Transport properties at high magnetic fields fields: old facts and new results

Transcription

1 Alpha - meeting Vienna, April 24 Transport properties at high magnetic fields fields: old facts and new results E. Bauer Institut für Festkörperphysik, Technische Universität Wien 28. April 24

2 Contents Basic features Boltzmann Equation Magnetic Field influence Examples... Field effect on stable nonmagnetic and magnetic systems Field induced phase transitions Itinerant metamagnetism Field effect in Kondo and heavy fermion systems Quantum phase transitions Field effects in superconductors

3 About transport properties External forces cause charge and/or energy flow across a sample. Who transports charge or energy? Who is responsible? Charge and energy are carried by: electrons and quasi-particles like phonons (lattice vibrations) magnons (spin waves), plasmons, etc. Forces: electrical fields, temperature gradients, magnetic fields. Combination of external forces: electrical resistivity thermal conductivity Seebeck coefficient Peltier-effect magnetoresistance Hall-effect,...

4 Ohm s Law in Magnetic Fields j = σ (T, B) E (1) j... electrical current density, E... electric field, σ... temperature and magnetic field dependent conductivity tensor. Magnetic field breaks symmetry, but Onsager-relations holds for σ : Specific electrical resistivity: σ ij ( B) = σ ji ( B). (2) ρ(t, B) := E I I 2 = I ( σ ) 1 I I 2 (3) Magnetoresistance: ρ ρ := ρ(t, B) ρ(t, B = ) ρ(t, B. (4) = )

5 The Boltzmann Equation, Fermi-Dirac distribution function to describe material in equilibrium: f (E ( k )) = 1 [ ] 1/k B T = β (5) E( 1 + exp k) µ k B T External ( ) forces: development of f in phase-space function f k, r, t ; Volume elements non-interacting incompressible liquid: d 3 kd 3 r d 3 k d 3 r remains constant: (due to F at period t) f ( k + kdt, r + rdt, t + dt ) = f ( k, r, t ) (6)

6 Due to collisions: Electrons scattered into - or out of the 6-dimensional volume elements d 3 kd 3 r and d 3 k d 3 r within time dt. ( ) ( ) ( ) f k + kdt, r + rdt, t + dt = f k, f r, t + (7) t Taylor series in dt of l.h.s. and comparing coefficients Boltzmann equation ( d dt + ) k k + r r f ( k, r, t ) = ( ) f t C C (8) External forces time-independent ( k k + r r ) f ( k, r ) = ( ) f t C (9)

7 Using r = v( r) = 1 h k ɛ( k), (1) f ( k, r, t ) Boltzmann equation F = h [ k = e E 1 ] + c ( v H), (11) f (ɛ) (in the case of equilibrium), (12) fied term { [ }}{ e E + 1 ( ( k ɛ) H h hc ) ] ( ) k f k, r + 1 h ( k ɛ ) ( ) r f k, r collision term {( }}) { f t C = (13)

8 External forces perturbe conduction electron system but internal interactions cause stationary state! Stationary state attained exponentially relaxation time approach: ( ) f = f f. (14) t τ C Assume ( ) f = f f = f 1 t C τ τ = 1 τ Φ f ɛ. (15) Φ measures deviation from the equilibrium state f. Thus: [ e E + 1 ( ( k ɛ) H hc ) ] 1 ( ) h k f k, r + 1 h ( k ɛ ) ( ) r f k, r = = 1 τ Φ f ɛ. (16)

9 After tedious mathematics Boltzmann equation: e E v k f ɛ + = 1 τ Φ f ɛ. e h 2 c [ H ( k ɛ k Φ)] f ɛ v k [ɛ µ] f ɛ r ln T = (17) With H =, and µ µ(t ): Φ = τ [ ee ] (ɛ µ) r ln T v k. (18) If µ = µ(t ): Φ = τ [ e E ( ɛ µ T + dµ ) ] r T v k. (19) dt

10 Finally f ( k, r ) Knowledge of f = f (ɛ) τ ( k, r ) [ ee ] f (ɛ µ) r ln T v k ɛ. (2) allows description of the heat current density q (sum of all particles with respective kinetic energy). q V ( ) 4π 3 v r (ɛ µ) f k, r d 3 k (21) Similarly: electrical current density j ( charge times velocity). j = V ( ) 4π 3 e v r f k, r d 3 k (22) Unperturbed distribution function does not contribute to current densities!

