Pavel D. Grigoriev L. D. Landau Institute for Theoretical Physics, Russia Consider very anisotropic metal, such that the interlayer tunneling time of electrons is longer than the in-layer mean scattering time and than the cyclotron period. Does the interlayer magnetoresistance has new qualitative features? Do we need new theory to describe this regime? [1] P. D. Grigoriev, Phys. Rev. B 83, (2011). [2] P. D. Grigoriev, JETP Lett. 94, 48 (2011) [arXiv: ]. The answer is yes, and we consider what are these differences.

Coherent and incoherent interlayer electron transport 2D electron gas The coherent interlayer tunneling conserves the in-plane electron momentum p ||. This gives the well-defined 3D electron dispersion ε(p)=ε  (p  ) +2t z cos(k z d) and Fermi surface as a warped cylinder. This also assumes t z  >>h, where  is the in-plane mean free time. Examples: most anisotropic metals. The incoherent interlayer electron tunneling does not conserve the in-plane momentum. The 3D electron dispersion and FS do not exist (only in-layer 2D). Examples: compounds with extremely small interlayer coupling, where the interlayer electron transport goes via local crystal defects or by the absorption of bosons. “Weakly coherent” interlayer magnetotransport: p  is conserved in interlayer tunneling, but the tunneling time is longer than the cyclotron and/or mean free times. The 3D FS and electron dispersion are smeared. Examples: all layered metals with small t z in strong magnetic field. Candidates: some organic metals, heterostructures, high-Tc cuprates. Are the standard formulas for magnetoresistance applicable in this case? Does this regime contains new physics? 3

Generally accepted opinion This conclusion is incorrect and was obtained because the authors have used oversimplified model for the electron interaction with impurities. P. Moses and R. H. McKenzie, Phys. Rev. B 60, 7998 (1999). They have used the following 2D electron Green’s function disorder wrong Introduction

Motivation 2 This question is rather general. The weakly coherent regime appears in very many layered compounds: high-Tc cuprates, pnictides, organic metals, heterostructures, etc. Magnetoresistance (MQO and AMRO) are used to measure the quasi-particle dispersion, FS, scattering,.. Experimentally observed transitions coherent – weakly coherent – strongly incoherent interlayer coupling show many new qualitative feature: monotonic growth of interlayer magnetoresistance, different amplitudes of MQO and angular dependence of magnetoresistance, etc.

Motivation (monotonic growth) Monotonic growth of interlayer magnetoresistance, observed in many layered compounds when magnetic field is  layers (parallel to electric current)  -(BEDT-TTF) 2 SF 5 CH 2 CF 2 SO 3 F. Zuo et al., PRB 60, 6296 (1999). W. Kang et al., PRB 80, (2009) 2a

The coherent regime of interlayer magnetotransport is well understood. For axially symmetric dispersion and in the first order in t z it simplifies to: [ R. Yagi et al., J. Phys. Soc. Jap. 59, 3069 (1990)] This gives angular magnetoresistance oscillations (AMRO): If the electron dispersion ε(p) is known, the background conductivity is given by the Shockley tube integral (solution of transport equation): Yamaji angles 4 Introduction

5 Harmonic expansion of Fermi momentum Harmonic expansion of the angular dependence of FS cross-section area (measured as the frequency of magnetic quantum oscillations): [First order: C. Bergemann et al., PRL 84, 2662 (2000); Adv. Phys. 52, 639 (2003). Second order relation between k  and A  : P.D. Grigoriev, PRB 81, (2010). ] One can derive the relation between the first coefficients k  and A  ! Introduction

The model of weakly coherent regime The Hamiltonian contains 3 terms: The 2D free electron Hamiltonian in magnetic field summed over all layers: the coherent electron tunneling between any two adjacent layers: and the point-like impurity potential: where 2D electron gas is the same as in the coherent regime, but the parameters and approach to the solution differ Model

Calculation of interlayer conductivity in the weakly incoherent regime 2D electron gas  zz The interlayer transfer integral t z <<  0 is the smallest parameter. We take it into account in the lowest order (after the magnetic field and impurity potential are included as accurately as possible). Interlayer conductivity is calculated as the tunneling between two adjacent layers using the Kubo formula: where the spectral function Approach to the solution of the problem 8 includes magnetic field and impurity scattering. The impurity distributions on two adjacent layers are uncorrelated, and the vertex corrections are small by the parameter t Z /E F, =>

The electron Green’s function in 2D layer with disorder in B z The point-like impurities are included in the “non-crossing” approximation, which gives: where Tsunea Ando, J. Phys. Soc. Jpn. 36, 1521 (1974). The density of states on each Landau level has the dome-like shape: Density of states E D(E) Bare LL Broadened LL Landau level width In strong magnetic field the effective electron level width is much larger than without field: 9

Monotonic part of conductivity for B || z The averaging over impurities on two adjacent layers is not correlated. For B = B Z we get In strong magnetic field we substitute the Green’s function from the non- crossing approx. and obtain the monotonic part of interlayer conductivity where and In weak magnetic field this gives 10 In the SC Born approximation

The shape of LLs is not as important as their width! The inclusion of diagrams with intersection of impurity lines in 2D electron layer with disorder only gives the tails of the DoS dome. The width of this dome remains unchanged and ~B z 1/2 : DoS E D(E) bare LL broadened LL DoS E D(E) bare LL broadened LL Therefore, we can take the DoS: The corresponding Green’s function is 20 The conductivity is not sensitive to the shape of LLs, but strongly depends on their width. where and  0 is the electron level width without magnetic field which gives

