Landau level spectrum of Bloch electrons in a
honeycomb lattice
R. Rammal
To cite this version:
R. Rammal. Landau level spectrum of Bloch electrons in a honeycomb lattice. Journal de Physique, 1985, 46 (8), pp.1345-1354. <10.1051/jphys:019850046080134500>. <jpa00210078>
HAL Id: jpa-00210078
https://hal.archives-ouvertes.fr/jpa-00210078
Submitted on 1 Jan 1985
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J.
Physique 46 (1985)
1345-1354
AOÛT 1985,
1345
Classification
Physics Abstracts
71.10
-
71.25
-
74.10
Landau level spectrum of Bloch electrons in
a
honeycomb lattice
R. Rammal
Centre de Recherches sur les Très Basses Températures, CNRS, B.P. 166X, 38042 Grenoble Cedex, France,
and Department of Physics, University of Pennsylvania, Philadelphia, PA 19104-3859, U.S.A.
(Reçu le
25 juin 1984,
accepté
le 2 avril
1985)
On analyse le spectre d’énergie du modèle des liaisons fortes, sur un réseau en nid d’abeilles, en présence
Résumé.
d’un champ magnétique uniforme. Le graphe du spectre est montré pour différents flux réduits rationnels à travers
les cellules élémentaires. On montre que ce spectre possède des propriétés récursives analogues à celles du réseau
carré et du réseau triangulaire. Les propriétés spécifiques (bandes interdites, sous-bandes, etc.) obtenues sont
attribuées à une propriété de frustration et sont comparées avec celles des réseaux de Bravais. Nos résultats ont
une implication directe pour les mesures récentes du champ critique supérieur d’un réseau supraconducteur en
nid d’abeilles. Nous comparons aussi la structure du bord du spectre dans les trois réseaux : carré, triangulaire et
nid d’abeilles.
2014
The energy spectrum of a tight-binding honeycomb lattice in the presence of a uniform magnetic
Abstract.
field is analysed. The graph of the spectrum over a wide range of rational reduced flux 03A6/03A60 through elementary
hexagonal cells is plotted. The energy spectrum is found to have recursive properties similar to those discussed
previously on the square and triangular lattices. New features of the spectrum are also obtained. Specific properties
(gaps, subbands, etc.) are shown to be a direct consequence of frustration and are compared with the spectrum of
Bravais lattices. Our results are shown to be relevant for the recent measurements of the upper critical field of a
superconducting honeycomb network. A comparison of the structure of the edge of the spectrum on square,
triangular and honeycomb lattices is also outlined.
2014
1. Introduction.
Recently, it has been shown that [1, 2] the magnetic
properties of regular superconducting networks are
controlled, within the framework of mean field theory,
by the so-called Harper’s equation [3]. More precisely,
the edge of the spectrum of the tight-binding model,
in the presence of a magnetic field [4] was identified
with the upper critical line of the superconducting
networks. This connection has been achieved experimentally [5] by a direct measurement of the critical
temperature of a regular two dimensional superconducting network. The observed well-defined structures of the critical line reflect the expected features of
this line.
In this paper we propose to investigate the energy
spectrum of a tight-binding model on a honeycomb
lattice (H) in the presence of a uniform magnetic field.
Our motivations are as follows :
a) Up to now, only Bravais lattice have been
studied : square (S) lattice [4, 6] and triangular lattice [7]. Previous work has shown that the spectrum
have various symmetry properties some of which
may be traced to the point group of the lattice. Others,
such as recursiveness are general features of the model
Hamiltonian used and the difference between rational
and irrational numbers [4, 8]. What happens on a nonBravais lattice such as the honeycomb lattice ?
On the other hand, the topological structure of the
lattice plays an important role in the band structure,
already in zero magnetic field. The occurrence of odd
rings in the lattice induces some particular features of
the density of states which may be traced to frustration [9] phenomena. What are the consequences of the
particular band structure at zero field on the Landau
level spectrum ?