11 Definition: K n = 1 (2π) 3 τ γ v γ v γ (ɛ γ µ) n ( f γ ɛ γ ) d 3 k (23) anisotropic materials: K n tensor of 2nd rank solid state physics used to determine τ γ Final results j = e 2 K E e T K 1 T (24) q = ek 1 E 1 T K 2 T (25)

12 Resistivity contributions If different scattering mechanisms independent from each other: Matthiessen-rule 1 Scattering processes are simply added: τ γ = 1 τ γ (1) Metals: Temperature dependencies: + 1 τ (2) γ ρ = ρ + ρ el + ρ ph + ρ mag (26) ρ : temperature independent residual resistivity ρ el : electron - electron scattering; AT 2 ; A el. density of states ρ ph : electron - phonon scattering; T 5 at low T and T at high T ; ρ mag : electron scattering on magnetic moments; various temperature dependencies.

13 Magnetic field effects f t + 1 h F k f + v γ f = Collision term expressed by: ( ) f t C. (27) ( ) f = {f(γ )[1 f(γ)]p γ t γ f(γ)[1 f(γ )]P γ γ }, C γ (28) P γ γ... quantum-mechanical transition rate per unit time from state γ to state γ. With f = f o + f 1, and the Ansatz for f 1 ( ) fo f 1 := Φ γ, (29) ɛ γ

14 A linearisation yields (C + Ω)Φ γ = e ( ) fo v γ ɛ E, (3) γ introducing C, the collision operator, and Ω, the magnetic operator. Operators are given by where CΦ γ = 1 k B T ΩΦ γ = ē f ( o v γ B h ɛ γ W γγ [Φ γ Φ γ ] and (31) γ ) k Φ γ, (32) W γγ = f o (γ )[1 f o (γ)]p γ γ = W γ γ (33) C is symmetric, while Ω is antisymmetric.

15 Formal solution Current density Φ γ = e (C + Ω) 1 ( fo ɛ γ ) v γ E. (34) j = e ( ) fo v γ Φ γ ɛ γ γ = e ( ) ( ) 2 fo v γ [C + Ω] 1 fo v γ ɛ γ γ ɛ E. (35) γ }{{} = σ

16 Classical Magnetoresistance Classical magnetoresistance due to Lorentz force on moving charges always present. For B magnetic operator Ω enters the Boltzmann equation. Components: magnetic field B; group velocities v γ ; gradient k in reciprocal space. BUT NOT: transitions probabilities represented by τ γ. Only electrons close to the Fermi surface contribute to the current density: Importance of the topology of this surface on operator Ω.

17 Kohler s rule ρ/ρ [%] YAl 2 Dr/r [%] B/r(,T) [T/mWcm] B=2T B=4T B=6T B=8T B=1T T [K] Magnetoresistance can be expressed as ρ (B, T ) = F (B/ρ (B =, T )). ρ Justified only in the singleband model; relaxation time of the form τ γ = τ(ɛ( k)). good approximation in many cases where one scattering mechanism dominates.

18 Classical Magnetoresistance, an example Dr/r [%] YAl 2 Fit-parameters: a = 7.6 T 2 (µωcm) -2 b = 1.5 B=2T B=4T B=6T B=8T B=1T Fit T [K] ρ(, T ) follows the Bloch Grüneisen equation ρ ρ = B 2 a [ρ(b =, T )] 2 + bb 2 a, b field and temperature independent parameters (characterising two different conduction bands) ρ(, T ) = ρ + c 2 Θ D ( T Θ D ) 5 ΘD /T z 5 dz (exp(z) 1)(1 exp( z))

19 Magnetoresistance in ferromagnetic materials ρ m (B, T ) = C m 1 J(j + 1) ( J 2 z J z 2 + η J z sinh 2 η T T c ) ρ [µωcm] B 2/3 [T 2/3 ] B= T B=2 T B=4 T B=6 T B=8 T B=1 T b Dr/r [%] Dr/r [%] T [K] GdAl T [K] Relaxation time field dependent! T C T C a ρ m ρ spd = C 1 h 2 (T/T c 1) 2 T < T c ( ρ m C 2 = C 1 h ρ spd (1 T/T c ) minimum at T = T c ρ m ρ spd = C 1 ( h C 2 ) 2/3 ) 1/2

