# Quantum Limit in a Parallel Magnetic Field in Layered Conductors

###### Abstract

We show that electron wave functions
in a quasi-two-dimensional conductor
in a parallel magnetic field are always
localized on conducting
layers.
Wave functions and electron spectrum in a quantum limit,
where the ”sizes” of quasi-classical electron orbits are of
the order of nano-scale distances between the layers,
are determined.
AC infrared measurements to investigate Fermi surfaces
and to test Fermi liquid theory in Q2D organic and high-T
materials in high magnetic
fields, ,
are suggested.

PACS numbers: 74.70.Kn, 73.43.-f, 75.30.Fv

^{†}

^{†}preprint: Lebed-Bagmet

Layered quasi-one-dimensional (Q1D) and quasi-two-dimensional (Q2D) organic conductors exhibit unique magnetic properties [1-6]. Recently [4], it has been understood that most of them can be explained in terms of effective space dimensionality crossovers for electron wave functions in a magnetic field. In their simplest forms, dimensional crossovers were suggested to explain field-induced spin-density wave phases [5-8,1] and to predict reentrant superconductivity (RS) phenomenon [8-10] .

In particular, in a Q2D conductor with electron spectrum,

(1) |

in a parallel magnetic field,

(2) |

quasi-classical electron trajectories, determined by the equations of motion,

(3) |

become periodic and restricted along -axis:

(4) |

Periodic along -axis electron trajectories (4) correspond to ”two-dimensionalization” of electron wave functions [9]. It is important that quasi-classical (QC) dimensional crossovers happen at weak magnetic fields [5,9], where the ”sizes” of electron orbits (4) are much larger than the inter-plane distances, . For instance, these QC crossovers are responsible for novel type of cyclotron resonance (CR) on open orbits [11-15]. In particular, Kovalev et al. [15] have suggested a new method to investigate Q2D Fermi surfaces (FS) by means of CR [11-14] and studied FS in organic conductor -(ET)I. For theoretical justification of the method, they used QC kinetic equation, which is appropriate under experimental conditions [15], where at .

Meanwhile, in high experimental fields, , typical ”sizes” of electron orbits (4) become less than inter-layer distances [8],

(5) |

in a number of Q2D organic and high-T materials. Under condition (5) [which we call quantum limit (QL)], theoretical methods used so far [5-15] are not justified. On the other hand, it is known that existence or not of Q2D Fermi surfaces is one of the main problems in the area of high-T and organic materials [16]. In this context, it is important to suggest quantum mechanical variant of Kovalev et al. method [15] to investigate Q2D FS in high fields (i.e., in QL case (5)), where the method is less sensitive to impurities existing in doped high-T materials.

The main goal of our Letter is to determine electron spectrum and wave functions in a Q2D conductor (1) in a parallel magnetic field (2). We show that, in contrast to the extended Bloch waves [17,18], all wave functions are localized on conducting planes and are characterized by some quantum numbers at . Quantization law, obtained in this Letter, is qualitatively different from well-known Landau level quantization [17,18] in a perpendicular magnetic field. As a result ac infrared properties are shown to be unusual. In particular, we use our common results to extend QC method [15] to study Q2D FS to QL case (5). We hope that this allows to test the existence of FS in numerous Q2D organic and high-T compounds.

To determine electron wave functions in Q2D conductor (1) in parallel magnetic field (2), we make use of QC description of electron motion within conducting -planes and solve fully quantum mechanical problem for electron motion between the planes. After QC Peierls substitutions for in-plane momenta, [17], one can represent electron Hamiltonian in the form: ,

(6) |

where the last term introduces potential energy of crystalline lattice along -axis; ; is Dirac delta-function. Note that Hamiltonian (6) is exact one for an isotropic Q2D case. As it follows from general theory [17], the above mentioned method disregards only corrections of the order of to electron energy for arbitrary function . It is seen from Eqs. (4),(5) that QL condition corresponds to , and, thus, these corrections are of the order of , where is a characteristic energy scale in a magnetic field. Therefore, Hamiltonian (6) allows to study both QC and QL (5) dimensional crossovers.

