arXiv:cond-mat/9607199 v1 27 Jul 1996
To be published inSynthetic Metals,
Proceedings of the International Conference on Synthetic Metals, Snowbird, UT, July 28 – August 2, 1996 E-print cond-mat/9607199, July 27, 1996
Temperature Evolution of the Quantum Hall Effect in Quasi-One-Dimensional Organic Conductors
Hsi-Sheng Goan and Victor M. Yakovenko
Department of Physics and Center for Superconductivity Research, University of Maryland, College Park, MD 20742, USA
Abstract
The Hall conductivity in the magnetic-field-induced spin-density-wave (FISDW) state of the quasi-one- dimensional organic conductors (TMTSF)2X at a finite temperature is calculated. The temperature dependence of the Hall conductivity is found to be the same as the temperature dependence of the Fr¨ohlich current of a reg- ular charge/spin-density wave. Predicted dependence σxy(T ) can be verified experimentally in the (TMTSF)2X compounds if all components of the resistivity tensor are measured and the conductivity tensor is reconstructed.
Keywords: Many-body and quasiparticle theories; Transport measurements, conductivity, Hall effect, magneto- transport; Magnetic phase transitions; Organic conductors based on radical cation and/or anion salts; Organic superconductors.
Organic metals of the (TMTSF)2X family, where TMTSF is tetramethyltetraselenafulvalene and X rep- resents an inorganic anion, are highly anisotropic, quasi-one-dimensional crystals that consist of parallel conducting chains. The overlap of the electron wave functions and the electric conductivity are the highest in the direction of the chains (the a direction) and are much smaller in the b direction perpendicular to the chains. In this paper, we neglect the coupling between the chains in the third, c direction, which is weaker than in the b direction, and study the properties of a single layer (the a-b plane), modeling (TMTSF)2X as a system of the uncoupled two-dimensional layers.
When a strong magnetic field is applied perpendic- ular to the a-b plane, the magnetic-field-induced spin- density-wave (FISDW) appears in the system (see Ref.
[1] for a review). In the FISDW phase, the Hall con- ductivity per one layer, σxy, is quantized at zero tem- perature as
σxy= 2N e2/h, (1)
where e is the electron charge, h = 2π¯h is the Planck constant, and N is an integer that characterizes the FISDW. However, at a finite temperature, because elec- trons are thermally excited above the FISDW energy gap, the Hall conductivity is not quantized. In this pa- per, we calculate temperature dependence of the Hall conductivity in the FISDW state.
To model (TMTSF)2X, let us consider a 2D system that consists of many chains, parallel to the x axis and
equally spaced along the y-axis with the distance b.1) The chains are coupled through the electron tunneling of the amplitude tb. To calculate the Hall conductivity, suppose that a magnetic field H is applied along the z axis perpendicular to the (x, y) plane, and an electric field Ey is applied perpendicular to the chains. The electron Hamiltonian in the FISDW state is:2)
H = −ˆ ¯h2 2m
∂2
∂x2+2∆ cos(Qxx)+2tbcos[kyb−G(x−vEyt)], (2) where m is the electron mass, Qxand ∆ are the wave vector and the amplitude of the FISDW potential, ky
is the electron wave vector across the chains, t is the time,
G = ebH/¯hc (3)
is the wave vector of the magnetic field,
vEy = cEy/H (4)
is the drift velocity in the crossed electric and mag- netic fields, and c is the velocity of light. Hamilto- nian (2) is written in the mixed representation, where an electron wave function depends on the coordinate x along the chains and the momentum ky across the chains. For simplicity, we set the FISDW wave vector across the chains, Qy, to zero, and neglect the next- nearest-neighbor hopping term 2t′bcos(2kyb) in Hamil- tonian (2). The electric and magnetic fields are intro-
1)The x and y axes correspond to the a and b axes of (TMTSF)2X.
2)We pay no attention to the spin indices, because they are not important for our purposes.
stitution, ky→ ky− eAy/c¯h, in the gauge
Ax= Az= 0, Ay = Hx − Eyct. (5) It follows from Eq. (2) that, in the presence of the mag- netic field, the hopping across the chains becomes a pe- riodic potential along the chains with the wave vector G (3) proportional to the magnetic field. We will refer to this periodic potential as the “hopping potential”.
Due to the presence of the electric field Ey, the hop- ping potential moves along the chains with the velocity vEy (4), whereas the FISDW potential is assumed to be pinned and does not move.
Let us linearize the longitudinal dispersion in Hamil- tonian (2) near the Fermi energy and focus on the elec- trons whose momenta are close to the Fermi momenta +kF and −kF. Let us count their momenta from +kF
and −kFand denote their wave functions as u and w.
