Angle dependence of interlayer magnetoresistance in strongly - PowerPoint PPT Presentation

Angle dependence of interlayer magnetoresistance in strongly anisotropic quasi-2D layered metals Pavel D. Grigoriev 1 , Taras I. Mogilyuk 2 1 L. D. Landau Institute for Theoretical Physics 2 NRC Kurchatov Institute Main question: How the angular dependence of interlayer MR depends on the shape of Landau levels? Are the standard formulas of angular dependence of magnetoresistance applicable in strongly anisotropic quasi-2D layered metals? Publication: arXiv:1309.3161 (Phys Rev B in press)

2 Introduction Layered quasi-2D metals (Examples: heterostructures, organic metals, all high-T c superconductors) θ Electron wave functions overlap leads to 2D electron gas the finite interlayer transfer integral t z Magnetic Electron dispersion in the tight-binding field B 2D electron gas approximation is highly anisotropic: σ zz ε( p )=ε ll ( p ll )+2t z cos( p z d /ℏ), 2D electron gas (coherent-tunneling, t z <<E F conserving p || ) ε ll ( p ll )= p 2 ll /2m ll FS B Fermi surface in layered Q2D Landau levels metals is a warped cylinder. Extremal The size of warping W=4 t z ~ cross ћω c sections Two close frequencies => beats of MQO

3 Introduction Angle-dependent magnetoresistance oscillations (AMRO) in quasi-2D metals. Fermi surface First theory: First observation: K.J. Yamaji, M.V. Kartsovnik, P. A. LLs Phys. Soc. Kononovich, V. N. Jpn. 58, 1520, Laukhin, I. F. Schegolev, ( 1989). JETP Lett. 48 , 541 (1988). B 3D =e 2 τ ∑ 2 , v z =∂ ε /∂ p z σ zz v z FS For axially symmetric dispersion and in the first AMRO order in t z the Shockley tube integral gives: [ R. Yagi et al., J. Phys. Soc. Jap. 59 , 3069 (1990)] gives AMRO gives damping of AMRO by disorder Yamaji angles

Introduction 4 Theory of magnetoresistance is not as simple While the thermodynamic potential is a linear functional of the DoS, conductivity has nonlinear dependence on DoS. Usually one separates monotonic (background) classical MR and MQO: σ ( B ) =σ cl ( B ) +~ σ ( B ) . In quasi-2D metals the oscillations (MQO) are strong, and such separation in the theory is incorrect even if MQO are smeared by T or long-range disorder. Accurate calculation of longitudinal MR in the presence of MQO explains the monotonic growth of MR even in the one-particle approximation (electrons + B + disorder). 4

5 Motivation Motivation Layered compounds are very common: high-Tc cuprates, pnictides, organic metals, intercalated graphites, heterostructures, etc. Magnetoresistance (MQO and AMRO) is used to measure the quasi- particle dispersion, Fermi surface, effective mass, mean scattering time.. It is an important complementary tool to ARPES. Experimentally observed dimensional crossover: 3D -> quasi-2D -> 2D shows many new qualitative features: (1)monotonic growth of R zz (B z ), (2)different damping of MQO, (3)different angular dependence of MR

The two-layer tunneling Model 6 model The Hamiltonian contains 3 terms: Magnetic 2D electron gas field B 1 3 2 2D electron gas 1. The 2D free electron Hamiltonian in σ zz magnetic field summed over all layers: 2D electron gas 3. The coherent electron tunneling between any two adjacent layers: 2. The short-range where impurity potential:

7 Impurity averaging Contain extra σ zz i±1 2D electron gas power of t Z / E F x 2D electron gas x t z t z x 2D electron gas i The impurity distributions on two adjacent layers are uncorrelated, and the vertex corrections are small by the parameter t Z / E F => 4 e 2 t z 2 d L x L y ∫ d 2 rd 2 r' ∫ dε ′ ( ε ) ] ⟨ Im G ( r, r ', j, ε )⟩ ⟨ Im G ( r ', r, j+ 1, ε )⟩ . 2π [ − n F σ zz = Vertex corrections can be ignored The calculation of interlayer conductivity reduces to 2D electron Green’s function

8 Calculation of the angular dependence of MR [started in P. D. Grigoriev, Phys. Rev. B 83, 245129 (2011) ]. The impurity averaging on adjacent layers can be done independently: 2 d σ zz = e 2 t z L x L y ∫ d 2 rd 2 r' ∫ dε 2π ⟨ A ( r,r',j, ε )⟩ ⟨ A ( r',r,j+1,ε ) ⟩ [ − n' F ( ε ) ] , A ( r, r ', j, ε ) =i [ G A ( r, r ', j, ε )− G R ( r, r ', j, ε ) ] . where the spectral function B= ( B x , 0, B z ) = ( B sin θ, 0, B cos θ ) Λ ( r ) =− yB x d= − yBd sin θ, In tilted magnetic field A= ( 0, xB z − zB x , 0 ) the vector potential is , the electron wave functions on adjacent layers acquire the coordinate-dependent phase difference and the Green’s functions G R ( r, r', j+ 1, ε ) =G R ( r, r', j, ε ) exp { ie [ Λ ( r ) − Λ ( r' ) ] } , acquire the phase 2π d 2 r [ − n' F ( ε ) ] [ G 2 ( r,ε ) cos ( h / 2π sin θ ) ] ] . h / 2π sin θ ) − Re [ G R 2 e 2 t z 2 d 2 ( r,ε ) exp ( ∫∫ dε eByd ieByd σ zz = ℏ The expression for conductivity New term! G R G R G R G A has the form: 8