20 Magnetoresistance in antiferromagnetic materials 1 c 2 [1] T N = 5K α =.4 α =.6 α =.8 above T c : similar to ferromagnets ρ m /ρ spd B 2 For T < T N T/T N more complicated Scenario (many different spin arrangements in AFM) ρ m ρ spd = χ 2 (T/T N )h 2 h = (gµ B /2k B T N )B

21 Magnetoresistance of antiferromagnetic DyAg ρ/ρ [%] DyAg 2 T 4 T T f =46 K c T N =56 K T/T N T [K] T 8 T 1 T antiferromagnetic order below T N1 = 46.5 K; spin arrangement according to (π, π, ) sinusoidally modulated spin arrangegement for T N 1 < T < T N2 = 56 K. Differences for moments parallel or perpendicular to external fields; no influence in the latter case for perfect AFM; strong influence for the former: disorder increases positive magnetoresistance; above critical field: field induces FM; resistivity starts to decrease χ 2 in fair agreement with above calculations.

22 Field-induced magnetic phase transitions First-order phase transitions (FOMPTs) characterised by spin-flip processes. accompanied by dramatic changes of structural, electrical and magnetic properties huge changes of the resistivity, matching that of giant magnetoresistance systems (GMR) Sommerfeld coefficient γ at FOMPT varies dramatically substantial changes of the density of states at the Fermi energy.

23 ρ [a.u.] Temperature and field dependent resistivity of Dy 3 Co Dy 3 Co H // c j // c H // (ab) j // c (a) (b) T 2 T 4 T 6 T 8 T 1 T 12 T T [K] Dy 3 Co: T N1 = 44 K, T N2 = 32 K. increasing fields: phase transitions merge together and vanish finally; strong anisotropy field applied at low temperatures: (H//c): transition at T m for H = 1.5 T; vanishes at 3.5 T (H c): transition at T m for H = 4 T; vanishes at 6 T

24 Magnetoresistance of Dy 3 Co ρ/ρ [%] Dy 3 Co : H // (ab), j // c K.65 K µ H [T] K ρ/ρ [%] K Y Axis 2. K 3. K 5.7 K 24 K 31.8 K µ Η [T]

25 GMR model ρ/ρ [%] Dy 3 Co : H // (ab), j // c.45 K field [T] Boltzmann equation: 1/ˆρ ˆσ = e2 8π 3 h s τ s ( v s ) 2dS F v s, Integral over constant energy surface in k-space (energy E F ). GMR at the FOMPTs due to superzone boundaries: vanishing of parts of Fermi surface near to antiferromagnetic zone boundaries resistivity increases at antiferromagnetic zone boundary El. velocity v s changes between ferro - and antiferromagnetic phase ( v s AF M < v s F M. Below 3 K: magnetic fields creates - strong history dependent - remanent magnetization; no sample history above 3 K.

26 Magnetoresistance of Dy 3 Co H crit [T] critical field [T] DH = DH (1 - at 1/2 ) increasing ext. fields T [K] decreasing ext. fields T [K] Modelling of thermal activated domain wall displacement Temperature dependent width of hysteresis loop (H c): H(T ) = H (1 αt 1/2 ), At lower temperatures: no activation type behaviour; domain wall motion from quantum tunnelling of the magnetization.

27 Itinerant Electron Metamagnetism m Co (m B ) Er 1-x Y x Co 2 Tm 1-x Gd x Co 2 RCo 2 m Co (m B ) µ H ext (T) Er Ho Dy Tb Gd Co µ H fd (T) LuCo 2 YCo 2 Strong energy dependent density of states (DOS) and significantly enhanced Stoner factors itinerant electron metamagnetism, IEM,. Examples: YCo 2 oder LuCo 2 ; metamagnetic phase transition from para- to ferromagnetism above critical field (about 7 T). Magnetic susceptibility exhibits maximum

28 IEM, recent examples LaCo 9 Ge LaNi 9 Ge 4.95 ρ(b)/ρ() T =.5 K T = 2.1 K T = 4 K.75 T = 6 K T = 8 K.7 T = 1 K T = 15K T = 2 K B [T] T = 25 K T = 3 K ρ(b)/ρ() T = 2.1 K T = 4 K T = 6 K T = 8 K T = 1 K T = 12 K T = 14 K T = 16 K T = 18 K T = 2 K B [T]

29 Itinerant Electron Metamagnetism II For RECo 2 with permanent magnetic moments: the molecular field H m may induce a magnetic moment! r (mwcm) a) TmCo 2 ErCo 2 HoCo 2 DyCo 2 TbCo 2 T C T (K) T (K) r (mwcm) 9 6 TmCo 2 H m = C(g J 1)Jµ B g J... Lande factor, J... total angular momentum, C... coupling constant. Co moment will be induced if H m > H cr 7 T. For ErCo 2 : H m 126 T first order phase transition at about 33 K with Co moment 1 µ B.