In this representation, a complete electron wave func- tion is a spinor (u, w), and the Hamiltonian is a 2 × 2 matrix, which can be expanded over the Pauli matrices ˆ
τ1, ˆτ2, ˆτ3, and the unity matrix ˆ1 (which we will not write explicitly in the following formulas). It is well known [1, 2, 3] that the FISDW wave vector depends on the magnetic field in the following manner:
Qx= 2kF− NG = 2kF− NebH/¯hc, (6) where N is an integer that characterizes the FISDW.
Taking into account Eq. (6), Hamiltonian (2) can be rewritten in the spinor representation as
H = −i¯hvˆ Fτˆ3
∂
∂x+∆ˆτ1eiˆτ3N Gx+2tbcos[kyb−G(x−vEyt)], (7) where vF= kF/m is the Fermi velocity. The last term in Eq. (7) can be eliminated by chiral transformation of the electron wave function:3)
u w
→ exp
iˆτ3
2tb
¯hωc
sin[kyb − G(x − vEyt)] u w
, (8) where
¯hωc = ¯hvFG = ebHvF/c (9)
is the characteristic energy of the magnetic field (the cyclotron frequency). In representation (8), Hamilto- nian (7) becomes
H=−i¯hvˆ Fτˆ3
∂
∂x+ ∆ˆτ1exp(iˆτ3N Gx)
× exp
iˆτ3
4tb
¯hωc
sin[kyb − G(x − vEyt)]
. (10)
Expanding the periodic function in the last term of Eq.
(10) into the Fourier series, we get the following expres- sion:
H=−i¯hvˆ Fτˆ3 ∂
∂x+ ∆ˆτ1eiˆτ3[N (kyb+GvEyt)]
×X
n
an+Neiˆτ3n[kyb−G(x−vEyt)], (11)
3)This kind of transformation was first introduced in Ref. [4] that started development of the FISDW theory.
Bessel functions: an= Jn(4tb/¯hωc).4) The last term in Eq. (11) is the sum of many sinusoidal potentials whose wave vectors are the integer multiples of the magnetic wave vector G. Each of these periodic potentials mixes the +kFand −kFelectrons and opens an energy gap at the electron wave vector kxshifted from ±kFby an in- teger multiple of G/2. The distance in energy between the gaps is equal to ¯hωc (9).
The term with n = 0 in the sum in Eq. (11) does not depend on x and opens the gap right at the Fermi level.5) When the temperature T is much lower than the distance between the energy gaps ¯hωc:
T ≪ ¯hωc, (12)
only the gap at the Fermi level is important, whereas the other gaps may be neglected. Condition (12) is al- ways satisfied in the relevant temperature range 0 ≤ T ≤ Tc (where Tc is the FISDW transition tempera- ture) in the weak coupling theory of the FISDW, where Tc ≪ ¯hωc. Thus, let us omit all the terms in the sum in Eq. (11), except the term with n = 0:
H = −i¯hvˆ Fτˆ3
∂
∂x+ ∆effτˆ1eiˆτ3[N (kyb+GvEyt)], (13)
∆eff = aN∆. (14)
This is the so-called single-gap approximation [5]. It was shown explicitly in Ref. [6] that omission of the gaps located deeply below the Fermi energy does not change the value of the Hall conductivity, at least at zero temperature.
By the above sequence of manipulations, we have combined the two periodic potentials in Eq. (2) into the single effective potential (13) that opens a gap at the Fermi level. It follows from Eq. (13) that the phase ϕ of this effective potential changes in time:
ϕ = −NGv˙ Ey, (15)
which means that the effective potential moves along the chains. Since, at zero temperature, all electrons are confined under the energy gap opened by this potential, the motion of the potential induces the Fr¨ohlich current [7] along the chains:
jx= − e
πbϕ.˙ (16)
Substituting Eqs. (15), (3), and (4) into Eq. (16), we find the quantum Hall effect (QHE) in agreement with Eq. (1):
jx=2N e2
h Ey. (17)
4)General expression (11) is valid even when the FISDW has a nonzero transverse wave vector and the transverse dispersion law of the electrons is more complicated, but the expression for the expan- sion coefficients anwould be different in that case.
5)Since, by introducing the ± electrons, we have already subtracted the wave vectors ±kF, the actual wave vector that corresponds to this term is 2kF.
Figure 1: The reduction factor f of the Hall conduc- tivity as a function of the ratio of the energy gap at the Fermi level ∆eff to the temperature T , as given by Eq.
(20).
To avoid confusion, we wish to emphasize that here the FISDW is assumed to be immobile, unlike in Ref.
