Electronic Transport. Peter Kratzer Faculty of Physics, University Duisburg-Essen

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1 Electronic Transport Peter Kratzer Faculty of Physics, University Duisburg-Essen

2 molecular electronics

3 = e2 n m Paul Drude ( ) molecular electronics

4 = e2 n m Paul Drude ( ) g = e2 h N ch molecular electronics

5 = e2 n m Paul Drude ( ) g = e2 h N ch thermoelectric transport: 3 transport coefficients hot κ σ α molecular electronics cold

6 magneto-thermopower = e2 n m MTJ 1 insulating heat sink 0 upper contacts lower contact T heat source Paul Drude ( ) g = e2 h N ch thermoelectric transport: 3 transport coefficients hot κ σ α molecular electronics cold

7 Theoretical approaches QM? scattering nonequilibrium? 1. Landauer-Büttiker approach coherent 2. Kubo-Greenwood approach coherent+ incoherent linear response 3. Boltzmann transport equation semiclassical incoherent

8 Linear response theory E hh~j(t)~j(0)ii Kubo-Greenwood approach: electrical current is understood as the (linear) response to a (small) electric field fluctuations of a system in equilibrium tell us about its response properties conductivity is related Zto the auto-correlation of the current at different times 1 1 Z µ =lim!!0 V! = dt e i!t d hhj µ (t i~ )j (0)ii 0 0 G(r, t; r 0,t 0 )=hr 0 e ih(t quantum-mechanical treatment: use propagator Z 1 1 Z µ =lim!!0 V! = dt e i!t d hhg (t i~, 0)j µ G(t i~, 0)j ii 0 0 t0 )/~ ri

9 Landauer-Büttiker approach transport through a bottleneck between two reservoirs µ L µ R scattering region (elastic, momentum relaxation but no energy relaxation) k in = e ikz + re ikz all dissipative processes take place in the reservoirs; yet the reservoirs remain fully in equilibrium (at chemical potentials µ and ) k out = te ikz L µ µ R L conductance g = e 2 /h (transmission probability) generalization to finite voltages possible

10 Boltzmann transport theory non-equilibrium distribution function of both positions and momenta f( r, k, t) = f( r vdt, k F / dt, t dt) + f t drift in r an k space + scattering f( r, k, t) t + v k f( r, k, t) r + F f( r, k, t) k coll = f t dt coll f(r, k, t) f k = f 0 ( (k)) + g k

11 linearized Boltzmann equation stationary and spatially homogeneous state drift of the distribution in k-space due to external forces is counteracted by relaxation due to scattering v k qe + (k) µ T T f 0 = f t scattering rate Pkk from quantum theory f t coll = k coll f k (1 f k )P k k (1 f k )f k P kk f t coll = k (g k g k )P kk if Pkk = Pk k inelastic elastic scattering

12 Boltzmann eq.: solutions(i) small deviations from equilibrium have the general form with (yet unknown) vector-valued function χ(ε) g k = q f 0 ( )v k solution for a given direction g k g k = g k v v useful for semiconductors and nanostructures! Solution has the form of a relaxation toward equilibrium f = g k v = g k P kk 1 t coll k v k with relaxation time determined from from the implicit equation 1 1 = k v P kk k v v = v A. Anselm, Introduction to Semiconductor Theory, MIR Publishers, 1981 k

13 Boltzmann eq.: solutions(ii) in bulk metals: anisotropy of transport is of interest! introduce vector mean free path Λk such that solve iteratively scattering-out term k = k v k + k P kk k g k = q f 0 k E scattering-in term relaxation time approximation k = k v k 1 k = P. T. Coleridge, J. Phys. F Met. Phys. 2, 1016 (1972) k P kk

14 Example: impurity scattering in dilute alloy direction-dependent relaxation time Cu(Ni) Cu(As) I. Mertig, Rep. Prog. Phys. 62, 237 (1999)

15 coupled charge and heat transport currents in the conserved quantities as response to generalized forces S I N L NN L NE S N = E = I E relation to conventional transport coefficients: = q2 T L l NN A = µ qt 1 qt L EN L NE L NN The matrix L of the Onsager coefficients is symmetric and positive definite Rate of entropy production L EE S E el = L EE L NE L EN L NN ds dt = ( S/ N, S/ E) L NN L NE S/ N L EN L EE S/ E T T 2 S N = µ T 1 l T 2 A 0

16 conventional re-formulation alternative definition of transport coefficients j j Q electromotive force ( electric field ) and temperature gradient as driving forces Note: Onsager relations now imply L21 = T L12 Measurable quantities: = L 11 = L 11 L 12 L 21 L 22 E T = L 12 L 11 el = L 22 L 21 L 12 L 11 N.W. Ashcroft & N.D. Mermin, Solid State Physics, Chapter 13

17 Transport coefficients moments of the distribution function are calculated from the solution of the Boltzmann equation L ( ) = d 3 k 4 3 f0 relation with the standard transport coefficients (k) µ k v k L 11 = q 2 L (0) ; L 21 = ql (1) ; L 22 = L (2) /T example: Seebeck coefficient and power factor for Bi2Te3 G.K.H. Madsen & D.J. Singh, Comp. Phys. Commun. 175, 67 (2006)