Arbitrary solution of the Schrodinger equation for Hamiltonian (6) can be written as

(7) |

which corresponds to free electron motion within -planes. After substitution of Eq. (7) into Hamiltonian (6), it can be rewritten as follows:

(8) |

By expanding in-plane energy in powers of , it is easy to make sure that Schrodinger equation for Hamiltonian (8) with the same accuracy can be expressed as:

(9) |

It is possible to prove [19] that, if one uses tight binding approximation for solutions of Eq.(9),

(10) |

[where is wave function of individual m-th layer at , corresponding to energy , ], then one disregards only corrections of the order of to electron energy.

Therefore, equation

(11) |

which can be derived after substitution of wave-functions (10) into Hamiltonian (9), has the same accuracy as Hamiltonian (6) and, thus, can be used to describe QL dimensional crossovers (5). At given in-plane momenta and , Eq.(11) is equivalent to the so-called Stark-Wannier ladder equation in electric field [20]. Using Ref.[20], one can express wave functions and energy levels in the following way:

(12) |

where is Bessel function of -th order [21]. [An important difference between wave functions and energy spectrum (12) and that in Ref. [20] is that the envelope functions, , and energy levels, , in Eq.(12) depend on and .]

Eq. (12) represents the main result of our Letter. In contrast to textbook extended Bloch waves with complex envelope, [17], the envelope functions in Eq. (12) are real functions localized on the N-th conducting layer (see Fig.1). Therefore, one concludes that, in a parallel magnetic field, all wave functions are localized on layers with energy gap between two neighboring wave functions being . Eq.(12) is valid both in QC and QL cases.

Below, we show that quantization law (12) leads to unusual ac infrared properties and suggest a method to investigate Q2D FS. For these purposes, we calculate ac conductivity component, perpendicular to conducting layers, , using known wave functions and energy spectrum (12). Let us first find matrix elements of momentum operator, , responsible for interactions between electrons and electric field, . It is possible to make sure that the matrix elements are non-zero only for wave functions with the same in-plane momenta, and , and energies :

(13) |

[In other words, only optical transitions between electrons with the same in-plane momenta localized on neighboring conducting layers are allowed].

To calculate , we make use of the following extension [22] of Kubo formalism:

(14) |

where is Fermi-Dirac distribution function, is a volume. After substituting matrix elements (13) and energy spectrum (12) in Eq.(14) and straightforward calculations, one obtains:

(15) |

[Integration in Eq. (15) is made along 2D contour ; ; we use the approximation since ].

It is convenient to write explicitly real and imaginary parts of conductivity (15):

(16) |

(17) |

where is the maximum value of energy gap, , on the contour of integration (see Fig.2); integral in Eq.(17) is determined as its principle value.

The main difference between Eqs.(16),(17) and the results of Ref.[17] is that Eqs.(16),(17) are valid both in QC and QL (5) cases, whereas the results [17] are essentially QC. Another difference is that Eqs.(16),(17) describe ”optical” conductivity (i.e., conductivity in the absence of impurities), in contrast to kinetic equation result [17]. From Eqs.(16),(17), it follows that ac properties in a parallel magnetic field are unusual. Indeed, integration of -function in Eq.(16) results in non-zero value of real part of conductivity for ac frequencies (see Fig.2). Therefore, electrons absorb electromagnetic waves at (in the absence of impurities!), in contrast to text book properties of metals [18].

Let us demonstrate that real part of conductivity (16) diverges at resonant frequency,

(18) |

Indeed, in the vicinity of its maximum with being momentum component perpendicular to at point, where takes its maximum (see Fig.2). In this case, integral (16) can be estimated as

(19) |

Therefore, by measuring at different directions of the field one can determine angular dependence of (see Ref.[15] and Fig.(2)). We stress, however, that the physical meaning of resonant frequency (18) at high magnetic fields (5), where electrons are almost completely localized on conducting layers (see Figs.1,2), is completely different from its kinetic equation interpretation [15,23].

To summarize, wave functions and electron spectrum of a Q2D conductor in a parallel magnetic are determined. A method to test FL picture in Q2D organic and high-T materials is suggested. We hope that this method is a useful experimental tool to study Fermi-liquid versus non Fermi-liquid behavior in low-dimensional compounds, especially as there have been claimed inconsistencies [24] between angular resolved photo-emission methods [16] and magneto-optical measurements in a perpendicular magnetic field [24]. We also think that QL dimensional crossovers and quantization law (12), suggested in the Letter, will be useful for studies of RS superconductivity [8-10] and for explanations of still unexplained phenomena observed in high parallel magnetic fields [25,26].