[8] where the influence of the FISDW motion on the QHE was studied. The effective potential (13) moves, because it is a combination of the stationary FISDW potential and the moving hopping potential (2).
Eq. (16) is a good starting point to discuss the temperature dependence of the QHE. According to the above consideration, the Hall conductivity is the Fr¨ohlich conductivity of the effective periodic potential (13). Thus, the temperature dependence of the QHE must be the same as the temperature dependence of the Fr¨ohlich conductivity. The latter issue was stud- ied in the theory of a regular charge/spin density wave (CDW/SDW) [9, 10]. At a finite temperature T , the electric current carried by the CDW/SDW condensate is reduced with respect to the zero-temperature value (16) by a factor f (T ). The same factor reduces the condensate Hall effect at a finite temperature:
σxy(T ) = f (T ) 2N e2/h, (18)
f (T ) = 1 − Z ∞
−∞
dkx
¯hvF
∂E
∂kx
2
−∂nF(E/kBT )
∂E
, (19)
where E =p(¯hvFkx)2+ ∆2eff is the electron dispersion law in the FISDW phase, kBis the Boltzmann constant, and nF(ǫ) = (eǫ+ 1)−1 is the Fermi distribution func- tion. The last term in Eq. (19) reflects the fact that normal quasiparticles, thermally excited above the en- ergy gap, equilibrate with the immobile crystal lattice;
thus, only a fraction of all electrons is carried along the chains by the moving periodic potential, which re- duces the Hall/Fr¨ohlich current. Derivation of Eq. (19) is given in Appendix.
The function f (19) depends only on the ratio of the energy gap at the Fermi level, ∆eff (14), and the
Figure 2: Hall conductivity in the FISDW state as a function of the temperature T normalized to the FISDW transition temperature Tc.
temperature T and can be written as [10, 11]
f ∆eff
kBT
= Z ∞
0
dζ tanh ∆eff
2kBT cosh ζ
/ cosh2ζ.(20) The function f (∆eff/kBT ) is plotted in Fig. 1. It is equal to 1 at zero temperature, where Eq. (18) gives the QHE, gradually decreases with increasing T , and vanishes when T ≫ ∆eff. Taking into account that the FISDW order parameter ∆ itself depends on T and vanishes at the FISDW transition temperature Tc, it is clear that f (T ) and σxy(T ) vanish at T → Tc, where σxy(T ) ∝ f(T ) ∝ ∆(T ) ∝√
Tc− T . Assuming that the temperature dependence ∆eff(T ) is given by the BCS theory [5], we plot the temperature dependence of the Hall conductivity, σxy(T ), in Fig. 2.
The function f (T ) (19) is qualitatively similar to the function fs(T ) that describes the temperature re- duction of the superconducting condensate density in the London case. Both functions approach 1 at zero temperature, but near Tcthe superconducting function behaves differently: fs(T ) ∝ ∆2(T ) ∝ Tc− T . As ex- plained in Appendix, this is due to the difference be- tween the static and dynamic limits of the response function.
The QHE in the FISDW state at zero temperature was derived theoretically in Refs. [2, 6, 12]. An at- tempt to calculate the Hall conductivity in the FISDW state at a finite temperature was made in Ref. [11], but it failed to produce the QHE at zero temperature.
Various aspects of the QHE in (TMTSF)2X were re- viewed in Ref. [13]. Temperature dependence of the Hall resistance in (TMTSF)2X was measured in exper- iments [14]. However, to compare the experimental re- sults with our theory, it is necessary to convert the Hall resistivity into the Hall conductivity, which requires ex- perimental knowledge of all components of the resistiv- ity tensor.
We conclude that, at zero temperature, a FISDW
number N that characterizes the wave vector (6) of the FISDW. As the temperature increases, the Hall conductivity decreases, vanishing at the FISDW transi- tion temperature Tc. The function f (T ) that describes the reduction of the Hall effect with the temperature is the same as the temperature reduction function of the Fr¨ohlich current of a regular charge/spin-density wave.
This work was partially supported by the NSF un- der Grant DMR–9417451, by the Alfred P. Sloan Foun- dation, and by the David and Lucile Packard Founda- tion.
