Charge density waves in strong magnetic field: the phase diagram and the structure below Tc

1. 1 Charge density waves in strong magnetic field:the phase diagram and the structure below Tc P. D. Grigorev, D. S. Lyubshin 1NHMFL, Florida State University, Tallahassee, Florida 2D. Landau Institute for Theoretical Physics, Chernogolovka, Russia [cond-mat/0504779 (sent to Phys. Rev. B)] CDW in parallel magnetic field. Effect of Zeeman splitting on a phase diagram of CDW. 1. Mean field theory of CDW in magnetic field. 2. Phase diagram. 3. Properties of high-field CDW phases.

2. 2 Introduction to CDW (general). Nesting property of Fermi surface leads to the Peierls instability in the susceptibility (QN) and to a charge- or spin-density state. Fermi surface As result a gap in electron spectrum appears in CDW or SDW state and the metal becomes an insulator. Nesting vector QN For a review see, e.g., the book of Gruner “Charge density waves in solids”

3. 3 CDW and magnetic field (1) External magnetic field usually suppresses CDW If Tc>>H, magnetic field only leads to a quadratic decrease of Tc. [ Dieterich and P.Fulde, Z. Phys. 265, 239 (1973)] Recently a new material -(BEDT-TTF)2KHg(SCN)4 appeared with Tc≈8K where external magnetic field leads to a rather complicated phase diagram D. Graf, J. S. Brooks et al., PRB 69, 125113 (2004) (Per)2Au(mnt)2 , Tc ≈ 12K

4. 5 Pauli effect of magnetic field and the phase diagram of CDW Two spin sub-bands near Fermi surface E(k) Q Q EF k

5. 6 How to describe CDW below Tc ? All above results were obtained from the susceptibility calculation that is applicable only in normal phase or on the transition line Tc(H). Schematic picture of the phase diagram How Hk depends on T ? Does the high magnetic field CDW state have one or more harmonics? What is the structure of CDWx state ?

6. 7 Description of a quasi-1D metal (starting model) Dispersion relation of electrons in quasi-1D metals imperfect nesting term Hamiltonian where UCandUS are charge and spin coupling constants.

7. CDW-normal transition lines The normal-CDW transition line depends strongly both on the imperfect nesting term t’b and on the coupling constant ratio US/UC

8. 5 Pauli effect of magnetic field and the phase diagram of CDW Two spin sub-bands near Fermi surface E(k) Q Q EF k

9. 10 Cosine phase (picture) Charge modulations for two spin components: Spin and charge density modulations: Charge density Spin-up subband ++ - + x x - Spin density +- - Spin-down subband x x

10. 11 Double-cosine phase (picture) Spin up and down charge modulations: Spin and charge density modulations:

11. 8 Phase diagram of CDW in magnetic field =-US/UC=0.7 Phase diagram near the triple point at perfect nesting. Phase diagram in a whole region Double-cosine phase wins almost always CDW0 state is stable at field

12. Density of states in the CDW0, cosine and double-cosine phases Spin up DoS Spin down DoS

13. Previous theoretical models of high field CDWx phase. R. D. McDonald, N. Harrison, L. Balicas, K. H. Kim, J. Singleton, and X. Chi, Phys. Rev. Lett. 93, 076405 (2004)

14. 14 Spin asymmetry of the cosine phase The energy gaps + and-in the electron spectrum for two spin orientations are different. Any charge current in the cosine phase is accompanied by a spin current. This property can be used to 1) create a controllable spin current. 2) distinguish cosine and double-cosine phases

15. 15 Summary A mean field theory of charge density wave in high magnetic field is developed(without orbital effects, i.e. when magnetic field is parallel to the conducting layers or orbital effects are small). The properties of high-field phase and the phase diagram are studied. 1). The appearance of double-cosine has been proposed and proved. 2). Phase diagram of CDW in magnetic field has been calculated near the transition temperature (where the expansion in Q is valid). The cosine CDW state with shifted wave vector at perfect nesting appear only in a narrow region near the triple point. 3). The properties of high-field CDWx state are investigated.

16. 9 Critical fields on transition line as function of =Us/Uc

17. 4 CDW and magnetic field (2) Two spin subbands near Fermi surface • Zeeman splitting separates two optimal nesting vectors for two spin orientations. This leads to a shift of nesting vector in high field (analogue of paramagnetic limit in superconductors and FFLO state). Does not exist in SDW. • [R.H. McKenzie,cond-mat/9706235] E(k) Q Q EF     • Orbital effect of magnetic field. • [L.P. Gor’kov and A.G. Lebed, • J. Phys. (Paris) 45, L433 (1984)] • Leads to one-dimensionization of electron • spectrum [P.M.Chaikin, J. Phys. I France 6, • 1875 (1996)] and to the orbital quantization • in the case of imperfect nesting. [Figure from Montambaux et al., PRB 55, 207 (1985)]

18. Mean field approach We introduce the average (order parameter) where and the electron Green’s function Resulting Hamiltonian is quadratic in electron operators: Equation on Green’s function is

19. Cosine or double-cosine phase? Since CDWc and CDW2c phases have the same transition temperature, to determine which phase takes place we compare their free energies. To the second order inQ this rewrites: Q should be determined from the consistency equation for each phase.

20. Cosine phase

21. Cosine phase (2) Equation on Q Solution of this equation is Ratio of Q- and Q is

22. Double-cosine phase If one has Q1 and Q2 CDW wave vectors, one has a series  Q1 + n(Q1- Q2 ), of wave vectors. Equation on Green’s function is where We expand the solution in powers of Q up to the third term : The equation on Qbecomes :

23. Double-cosine phase (2) where The energy gap (order parameter)

24. Analogue in surface superconductivity Similar phase diagram appears in surface superconductor in parallel magnetic field O. V. Dimitrova and M. V. Feigel'man, Pis'ma v ZhETF78 (10), 637 (2003) Hamiltonian of this system is different from CDW:

25. Orbital quantization in CDW (continue) (dash lines) ky kx G Closed pocket of FS

26. Orbital effect of magnetic field. Calculation of magnetic field dependence of critical temperature at different values of t’ (imperfect nesting) [Zanchi et al., (1996)]. At strongly imperfect nesting an increase of magnetic field may lead to an increase of Tc and even to the appearance of Field-induced CDW (FICDW). Nesting becomes more perfect as magnetic field increases.

27. CDWx phase in high magnetic field Dependence of shift of CDW wave vector in high magnetic field. As magnetic field increases the CDW wave vector shifts from 2kF . This shift is due to Zeeman splitting. At H>>Tc, Result of susceptibility calculation near the transition temperature performed by Zanchi et al. in PRB 53, 1240 (1996)

28. Interplay between the effects of Zeeman splitting and orbital quantization. [PRB 68, 201101 (2003); Physica B 346-347, 368-372 (2004)] Qualitative explanation Series of phase transitions in tilted magnetic field (experiment) G

29. Orbital quantization in CDW Susceptibility as function of qx in magnetic field consists of a series of equidistant maxima. Susceptibility as function of qx , qy [G. Montambaux et al., PRL 55, 2078 (1985)]