30 Itinerant Electron Metamagnetism III r (mwcm) Er 1-x Y x Co T (K) x =. x =.1 x =.2 x =.3 x =.4 x =.5 x =.6 x =.7 decrease of H Co fd with x (the influence substitution on H cr is neglected) two distinct magnetic phase transitions, if H Co fd (x) < µ H cr at T = T C but H Co fd (T ) > µ H cr for T << T C (x = x cŕ ) the rare-earth sublattice only orders magnetically, when H Co fd < µ H cr at T < (x = x cr )

31 Inverse Itinerant Electron Metamagnetism IIEM H mol H extern RECo 2 compounds - ferrimagnets Effective field at the Co site: H eff = H m H ext H effective If H ext increases such that Hcritical critical H eff < H cr ErCo 2 is a ferrimagnet H m antiparallel to the Er moment. m Er > m Co, H m antiparallel to the external field H ext ; an abrupt collapse of the Co moment at H eff = H cr with an metamagnetic phase transition may be observed (inverse IEM).

32 IIEM Dr/r (%) Dr/r (%) x= x= Hinv (T) 2 1 bar 1.5 kbar 3.5 kbar 5. kbar 6.5 kbar 8. kbar T =.5K P (kbar) Er.6 Y.4 Co B (T) Er.7 Y.3 Co 2 Yamada et al: H cr increases as a function of pressure and metamagnetism vanishes above a critical pressure Inverse IEM observable in (Er, Y)Co 2 under pressure. Substitution decreases H m, thus H eff = H m H ext decreases and H eff < H cr becomes realistic (additionally improved by pressure because H cr grows).

33 Field dependent resistivity of Kondo systems ρ [µωcm] ρ [µωcm] ρ [µωcm] (b) (c) YbCu4Ag B = T B = 6 T B = 12 T B = T B = 1 T B = 2 T B = 3 T YbCu4Pd YbCu4Au T [K] (a) B = 4 T B = 8 T B = 12 T B = T B =.5 T B = 1. T B = 2. T B = 3. T B = 6. T B = 12 T Various features: YbCu 4 Ag: positive magnetoresistance ρ(t ) A T 2 large value of A (high density of states); A diminished by external fields YbCu 4 Au and YbCu 4 Pd negative magnetoresistance strongly field dependent influence of long range magnetic order and crystal field splitting

34 Magnetoresistance of Kondo systems 35 3 YbCu 4 Au 4 35 T=.45K T=2.K T=7.K T=15K r / r o T* =.4K r [µwcm] 25 2 Temp. T r [µwcm] 3 25 B /T+T * [T/K] Temp. YbCu 4 Au K 15.5 K B [T] B [T]

35 Magnetoresistance of Kondo systems - Theory Bethe - Ansatz solution of Coqblin - Schrieffer Hamiltonian, 2j+1 i=1 n i = 1. ρ() ρ(b) = 1 2j + 1 2j+1 i=1 1 sin 2 πn i 2j+1 n i... occupation numbers of 2j + 1 Zeeman levels. ρ(b)/ρ() Coqblin Schrieffer model B/B* j = 1/2 j = 3/2 j = 5/2 field dependent occupation number and magnetization required. ρ(b)/ρ() determined by single parameter, characteristic field B. Universal behaviour of ρ(b)/ρ() vs. B/B physics of Kondo impurities dominated by single energy scale (Kondo temperature T K ). Large magnetic fields: logarithms as features of asymptotic freedom, i.e. ρ(b) ln 2 [(B/T K ) 2 ].