Finally, the rich structure of the spectrum of Landau
levels results from the coexistence of two incommensurate periods. The first is given by the lattice structure
and the second is fixed by the magnetic field. The
relevant parameter in the band structure is the ratio
f 4&#x3E;14&#x3E;0’ where 00 denotes the quantum flux and 0
the flux through an elementary cell of the lattice. In
this respect, the Landau level spectrum can also be
=
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046080134500
1346
viewed as a frustration problem [10]. To what extent
does the band structure (closure of the gaps, band
touching, etc.) result from geometrical considerations
alone ?
b) In view of the experimental measurement of the
edge of the spectrum, using superconducting networks [5], it is useful to compare this edge on various
lattices. In particular, the reduction of the critical
temperature Teat different rationals f 0/00 is
sensitive to the topology of the network. This is clearly
visible when we compare the square and triangular
lattices. What is the corresponding situation on the
honeycomb lattice ?
This paper is organized as follows. In section 2, a
general formulation of the problem is given, as well as
a coherent set of notations. Section 3 is devoted to the
calculation of the band structure at zero and low
magnetic fields. Section 4 is actually the main part of
this paper. The spectrum in calculated at rational flux
and the density of states is expressed in each subband.
Various symmetry and recursiveness properties are
also discussed as well as the measure of the spectrum.
The structure of the edge spectrum is discussed in
relation with superconducting networks in section 5.
In particular, a comparison between square, triangular
and honeycomb lattices is outlined in this section.
In the last section we give some concluding remarks
and discuss some other illustrations of the resuts.
=
2. Notations and
a/,
=
figure
lb :
-
=
a -VI-3).
magnetic
In the presence of a uniform
field H, perthe
to
the
above
pendicular
planar lattice,
equation is
modified by phase factors depending on the magnetic
field strength. Let Yap be the phase factor between
atoms
A
a
and fl :
being the vector potential, equation (2. 1) becomes
:
general formulation.
The honeycomb lattice is made up of two dimensional
array of hexagonal unit cells, of side a, with atoms at
the vertices. Such a structure is encountered in solid
state physics in some strongly anisotropic crystals [11].
The unit cell is a rhombus of side
with angles
2 7T/3 and n/3 at its vertices. We take the x-axis parallel
to the long diagonal and the y-axis parallel to the short
diagonal of the rhombus. There are two atoms
(Fig. la) in each unit cell, A and B. In the reciprocal
space k
(kX’ ky), the first Brillouin zone is the
formed
hexagon
by the appropriate perpendicular
bisectors. Particular points of relative importance,
lying at the edge of the first Brillouin zone, are shown
on
Fig. 1. (a) The honeycomb lattice made up of hexagonal
cells of side a. A(x - a, y) and B(x, y) denote two atoms in
the unit cell. (b) The first Brillouin zone in the reciprocal space
k
(kX’ ky). Particular points lying at the edge of the zone
are shown : F(2 n/3a, 0) and K(2 nl3a, ± 2 7r/3
F(2 n/3a, 0)
As shown in section 4, the whole spectrum of equation (2. 3) depends only on the reduced flux f 0/00,
giving the ratio of the magnetic flux through an elemen6 Ha’,13-14) to the flux
tary hexagonal cell
for
hcle
quantum (00
electrons).
=
I(o
=
=
3.
Spectrum at zero and low magnetic fields.
3.1 ZERO
FIELD SPECTRUM.
-
In the absence of the
is given by :
magnetic field, the dispersion relation
and
respectively.
a model for the honeycomb lattice, we consider
where k
(k, ky) is the wave vector associated to the
stight-binding model with nearest-neighbour over- periodic solutions CPa tfa exp(ikx. x + iky.Y) of equalap only. If cp(r) is used to denote the wave function tion (2.1), (a A, B).
amplitude on site r, then the Schrodinger equation
The spectrum (two bands) is confined in the interval
reduces to a finite-difference equation
[ - 3, + 3] and the edges are reached for k 0.