AMRO for arbitrary shape of LL where for

Model 3: 10 The 2D electron Green’s function with disorder in B z calculation The point-like impurities are included in the “non-crossing” approximation, which gives: 0* ( r 2 ) Ψ n,k' y G ( r 1 ,r 2 ,ε )= ∑ 0 ( r 1 ) G ( ε, n ) , Ψ n, k y n,k y ,k' y where, if Landau levels do not overlap, Tsunea Ando, J. Phys. E+E g ( 1 − c i ) ± √ ( E − E 1 ) ( E − E 2 ) Soc. Jpn. 36, 1521 (1974) G R ( E, n ) = , 2 E E g 2 ∝ B, 2 N i =N i / N LL . 2 , E 2 =E g ( √ c i + 1 ) 2 , E 1 =E g ( √ c i − 1 ) c i = 2 π l Hz E g =V 0 / 2 π l Hz The density of states on each LL Density of states D(E) has a dome-like shape: = √ ( E − E 1 ) ( E 2 − E ) c i > 1 D ( E ) =− Im G R ( E ) , π 2 π | E | E g E 2 N i =N i / N LL . Bare LL c i = 2 π l Hz Broadened LL LL width = √ Γ B 4 ω c In strong magnetic field the effective level >>1 width is much larger than without field: Γ 0 π Γ 0

11 Why non-crossing approximation is applicable in 2D? Four-site approximation Tsunea Ando, J. Phys. Soc. Japan 37, 622 (1974)] Diagrams with intersections of impurity lines:

Density of states Noncrossing approximation: Lorentz shape of LL: Gauss shape of LL: AMRO for various shapes of LL AMRO in Lorentz case: AMRO in Gauss case: for AMRO in dome case: for

The normalized density of states at for various LL shapes. ℏ ω c / Γ= 10 Lorentzian (solid green curve), dome-like (dashed red curve), and Gaussian (dotted blue curve). The dash-dotted purple curve gives the normalized Density of states for Lorentzian LL shape at twice larger value . ℏ ω c / Γ= 20.

Comparison of AMRO for various LL shapes Lorentz shape Gaussian shape The angular dependence of normalized interlayer conductivity for Lorentzian LL ω c τ 0 = 10 With four different values (thin solid green curve), 10/3 (dashed red curve), 5/3 (dotted blue curve), and 1 (dash-dotted purple curve). The other parameters k F d= 3, μ= 605 K,T= 3 K,B 0 ≈ 11.6 T, m * =m e which for cyclotron mass and for θ= 0 corresponds to . ℏ ω c = 10 K

Gaussian shape Dome-shaped AMRO for the dome-shaped and gauss density of states of LL, corresponding to the self-consistent Born approximation.

Density of states Noncrossing approximation: Lorentz shape of LL: Gauss shape of LL: AMRO for various shapes of LL AMRO in Lorentz case: AMRO in Gauss case: for AMRO in dome case: for

SPIN CURRENT AND THE INFLUENCE OF ZEEMAN SPLITTING ON AMRO. SPIN CURRENT AND THE INFLUENCE OF ZEEMAN SPLITTING ON AMRO. The monotonic part of the spin-current conductivity, determined as the difference between the monotonic parts of conductivities between with spin up and down σ zz ( μ+gμ B B ) −¯ σ zz ( μ ) s zz ≡¯ σ zz ↑ −¯ σ zz ↓ =¯

Lorentz case Gauss case The angular dependence of the monotonic part of spin-current conductivity 0 ω c τ 0 = 10 for four different values of s zz / σ zz (solid green line), 2.0 (dashed red line), 1.0 (dotted blue line) and 0.5 (dash-dotted purple curve).

19 Conclusions 1: In the present theoretical study we take MQO into account and consider their influence on the angular dependence of interlayer conductivity. 2: The angular dependence of MR depends on LL shape. For Gaussian LL shape AMRO are much stronger. Angular dependence of magnetoresistance is calculated. 3: We also estimated the spin current, which appears because of AMRO. For typical parameters of the organic ∼ metal in the field B 10T the spin current is about 2% of the zero-field charge current, but it may almost reach the charge current for special tilt angles of magnetic field. The angular oscillations of the spin current are stronger and ∼ shifted by the phase π/2 as compared to the usual charge- current AMRO.

Thank you for your attention !

ARPES data and Fermi-surface 22 Motivation shape The Fermi surface of near optimally doped (a) integrated intensity map (10-meV window centered at EF ) for Bi2212 at 300 K obtained with 21.2-eV photons (HeI line); (b),(c) superposition of the main Fermi surface (thick lines) and of its (p,p) translation (thin dashed lines) due to backfolded shadow bands; (d) Fermi surface calculated by Massidda et al. (1988). Drawback 2: Ambiguous interpretation.