Comparison with experiments on interlayer MR R zz (B) ( magnetic field dependence: background and MQO) Sometimes, MR grows too strongly with increasing B z On experiments MR grows with B z even in the minima of MQO! 11 Theory on MR W. Kang et al., PRB 80, (2009) New Old F. Zuo et al., PRB 60, 6296 (1999).  -(BEDT-TTF) 2 SF 5 CH 2 CF 2 SO 3 MR growth appears also at large t z as in B PRL 89, (2002); Result 1

Physical reason for the decrease of interlayer conductivity in high magnetic field 12 BZBZ 1 2 The impurity distributions on adjacent layers are different. When an electron tunnels between two layers, its in-plane wave function does not change, but the energy shift due to impurities differs by the LL width  W  (  0  C ) 1/2 ~ B Z 1/2 Why  W ~ B Z 1/2 ? Because the area where  e  0, approximately, S ~1/B Z, and the number of effectively interacting with the electron impurities c i  SN i ~1/B Z, fluctuates as c i 1/2 ~ B Z -1/2, => the average shift of electron energy due to impurities W=SN i V 0 fluctuates as W/c i 1/2 ~(SN i ) 1/2 V 0 ~ B Z 1/2.

The same physical conclusion comes from more accurate averaging of electron Green’s function BZBZ 1 2 The impurity potential shifts the energy of each electron state, given by W=Re . This shift is random with the distribution The averaging of electron Green’s function over impurities must include this averaging over the energy shift, which increases the effective imaginary part of the electron self energy: The interlayer conductivity contains averaged electron Green’s functions 13

amplitude of MQO differs because the Dingle temperature increases with field Magnetic quantum oscillations of conductivity in the weakly incoherent regime 2323 where New result Old result Comparison of the results on R zz of standard theory (coherent regime) and new theory (weakly incoherent) : Result 2. MQO of interlayer conductivity are given by Dingle temperature background MR grows with B z

Calculation of the angular dependence of MR In tilted magnetic field the vector potential is, the electron wave functions on adjacent layers acquire the coordinate-dependent phase difference and the Green’s functions acquire the phase where the spectral function The expression for conductivity has the form: GRGAGRGA GRGRGRGR The impurity averaging on adjacent layers can be done independently: 21

Angular dependence of magnetoresistance in the weakly incoherent regime 22 Result 3. Angular dependence of interlayer conductivity is given by old expression: B=5T B=10T New result Old result *1/2 The difference comes from the high harmonic contributions and from the prefactor where but  depends on B z : and the prefactor acquires the angular dependence:  zz (  )

Comparison with experiment on angular oscillations of magnetoresistance (AMRO) “Clean” sample “Dirty” sample Theory (qualitative view): new result Old result Experiment: Appendix 1. M. Kartsovnik et al., PRB 79, (2009) 24 P. Moses and R.H. McKenzie, Phys. Rev. B 60, 7998 (1999).

Further work 1.The crossover 2D --> quasi-2D --> 3D ( t z ~  0 ) 2.The crossover weak --> strong magnetic field (  C ~  0 ). 3.Very high field, when the growth of R zz (B) is faster than ~B 1/2. 4.Change in angular dependence of harmonic amplitudes of MQO 5.Influence of chemical potential oscillations and electron reservoir. 6.Quasi-1D anisotropic metals. Above analysis is applicable to the high-field limit  C >  0, t z. There is still much work to do: 25

Summary 26 In the “weakly coherent” regime of interlayer conductivity, i.e. when the interlayer tunneling time is longer than the electron mean free time in the layers, the effect of impurities is much stronger and the Landau level width is much larger than in the standard 3D theory. This strongly changes the angular and field dependence of magnetoresustance: Thank you for attention! 1.The background interlayer MR grows ~B 1/2 with increasing field B|| . 2.The Dingle temperature grows ~B 1/2, which leads to the weaker increase of the amplitude of MQO with increasing B. 3.The angular dependence of MR changes: additional (cos  ) -1/2 factor appears and the maxima of AMRO are weaker. [1] P. D. Grigoriev, Phys. Rev. B 83, (2011). [2] P. D. Grigoriev, JETP Lett. 94, 48 (2011) [arXiv: ].

Strongly incoherent interlayer magnetotransport is very model-dependent Usually, the conductivity in this regime has non-metallic exponential temperature dependence (thermal activation or Mott-type). It has weak angular dependence of background magnetoresistance (contrary to the coherent case) [ A. A. Abrikosov, Physica C , 154 (1999); U. Lundin and R. H. McKenzie, PRB 68, (R) (2003); A. F. Ho and A. J. Schofield, PRB 71, (2005); V. M. Gvozdikov, PRB 76, (2007); D. B. Gutman and D. L. Maslov, PRL 99, (2007) ; PRB 77, (2008); etc.] Exception gives the following model [PRB 79, (2009)]: E0E0 12 The interlayer transport goes via local hopping centers (resonance impurities). Resistance contains 2 in-series elements: The hopping-center resistance R hc is almost independent of magnetic field and has nonmetallic temperature dependence. The in-plane resistance R || between nearest hopping centers depends on  magnetic field and has the metallic temperature dependence. 6