Close to this point we have two parabolic bands and
As
=
an
=
=
=
denotes the associated eigenvalue, r’ denotes
neighbours of r and V the strength of the overintegral. In the following, we take V 1 for
where
c
nearest
lap
simplicity.
=
the curves of constant energy are circular.
As can be seen the adjacent band extrema are at the
comers of the first Brillouin zone (points K). The
bottom of the valence band (- in Eq. (3.1)) and the
top of the conduction band (+ in Eq. (3.1)) lie at
k=0.
1347
Eigenstates corresponding to e 0 (point K) are
on figure 2. Similarly the associated eigenstates to point F are shown on figure 3. Note that,
close to points K, the dispersion relation becomes
linear instead of quadratic. The density of states
n(s) associated with equation (3.1) is given [12] by
the following expression
=
shown
Degenerate eigenmodes associated with the energy
zero magnetic field. Here a and b denote two
arbitrary numbers and w is the complex conjugate of
m
exp(2 in/3).
Fig.
2.
level
e
-
=
0 at
=
Here
is the
argument of the complete elliptic integral K(k) of
first kind.
Note that in addition to the Van Hove’s singularities (at s
± 1, ± 3), which are typical of two
dimensional lattices, the density of states n(E) vanishes
0 where a non-analytical singularity appears
at B
=
=
0 may be traced
The origin of this singularity at 8
back to the linear dispersion relation close to points K,
resulting from the degeneracy of the two bands at
these points.
=
3.2 Low MAGNETIC FIELD LIMIT. - The calculation
of the energy spectrum at low magnetic field can be
carried out using a semiclassical theory (Onsager
quantization scheme [13]) or a continuum approximation [14] for the tight-binding equations (Eq. (2.3))
in the presence of a small magnetic field. Note that
[15] the semiclassical method may break down near
saddle points. In the present case, the Onsager’s
scheme can still be applied near the Van Hove singularities, but it will be shown to break down near the
non-analytic singularity s = 0.
Near the
3.2.1 Onsager’s quantization scheme.
band edges s
± 3, the quantization rule gives the
following Landau levels :
-
=
Fig.
zero
Eigenmodes associated with
magnetic field.
3.
-
s
=
+ 1 and - 1 at
Similarly, near the singular point s = 0, one obtains
This result, implying a square-root departure of g as a
function of the magnetic field must be contrasted with
the usual linear behaviour given by equation (3.4).
However, as shown below, the result of equation (3..5)
is not
entirely correct
: n
+12
must be
replaced by n.
When this correction is taken into account, equation (3 . 5) gives accurate results, as shown in figure 4.
3. 2. 2 Continuum approximation.
In this approach,
the tight-binding equations (Eq. (2.3)) are linearized
in the presence of a vanishing magnetic field. In the
following, we choose for convenience the Landau
H(O, x, 0).
gauge A
Let us consider first the case of the band edge B
3.
in
account
the
translation
into
Taking
symmetry
y direction, solutions of equations (2.3) can be
chosen to be of the form cp(x, y) = tJI(x) exp(iky. y).
One obtains a system of two coupled equations
-
=
=
where y
2 noloo and n is a positive integer (n &#x3E; 0).
As shown in figure 4, equation (3.4) reproduces
quantitatively the Landau level structure near the
band edges.
=
1348
Close to
u
=
x -
e
a/2,
=
3, one can choose ky
a more
0 and look at solutions of equation (3.6) where t/J A
symmetric equation is obtained by adding the above two equations :
=
Now, the two sides of equation (3. 7) are expanded in ula and y. This leads to
=
t/JB
=
0. Putting
:
The solution of this set of equations can be expressed
in terms of normalized harmonic oscillator functions.
In particular, the corresponding eigenvalues are
given by
where n takes on all positive integer values.
The (partial) breakdown of Onsager’s rule is simply
caused by the band degeneracy at e
0. This is a
the
structure
of
lattice
consequence
partially neglected
in the semi-classical approach.