18 Thermoelectric transport in a Si nanowire image from: J. D. Weisse et al. NanoLett. 11, 1300 (2011)

19 Si nanowire infinitely extended in (001) direction square cross section, diameter 1.6 nm, atomistic model electronic structure from sp 3 tight-binding model* for Si with second-nearest neighbor hopping, plus screening of the 2NN matrix element by the first neighbor phonons from Brenner s force field model electron-phonon coupling determined by tight-binding Hamiltonian Si 170 H 56 supercell (10 repeats) *G. Grosso and C. Piermarocchi, Phys. Rev. B 51, (1995)

20 conduction bands phonon band structure q valence bands direct band gap 2.19 ev

21 microscopic expression for the scattering thermal phonon population (Bose-Einstein) crystal momentum conservation along (001) phonon eigenvector electronic wavefunctions

22 relaxation time Results of atomistic modeling: relaxation time increased in higher subbands strongly wave-vector dependent relaxation time carriers in higher bands may be long-lived

23 Transport properties non-monotonic behavior of carrier mobility and of the thermopower I. Bejenari and P. Kratzer, Phys. Rev. B 90, (2014)

24 Relation between Kubo- Greenwood and Landauer- Büttiker approaches

25 diagonal conductivity can be reduced to a Fermi-surface properties µµ = ~ 4 V Z +1 Tr j µ G + (") G (") j µ G + (") G (") resolvent operator spectral density G ± (r, r 0, ") = X a a(r 0 1 ) " Ĥ ± i a (r) G ± (r, r 0, ") =G + (") G (") = 2 i X a a(r 0 ) a (r) (" " F ) j I µ = 1 v at IJ µ E J layer-resolved conductivity (between layers I and J) IJ µµ = ~ 4 Z I d 3 r Z J d 3 r 0 Tr [j µ G(r I,r 0 J, " F ) j µ G(r 0 J,r I, " F )]

26 scattering theory at EF For small voltages, we may restrict ourselves to EF using the theorem of Baranger and Stone, the conductance, defined by I tot = Aj I z = gu U = E z d? can be expressed as g = ~ 2 g = e2 h Z X X k 0 in ds I Z ds J Tr j z G + (r I,r 0 J, " F ) j z G (r 0 J,r I, " F ) S fi = t fi s G + (r, r 0, ") =G (r 0,r,") = X k 0 in The carriers are scattered elastically by the interface/ the constriction, described by the unitary S-matrix q v f (k) v i (k 0 ) = A fi v f (k)v i (k 0 ) This proves Landauer s expression for the conductance k out T kk 0 T kk 0 = t kk 0 2 v f (k) v i (k 0 ) J X k out I z k 0 in(r 0 J) kout (r I )A kk 0

27 Implementation for a heterostructure For transmission through planar interfaces, k is preserved, and different k are treated as "transmission channels" T (E) = 1 A BZ Z ~ kk 2BZ d 2 k k T ( ~ k k,e) The transmission probability is calculated in a matrix representation in terms of atomic sites i,j Conductance Z g = e 2 /~ T ( ~ k k, ") = (@ E f 0 ) T (E)dE X i2i,j2j Tr h M i G ij( ~ k k, ")M j (G ij ) ( ~ i k k, ") E μ L μ R Seebeck coefficient S = 1 et R (@E f 0 ) T (E)(E µ)de R (@E f 0 ) T (E)dE T L T R U. Sivam & Y. Imry, PRB 33, 551 (1986)

28 Transport in DFT codes in codes with local-orbital basis: atomic orbitals define the left and right leads molecular electronics: self-energies in the leads replace the current operators (AITRANSS code module in FHI-aims) in plane-wave codes: extra step required solve Lippmann-Schwinger eq. for the potential of the scattering region (atomic projector-wavefunctions are used to enrich the basis set) H.J. Choi & J. Ihm, PRB 59, 2267 (1999) A. Smogunov, A. Dal Corso & E. Tosatti, PRB 70, (2004) in Green's-function based methods, e.g., the Korringa-Kohn- Rostoker (KKR) method: layer-resolved Greens functions available relativistic four-component generalization of current operators

29 Magnetotransport

magneto-thermopower (MTP) tunneling-mtp MTP in multilayers L. Gravier et al.