36 Magnetoresistance of Kondo systems r [µwcm] T=2.K T=3.K T=4.K T=5.K T=7.K T=9.K T=13K T=2K Schlottmann's model curves B [T] YbCu 4 Au J=3/2 B (T ) = B () + k BT gµ g... Lande factor, µ... mag. moment of Kondo ion; T+T* [K] YbCu 4 Au J=3/2 linear regression bo=.53 b1= T [K] r =r(.5k,12t)=11.16µwcm B ()... Kondo field Kondo temperature via T K = B () gµ k B

37 Quantum phase transitions Central aspect: Quantum fluctuations hω in relation to k B T T thermally disordered quantum critical kt ~ 67 c at B c : quantum phase transition into disordered regime; path a : no real phase transition; system always disordered; 3 regions exhibit different properties; path b : divergence of order parameter fluctuations scaling properties. classical critical ordered (a) B c (b) quantum disordered QCP thermal fluctuations dominate near finite phase transition at each temperature (critical slowing down) classical region narrows for T c. path a : real phase transition; crossover from quantum critical behaviour away from phase boundary to classical behaviour at phase boundary. B

38 Non-Fermi-liquid (NFL) behaviour due to a phase transition at T = CeCu 6-x Au x AFM NFL FL CeCu 5.9 Au.1 Metals at low temperature: Fermi liquid behaviour; e.g., C p /T (T ) const.; χ const. ρ = ρ + AT 2. Occurrence of QCP by tuning parameters such as pressure, field, substitution,... QCP causes strong fluctuations, even at finite temperatures; FL behaviour breaks down appearance of NFL features.

39 Non-Fermi-liquid (NFL) behaviour due to a phase transition at T = - II Experiment and theory (e.g., Millis, 1993, Moriya and Takimoto, 1995) c/t ln(t/t ) and c/t = γ a T ; χ(t ) = χ χ 1 T m ; ρ T n ; n < 2

40 NFL & QCP, Coleman, 1999, 21 Hybridisation strength between f and conduction electron magnetic instability for W W c due to increasing interactions between quasi-particles; magnetic moments quenched at finite temperatures; do not play role at QCP; QCP is a spin-density wave instability of Fermi surface; NFL results from el. scattering off a critical spin-density wave; weak coupling, Gaussian behaviour of spin fluctuations. composite QP in real space (large Fermi surface) disintegrate at QCP, magnetic sublattice establishes; Local moments exists down to T K ; play active role in fluctuations at QCP ; strong coupling, non-gaussian spin fluctuations with hyperscaling.

41 Evolution of the valence state of YbCu 5 x Al x YbCu 5-x Al x ν(3k) ν(1 K) LaCu 5-x Al x, T = 12 K YbCu 5-x Al x, T = 3 K strong temperature dependence of the valence ν in the intermediate valence range valence unit cell volume [A 3 ] o YbCu 5-x Al x, T = 12 K x concentration dependent volume increase x onset of long range magnetic order

42 Resistivity of YbCu 5 x Al x ρ/ρ 273 K max T ρ, low [K] YbCu 5-x Al x x x = 1.7 x = 1.6 x = 1.5 x = 1.4 x = 1.3 x = T [K] strong decrease of the resistivity maximum with increasing Al content; T K lowers (Tρ max T K, Cox and Grewe, 1988) development of crystal field splitting decrease of T K due to changes of the electronic structure (unit cell volume increases)

43 NFL behaviour in Yb compound: Substitution effects in YbCu 5 x Al x C p /T [J/molK 2 ] YbCu 5-x Al x x = 1 x = 1.3 x = 1.4 x = 1.5 x = 1.6 x = 1.75 x = T [K] Cu/Al substitution: increase of the lattice spacings no pressure effect! electronic effects cause magnetic instability (via a decrease of N(E F ). Evolution of the electronic contribution to the specific heat from intermediate valence to long range magnetic order. Near to the critical concentration (x = 1.5): ln T behaviour of C p /T. Very large values of C p /T for. (indication for strongly renormalised effective masses)

44 Field and temperature dependent specific heat of YbCu 3.5 Al 1.5 C/T [mj mol -1 K -2 ] YbCu 3.5 Al T [K] T 2T 5T 8T 1T 13T Ch. Seuring et al., 21 at zero field above 3 mk: ln T behaviour, below 3 mk: C/T tends to flatten out; C/T decrease as a response to external magnetic fields; for higher fields: Recovery of a FL ground state with C/T = const.

45 Electrical resisitivity of YbCu 3.5 Al 1.5 ρ- ρ [m Ω] specific heat, susceptibility: x cr 1.5 YbCu 3.5 Al ±.2 ρ = ρ +ct ρ = ρ +ct T [K] B = T A = f(1/b) B [T] Ch. Seuring et al., 21 A [mω K -2 ] y H cr = at lowest temperatures: ρ(t ) = ρ + ct 1.3,.3 T.6 K: ρ(t ) = ρ + ct ; increasing magnetic field recovers a FL state, i.e., ρ AT 2, range, where T 2 law holds increases with increasing fields, coefficient A increases with decreasing fields and diverges for B ; agreement to predictions of the SCR model.