=
4. Landau level subband structure.
Some time ago, Hofstadter
[4] studied the
energy
spectrum of a single tight binding square lattice. The
same study was extended later for the triangular
lattice [7] and generalized [6] square lattice structures.
In all these cases, it was shown that the relevant parameter in the band structure is the ratio f
0/00.
For rational /(= p/q), the tight-binding band is
split up into q non-overlapping subbands, each
containing an equal number of states. For irrational
values of f the number of bands is infinite, but the
gaps that exist at rational values of f persist over
some finite range of f, through a succession of closures
and openings. The graph of the energy spectrum as a
function of magnetic field exhibits a recursive structure
in the plot whereby copies of the entire spectrum are
contained within the spectrum itself. Although a
rigorous proof of this property is not available, one
can see its plausibility by looking at the spectrum as a
composite of broadened and split Landau levels.
More general properties were also discovered :
i) periodicity in f along the field axis, with period 1
due to gauge invariance, ii) the closures of the gaps.
For instance on the square lattice, at f
1/2, the
spectrum comprises two bands which touch each
other. This means that at this energy the density of.
states has an isolated zero with a non-analytic behaviour. Other such touchings of bands can be observed
on the central horizontal axis. The case of the triangular lattice exhibits similar behaviour. In this case, a
gap closure due to touching of two bands is observed
when f
1/6 and 5/6.
=
Fig. 4.
Energy spectrum for broadened Landau levels.
Only low index levels are shown and the fine structure corresponds to just a few rational values of the reduced flux 0/00.
Solid lines correspond to Landau levels starting at zero field
from s
3 and 8
0 respectively (Eqs. (3.4) and (3.10)).
-
=
=
which is simply the Schrodinger equation for a simple
harmonic oscillator. The energy levels associated with
equation (3.8) reproduce the result of equation (3.4).
In order to perform the continuum approximation
for the degenerate level s
0, we use equation (3.6),
=
written in variable u
=
x -
al2 and for ky
to the point (kx
0, ky) of the first
Brillouin zone. In the limit *y 1, we expand the system as above and this yields
corresponding
=
=
=
1349
shown in section 2, a band touching occurs
already in the honeycomb lattice in the absence of a
magnetic field, at s 0. A natural question arises :
to what extent do the above properties of band touchings result from geometrical considerations alone ?
In this section we shall investigate this question, by
studyingttie band structure in the presence of a
magnetic field. For this, we focus our attention to
these properties instead of recursiveness, which is
also present in this spectrum.
As
4.1 REDUCTION TO ONE DIMENSION PROBLEM.
was noticed in section 2, the honeycomb lattice is a
non-Bravais lattice. Each unit cell contains two atoms
A and B. We choose as above the Landau gauge
A
H(O, x, 0) ,for convenience. Writing the tightbinding equations (Eq. (2. 3)) for sites B(x;y), A(x - a, y)
one obtains a system
and A(x + a/2, y ± a
of four eigenvalue equations. Eliminating the A sublattice sites, the following equation for B site is obtained
In equation (4.1), Fi(l i 6) denote phase factors,
gauge, they are given by the following expressions
resulting from the elimination of A sites. For the choosen
As
was
-
=
=
v’3/2),
It is reasonable to assume plane wave behaviour in the y direction, since the coefficients in the above equation
only involve x. Therefore we write
and then deduce the one dimensional eigenvalue equation
After
some
rearrangements and the use of equation (4. 2), this leads to the finite difference equation
a,J3/2.
Here m denotes an integer, and y
where t/lm
2n4&#x3E;/4&#x3E;o
OB(x) at x m(3 a/2), A B2 - 3 and K ky
is the magnetic field parameter. Only the ratio f
0/0 0 is involved in equation (4. 5) as expected.
