Magnetoresistance in strongly anisotropic layered metals in “weakly in coherent” regime

1. Pavel D. Grigoriev L. D. Landau Institute for Theoretical Physics, Russia Magnetoresistance in strongly anisotropic layered metals in “weakly incoherent” regime 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? The answer is yes, and we consider what are these differences. [1] P. D. Grigoriev, Phys. Rev. B 83, 245129 (2011). [2] P. D. Grigoriev, JETP Lett. 94, 48 (2011) [arXiv:1104.5122].

2. 3 The coherent interlayer tunneling conserves the in-plane electron momentum p||.This gives the well-defined 3D electron dispersion ε(p)=ε (p) +2tz cos(kzd) and Fermi surface as a warped cylinder.This also assumes tz>>h, where  is the in-plane mean free time. Examples: most anisotropic metals. 2D electron gas 2D electron gas Coherent and incoherent interlayer electron transport 2D electron gas 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 tzin 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. Introduction Generally accepted opinion P. Moses and R. H. McKenzie, Phys. Rev. B 60, 7998 (1999). This conclusion is incorrect and was obtained because the authors have used oversimplified model for the electron interaction with impurities. They have used the following 2D electron Green’s function disorder wrong

4. 2 Motivation 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.

5. 2a Motivation (monotonic growth) Monotonic growth of interlayer magnetoresistance, observed in many layered compounds when magnetic field is  layers (parallel to electric current) -(BEDT-TTF)2SF5CH2CF2SO3 F. Zuo et al., PRB 60, 6296 (1999). W. Kang et al., PRB 80, 155102 (2009)

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

7. Introduction 5 Harmonic expansion for the angle-dependence of FS cross-section area (MQO frequency) in Q2D layered metals. Harmonic expansion of Fermi momentum Harmonic expansion of the angular dependence of FS cross-section area (measured as the frequency of magnetic quantum oscillations): One can derive the relation between the first coefficients kmnand Amn ! [First order: C. Bergemann et al., PRL 84, 2662 (2000); Adv. Phys. 52, 639 (2003). Second order relation between kmnandAmn: P.D. Grigoriev, PRB 81, 205122 (2010).]

8. Model 7 The model of weakly coherent regime is the same as in the coherent regime, but the parameters and approach to the solution differ. The Hamiltonian contains 3 terms: 2D electron gas 1 32 2D electron gas The 2D free electron Hamiltonian in magnetic field summed over all layers: 2D electron gas the coherent electron tunneling between any two adjacent layers: and the point-like impurity potential: where

9. Approach to the solution of the problem 8 Calculation of interlayer conductivity in the weakly incoherent regime The interlayer transfer integral tz<<0is 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: 2D electron gas 2D electron gas szz 2D electron gas where the spectral function includes magnetic field and impurity scattering. The impurity distributions on two adjacent layers are uncorrelated, and the vertex corrections are small by the parameter tZ/EF, =>

10. 9 The electron Green’s function in 2D layer with disorder in Bz 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 D(E) E Landau level width Bare LL Broadened LL In strong magnetic field the effective electron level width is much larger than without field:

11. 10 Monotonic part of conductivity for B || z The averaging over impurities on two adjacent layers is not correlated. For B = BZ we get In weak magnetic field this gives In strong magnetic field we substitute the Green’s function from the non-crossing approx. and obtain the monotonic part of interlayer conductivity and where In the SC Born approximation

12. 20 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 ~Bz1/2: DoS DoS D(E) D(E) E E bare LL bare LL broadened LL broadened LL The conductivity is not sensitive to the shape of LLs, but strongly depends on their width. Therefore, we can take the DoS: and 0 is the electron level width without magnetic field where The corresponding Green’s function is which gives

13. Result 1 11 Comparison with experiments on interlayer MR Rzz(B)(magnetic field dependence: background and MQO) Theory on MR New Old On experiments MR grows with Bz even in the minima of MQO! W. Kang et al., PRB 80, 155102 (2009) Sometimes, MR grows too strongly with increasing Bz MR growth appears also at large tz as in F. Zuo et al., PRB 60, 6296 (1999). -(BEDT-TTF)2SF5CH2CF2SO3 B PRL 89, 126802 (2002);

14. 12 Physical reason for the decrease of interlayer conductivity in high magnetic field 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 ~ BZ1/2 BZ 1 2 Why W ~ BZ1/2? Because the area where e0, approximately, S~1/BZ, and the number of effectively interacting with the electron impurities ci SNi ~1/BZ , fluctuates as ci1/2~ BZ-1/2, => the average shift of electron energy due to impurities W=SNiV0 fluctuates as W/ci1/2 ~(SNi)1/2V0~ BZ1/2 .

15. 13 The same physical conclusion comes from more accurate averaging of electron Green’s function The impurity potential shifts the energy of each electron state, given by W=Re . This shift is random with the distribution BZ 1 2 The interlayer conductivity contains averaged electron Green’s functions 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:

16. 23 Result 2. Magnetic quantum oscillations of conductivity in the weakly incoherent regime MQO of interlayer conductivity are given by Dingle temperature background MR where Comparison of the results on Rzz of standard theory (coherent regime) and new theory (weakly incoherent) : amplitude of MQO differs because the Dingle temperature increases with field background MR grows with Bz New result Old result

17. 21 Calculation of the angular dependence of MR The impurity averaging on adjacent layers can be done independently: where the spectral function 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 The expression for conductivity has the form: GRGA GRGR

18. 22 Result 3. Angular dependence of magnetoresistance in the weakly incoherent regime Angular dependence of interlayer conductivity is given by old expression: but depends on Bz: where and the prefactor acquires the angular dependence: szz(q) Old result *1/2 B=10T B=5T New result The difference comes from the high harmonic contributions and from the prefactor

19. 24 “Dirty” sample “Clean” sample Appendix 1. Comparison with experiment on angular oscillations of magnetoresistance (AMRO) Experiment: Theory (qualitative view): new result Old result P. Moses and R.H. McKenzie, Phys. Rev. B 60, 7998 (1999). M. Kartsovnik et al., PRB 79, 165120 (2009)

20. 25 Further work Above analysis is applicable to the high-field limit C>0, tz. There is still much work to do: • The crossover 2D --> quasi-2D --> 3D (tz ~ 0) • The crossover weak --> strong magnetic field (C ~ 0). • Very high field, when the growth of Rzz(B) is faster than ~B1/2 . • Change in angular dependence of harmonic amplitudes of MQO • Influence of chemical potential oscillations and electron reservoir. • Quasi-1D anisotropic metals.

21. 26 Summary 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: The background interlayer MR grows ~B1/2 with increasing field B||. The Dingle temperature grows ~B1/2 , which leads to the weaker increase of the amplitude of MQO with increasing B. 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, 245129 (2011). [2] P. D. Grigoriev, JETP Lett. 94, 48 (2011) [arXiv:1104.5122]. Thank you for attention!

22. 6 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 317-318, 154 (1999); U. Lundin and R. H. McKenzie, PRB 68, 081101(R) (2003); A. F. Ho and A. J. Schofield, PRB 71, 045101(2005); V. M. Gvozdikov, PRB 76, 235125 (2007); D. B. Gutman and D. L. Maslov, PRL 99, 196602 (2007) ; PRB 77, 035115 (2008); etc.] Exception gives the following model [PRB 79, 165120 (2009)]: The interlayer transport goes via local hopping centers (resonance impurities). Resistance contains 2 in-series elements: E0 1 2 The hopping-center resistance Rhc 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.