One of us (AGL) is thankful to E.V. Brusse and P.M. Chaikin for numerous and useful discussions.

## References

- (1) T. Ishiguro, K. Yamaji, and G. Saito, Organic Superconductors (Second Edition, Springer-Verlag, Heidelberg, 1998).
- (2) See review articles in J. Phys. I (France) 6 (1996) and references therein.
- (3) See review S.E. Brown, M.J. Naughton, I.J. Lee, E.I. Chashechkina, and P.M. Chaikin in More is Different (Fifty Years of Condensed Matter Physics), N.P. Ong and R.N. Bhatt Eds. (Princeton Series in Physics, Princeton, 2001) and references therein.
- (4) See recent Letters A.G. Lebed and M.J. Naughton, Phys. Rev. Lett. 91, 187003 (2003); A.G. Lebed, N.N. Bagmet, and M.J. Naughton, Phys. Rev. Lett. 93, 157006 (2004), and references therein.
- (5) L.P. Gor’kov and A.G. Lebed, J. Phys. Lett. (France) 45, L-433 (1984); A.G. Lebed, Phys. Rev. Lett. 88, 177001 (2002).
- (6) P.M. Chaikin, Phys. Rev. B 31, 4770 (1985).
- (7) M. Heritier, G. Montambaux, and P. Lederer, J. Phys. Lett. (France) 45, L-943 (1984).
- (8) A.G. Lebed, Pis’ma Zh. Eksp. Teor. Fiz. 44, 89 (1986) [JETP Lett. 44, 114 (1986)].
- (9) A.G. Lebed and K. Yamaji, Phys. Rev. Lett. 80, 2697 (1998).
- (10) N. Dupuis, G. Montambaux, and C.A.R. Sa de Melo, Phys. Rev. Lett., 70, 2613 (1993).
- (11) T. Osada, S. Kagoshima, and N. Miura, Phys. Rev. B 46, 1812 (1992).
- (12) L.P. Gor’kov and A.G. Lebed, Phys. Rev. Lett. 71, 3874 (1993).
- (13) A. Ardavan, J.M. Schrama, S.J. Blundell, J. Singleton, W. Hayes, M. Kurmoo, P. Day, and P. Goy, Phys. Rev. Lett. 81, 713 (1998).
- (14) A. Ardavan, S.J. Blundell, and J. Singleton, Phys. Rev. B 60, 15500 (1999).
- (15) A.E. Kovalev, S. Hill, K. Kawano, M. Tamura, T. Naito, and H. Kobayashi, Phys. Rev. Lett. 91, 216402 (2003).
- (16) See, for example, H. Dong, M.R. Norman, T. Yokoya, T. Takeuchi, M. Randeira, J.C. Campuzano, T. Takahashi, T. Mochiku, and K. Kadowaki, Phys. Rev. Lett. 78, 2628 (1997).
- (17) E. M. Lifshitz and L. P. Pitaevskii , Statistical Physics, Part 2 (Butterworth-Heinemann, 1980).
- (18) A.A. Abrikosov, Fundamentals of Theory of Metals (Elsevier Science Publisher B.V., Amsterdam, 1988).
- (19) A.G. Lebed, unpublished.
- (20) E.E. Mendez and G. Bastard, Physics Today June 1993, 34.
- (21) I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (5-th Edition, Academic Press, Inc., London, 1994).
- (22) See, for example, G. Grosso and G.P. Parravicini, Solid State Physics (Academic Press, New York, 2000).
- (23) A.G. Lebed and N.N. Bagmet, Phys. Rev. B 55, R8654 (1997).
- (24) L.B. Rigal, D.C. Schmadel, H.D. Drew, B. Maiorov, E. Osquiguil, J.S. Preston, R. Hughes, and G.D. Gu, Phys. Rev. Lett. 93, 137002 (2004).
- (25) N.E. Hassey, M. Kibune, H. Nakagawa, N. Miura, Y. Iye, H. Takagi, S. Adachi, and K. Tanabe, Phys. Rev. Lett. 80, 2909 (1998).
- (26) N.E. Hassey, M.N. McBrien, L. Balikas, J.S. Brooks, S. Horii, and H. Ikuta, Phys. Rev. Lett. 89, 086601 (2002).