Appendix
We derived Eqs. (18) and (19) by calculating the single- loop Feynman diagram that represents the electromag- netic response of the electrons in the FISDW state to the electric field Ey. The expression for the diagram is sensitive to the ratio of the frequency ω and the wave vector q of the field Ey when both ω and q approach zero. Eqs. (19) and (20) correspond to the so-called dynamic limit where q/ω = 0 [10]. This limit is appro- priate in our case, because the electric field is, suppos- edly, strictly homogeneous in space (q = 0), but may be time-dependent (ω 6= 0). The effective periodic poten- tial (13) is also time-dependent. In the opposite, static limit ω/q = 0, we obtain the function
fs(T ) = 1 − ¯hvF
Z ∞
−∞
dkx
−∂nF(E/kBT )
∂E
, (21)
which describes the charge-density response to a static deformation of the CDW phase, ∂ϕ/∂x, as well as the superconducting condensate density in London super- conductors [9, 10]. In the latter cases, the CDW phase or magnetic field in the Meissner effect are stationary (ω = 0), but vary in space (q 6= 0). Comparing Eqs.
(19) and (21), one can see that f (T ) and fs(T ) are dif- ferent. We obtain Eqs. (19) and (21) by summing over the internal frequency of the loop first. Different, but equivalent expressions for f (T ) and fs(T ) were obtained in Ref. [10] by integrating over the internal momentum of the loop first.
The diagrammatic derivation is not very transpar- ent physically, so below we offer another derivation of Eq. (19), based on the ideas of Refs. [9, 15]. Let us consider a one-dimensional electron system subject to a CDW/SDW of the amplitude ∆0, which moves with a small velocity vDW. Let us calculate the Fr¨ohlich current proportional to vDW at a finite temperature T . We find the electron wave functions in the reference frame moving with the density wave and then Galileo- transform them to the laboratory frame [15]:
ψk±(x, t)=u±kei(kF+k+mvDW)x−i(kF+k)vDWt∓iEkt/¯h +w±kei(−kF+k+mvDW)x−i(−kF+k)vDWt∓iEkt/¯(22)h,
in vDW. In Eq. (22) and below, the index ± refers to the states above and below the CDW/SDW energy gap, not to the states near ±kF. The coefficients of superposition, uk and wk, are given by the following expressions:
|u+k|2= |w−k|2= ∆20
2Ek(Ek− ξk), (23)
|wk+|2= |u−k|2= Ek− ξk
2Ek
, (24)
where ξk = ¯hvFk and Ek =pξk2+ ∆20are the electron dispersion laws in the absence and in the presence of the CDW/SDW gap.
By analogy with the standard derivation of the su- perfluid density [16], let us assume that, because of in- teraction with impurities, phonons, etc., the electron quasiparticles are in thermal equilibrium with the crys- tal in the laboratory reference frame, so their distri- bution function is the equilibrium Fermi function nF. However, it is not straightforward to apply the Fermi function, because the two components of the eigenfunc- tion (22), which have the same energy in the reference frame of the moving CDW/SDW, have different ener- gies in the laboratory frame. Let us make a reasonable assumption that a state (22) is populated according to its average energy ¯E±k:
E¯±k=|u±k|2(±Ek+ ¯h(kF+ k)vDW)
+ |wk±|2(±Ek+ ¯h(−kF+ k)vDW). (25) The electric current I carried by the electrons is equal to
I = 2e¯hX
±
Z ∞
−∞
dk 2πnF
E¯k± kBT
(26)
×
|u±k|2 kF+ k m +vDW
¯ h
+ |w±k|2 −kF+ k m +vDW
¯ h
, where the factor 2 comes from the spin. Substituting Eq. (25) into Eq. (26) and keeping the terms linear in vDW, we find two contributions to I. The first contri- bution, I1, is obtained by replacing ¯Ek±by ±Ek in Eq.
(26), that is, by omitting vDW in Eq. (25). This term represents the current produced by all electrons moving with the velocity vDW:
I1= 2evDW2kF/2π. (27)
The second contribution, I2, comes from expansion of the Fermi function in Eq. (26) in vDW and represents reduction of the current due to thermally excited quasi- particles staying behind the collective motion:
I2=2emvDW
X
±
Z ∞
−∞
dk 2π
∂nF(±Ek/kBT )
∂Ek
×
vF(|u±k|2− |w±k|2) +¯hk
m(|u±k|2+ |wk±|2)
2
. (28)
The second term in the brackets in Eq. (28) is small compared to the first term and may be neglected. Sub- stituting Eqs. (23) and (24) into Eq. (28) and expressing the CDW/SDW velocity in terms of the CDW/SDW phase derivative in time, vDW = − ˙ϕ/2kF, we find the temperature-dependent expression for the Fr¨ohlich cur- rent:
I = I1+ I2= −ef(T ) ˙ϕ/π, (29) f (T ) = 1 −
Z ∞
−∞
dξk
ξk
Ek
2
−∂nF(Ek/kBT )
∂Ek
. (30) Eq. (30) is the same as Eq. (19). Dividing the current per one chain, I (29), by the interchain distance b, we get the density of current per unit length, jx (16).
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