30 anisotropic magnetoresistance (MR) tunneling-mr giant MR in multilayer structures Phenomenology Electric transport driven by a) voltage b) thermal gradient M M g(t ) S(T ) j j FM anisotropic magneto-thermopower (MTP) tunneling-mtp MTP in multilayers L. Gravier et al. PRB 73, (2006) magnetization All anisotropic effects require the inclusion of spin-orbit coupling in the DFT calculation!

magnetic tunnel junctions Heusler alloy Co2MnSi as ferromagnetic electrode Co Mn 2 1 majority spin 2 1 minority spin schematic of an MTJ Si ferromagnet T C = 985 K energy (ev)

31 magnetic tunnel junctions Heusler alloy Co2MnSi as ferromagnetic electrode Co Mn 2 1 majority spin 2 1 minority spin schematic of an MTJ Si ferromagnet T C = 985 K energy (ev) half-metallic band gap -5-5 upper contact -6 Γ X W K L Γ -6 Γ X W K L Γ electric current ferromagnet insulator ferromagnet magnetization T lower contact

32 transmission probabilities dependence on the interface termination MnMn/O MnSi/O CoCo/O MgO transport properties in the majority spin channel transmission CoCo MnSi MnMn energy (ev) majority spin conductance (e 2 /h) CoCo/O MnSi/O MnMn/O temperature (K) Seebeck coefficient (µv/k) CoCo/O MnSi/O MnMn/O temperature (K) Co-terminated interfaces allow for sizeable thermopower read-out at room temperature B. Geisler & P. Kratzer, PRB 92, (2015)

33 anisotropy of the MTP Dirac device: single Co segment embedded between Cu electrodes Can we use the thermopower to read out the orientation of the magnetization? Influence of the number of atomic layers in the Co segment? j M M T ~Mk(001) ' T ~Mk(100) S = S ~Mk(001) S ~Mk(100) Z de (@ E f 0 ) T (E)(E µ)

34 transmission probability [T (001) - T (100) ] x100 Transmission probability M (001) 4 ML 6 ML 10 ML M (100) E-E F (ev) Anisotropy of the magneto-thermopower can be read from the asymmetry of ΔT(E) Quantum well states show up as resonances below EF Sign and magnitude of the thermopower are affected by the resonance position, and hence by M.

35 anisotropic MTP S (001) - S (100) (µv/k) Thermopower and magneto-thermopower ΔS(T) for 4ML 6ML 10 ML of Co Temperature (K) Bulk-like samples (10ML) show negligibly small anisotropy ΔS(T). Both the sign of S and the anisotropy are enhanced in thin films. Seebeck coefficient S(T) (µv/k) M (001) M (100) z M x j Cu Co Cu M C 4 C 2 j V. Popescu & P. Kratzer, Phys. Rev. B 88, (2013)

36 multilayer spin valves Co layers may be magnetized alternatingly, or all in the same direction in plane Can we distinguish between these two cases by measuring the magneto-retistivity? the magneto-thermopower in a thermal gradient? Co Co Co Cu Cu Cu Cu V Co Cu Co Cu V. Popescu & P. Kratzer, New J. Phys. 17, (2015) Co Cu

37 effect of layer thickness on anisotropic MR on magneto-thermopower P AP Conductance g(t=0) (e 2 /h) Thickness of Co layer (ML)

38 effect of layer thickness on anisotropic MR on magneto-thermopower 5.0 p=4 ML p=6 ML p=8 ML parallel p=4 ML p=6 ML p=8 ML antiparallel 5.0 S (µv/k) S (µv/k) Temperature (K) Temperature (K) -15.0

39 Role of Quantum well resonances energy dependence of transmission probability enhances sensitivity! S (µv/k) p=4 ML p=6 ML p=8 ML Transmission probability p=4 ML p=6 ML p=8 ML T (K) E-E F (ev)

40 Role of Quantum well resonances energy dependence of transmission probability enhances sensitivity! p=4 ML p=6 ML p=8 ML S (µv/k) P AP T (K) E-E F (ev)

41 Figure of merit: MTP ratio By using the optimum layer thickness for Co, the sensitivity of thermal read-out can be much larger than the conventional, magneto-resistive read-out. (S AP -S P )/S P (%) p=4 ML p=6 ML p=8 ML MTP = S P(T ) S AP (T ) S AP (T ) 100% 0 MC V. Popescu & P. Kratzer, New J. Phys. 17, (2015) Temperature (K)

42 Summary Boltzmann transport theory: Band energies and velocities can be taken directly from ab initio calculations, while scattering integrals require knowledge of the particular scattering mechanism. While coherent superposition of electronic wave function is dropped, the scattering can still be treated quantum-mechanically. Energy-dependence of the relaxation time and scattering-in effects may be sizable, in particular in low dimensions. Landauer-Buttiker theory becomes equivalent to Kubo-Greenwood at small deviations from equilibrium mixed approach, including (spin) current operators Magnetotransport: Anisotropy and spin-valve effect not only in the conductance, but also in the thermopower. Magneto-thermopower can deliver higher sensitivity for electrical read-out of magnetic memories since the energy dependence of transmission enters (in addition to its magnitude).

43 Acknowledgments Voicu Popescu funding: priority program SPP1538 Spincaloric Transport Benjamin Geisler Igor Bejenari

Condensed Matter Physics 2016 Lecture 13/12: Charge and heat transport.

Condensed Matter Physics 2016 Lecture 13/12: Charge and heat transport. 1. Theoretical tool: Boltzmann equation (review). 2. Electrical and thermal conductivity in metals. 3. Ballistic transport and conductance