / d ) ( τ q / m )E σe Eletrial odutiity σ q τ / m For a lassial gas (Drude) For the Fermi-Dira distributio (Sommereld) We will disuss this i more detail, later.