46 Field induced NFL in YbCu 3.25 Al 1.75 : specific heat Sample oriented along easy axis with respect to the external magnetic field. C/T [J/molK 2 ] 3. YbCu 3.25 Al 1.75 T 1 T T 1.4 T 1.7 T 2 T 3 T 5 T 1 T Suppression of magnetic order at H cr = 1.4 T; H = H cr : C/T log T ; FL behaviour for H H cr ; large values of C/T, C/T (T ) strongly field dependent T [K]

47 Electrical resisitivity of YbCu 3.25 Al 1.75 ρ ρ [µωcm] ρ ct 1.4 +/-.1, 3-3 mk YbCu 3.25 Al 1.75 B = B cr = 1.4 T T [K] ρ ct, 3-52 mk YbCu 3.25 Al 1.75 B > B cr : r = r + AT B - B cr [T] A [µωcmk -2 ] T N for B = B cr = 1.4 T for B = B cr at lowest temp.: ρ(t ) = ρ + ct 1.4 ±.1,.3 T.55 K: ρ(t ) = ρ + ct ; increasing magnetic field recovers a FL state, i.e., ρ AT 2, range, where T 2 law holds increases with increasing fields, coefficient A increases with decreasing fields and diverges for B B cr ;

48 Crystal structure of CePt 3 Si Structure type: CePt 3 B Symmetry: P 4mm Ce: (1b).5,.5,.1468 Pt(1): (2c).5,,.654 Pt(2): (1a),, (fixed) Si: (1a),,.4118 Lattice parameter: a 4.7 Å c 5.44 Å CePt 3 Si structure has no inversion center! P4mm involves the absence of the mirror plane z z

49 Temperature dependent resistivity of CePt 3 Si ρ [µωcm] (a) CePt 33 Si Si LaPt 3 Si T [K] T mag ρ mag [µωcm] ρ [µωcm] 2 1 (b) T N = 2.2 K CePt 3 Si ρ = ρ + AT 2 ρ = 5.18 µωcm A = 2.35 µωcm/k T [K] dρ/dt [µωcm/k] Tc mid.75 K T c < T < T N : ρ = ρ + AT 2 with A = 2.23 µωcm/k 2 magnetic transition from dρ/dt at T mag 2.2 K; Kondo interaction in presence of crystal field splitting (3 doublets); LaPt 3 Si: metallic, θ D 16 K;

50 Temperature dependent specific heat of CePt 3 Si C p /T [J/mol K 2 ] CePt 3 Si LaPt 3 Si T N = 2.2 K 2 T c =.75 K T [K] S mag [J/mol K] T c.75 K; T N 2.2 K; T 3 - behaviour for T c < T < T N ; λ-like anomaly shifts to lower temperatures with increasing fields; entropy gain at T = T N about.22 R ln 2; ordering with small magnetic moments; LaPt 3 Si: γ = 9 mj/molk 2 θ LT D = 255 K.

51 Field dependence of CePt 3 Si C p /T [J/mol K 2 ].6 CePt 3 Si.5.4 µ H = T µ H = 1 T.3 µ H = 2 T µ H = 4 T (a) T [K] CePt 3 Si specific heat resistivity slope ~ -8.5 T/K (b) T [K] H c2 [T] dh c2 /dt H c2 8.5 T/K; H c2 () 5 T; high fields: γ.37 J/molK 2 ; coincidence with T 3 extrapolation; large H c2 values heavy quasiparticles as Cooper pairs; low temperature increase in C p /T : nuclear contribution from 1 95Pt.

52 Summary Transport in solids is sensitive to interactions with particles and quasiparticles (e.g., electrons phonons, magnons, ect. because electrons as sensors are directly built in the materials and do not behave as perturbation! Transport in solids allows to detect physical features such as superconductivity, long range magnetic order, metal to insulator transitions, Kondo interaction, etc. Transport in context of external parameters (pressure, fields) allows to distinguish between different interaction mechanisms such as long range magnetic order - Kondo interaction - crystal field splitting

53 and to establish the respective phase diagrams