It is instructive to compare equation (4.5) to similar equations obtained on square [4] and triangular [7]
lattices
=
=
=
=
=
=
and
Equation (4.5) is a second-order difference equation
whose solutions determine the broadening and fine
structure of the Landau levels for a given magnetic
field. The spectrum is confined to values of A between
1350
6 and + 6, i.e. - 3 6 .- + 3. A close inspection
of equation (4.5) shows that the spectrum is unchanged
under reflections E H - s for a given value of y. In
addition, the spectrum is invariant under translations :
f - f + n, n integer and exhibits a reflection symmetry about half integer values of f. All these symmetry
properties are also present on the square lattice
spectrum, reflecting translation and gauge invariance.
-
Because of the symmetries
4.2 RATIONAL FIELDS.
in the spectrum, we shall limit our discussion to
0 f 1/2. For rational values p/q of f (p, q integers prime relative to each other), the system (Eq. (4. 5))
becomes closed after translation by q periods thus
leaving 2 q separate equations. As was noticed by
several authors [7, 16], not all these equations are
independent, however. For f = p/q, equation (4.5)
becomes
-
’
Making the substitution :
one
obtains a system of q equations for amplitudes g(u), u
=
1, 2,
...,
q.
2 nplq, Indeed, one can easily verify that g(u) and g(q +
2 x and y
If we use Floquet’s theorem to construct these two functions through the relation
where 01
=
=
u) obey the
same
equation.
Hermitian system of q equations for q unknowns results. In our notations, the q x q secular determinant has
period 2 7r/q in 81, and the indices 81 and 82 appear in the constant term only. The non-zero matrix elements
of the secular determinant are given by
a
Expanding this secular determinant in terms of Similarly,
products of its elements, we know from its periodicity
2 nlq with respect to 01 that all non-constant terms
containing less than q factors exp(i01) cancel. The
only terms depending on 01 and 02 are therefore
easy to obtain.
The constant term
involving 02 is given by
the term
Therefore the
following form
involving (Jl only
is
given by
equation for A that results has the
1351
Pq(À) is a polynomial of degree q
containing 01 and 02- W(01, 82) is given by
where
in Å not
of the secular
equation
3 is a zero for this equation for
Furthermore, A
all p and q. This solution corresponds to a band
touching at s 0 in the spectrum. More generally,
the q bands in variable A transform into 2 q bands
0.
with a reflection symmetry about c
Some examples of polynomials P,(A) are given in
what follows, for simple values of f.
=
When 01 and 02 cover their range, W «(J l’ 02) varies
3 and W2
between W 1
+ 6. Intercepts of
the polynomial between these two values define
therefore the subbands for the rational field chosen.
In general, it is easy to verify the following form
=
-
=
-
=
=
Intercepts of P,,(A) with W 1 and W2 give the subband
edges. More precisely the edges are fixed by the following two conditions
and
For
instance, if
yields
the
p
=
1
following
2, equation (4.1b)
symmetrical bands :
0 s[ % /3 touching at
and q
=
four
,/3 8 ![ /6, and
0 and e +..,/3-. For small values of q, the cals
=
=
5.
culations can be achieved analytically up to q
For other values of q, we have calculated (using the
algorithm of Eq. (4.16)) the band edges for p/q given
by the first elements of the Farey series. We stopped the
20 because the band widths became
calculations at q
as narrow as 10-9. Figure 5 shows the spectrum
obtained in this manner. Broadening of the Landau
levels (Fig. 4) increases with f and a fine structure is
present between any two rational values of f. On
figure 4 are shown the Landau levels calculated in
section 3 in the portion of the spectrum (0 , s , 3,
0 f 1/2) shown in figure 5. These levels near
0 are clearly recognizable, their field
8=3 and c
dependence in the limit of low field matching exactly
the results of section 3 (indicated by full lines). Large
gaps separate the low index levels, and two major
gaps are obtained for 0 f - 1/2. A detailed discussion of the nesting hypothesis (recursiveness property) will be found in reference [7] and therefore not
discussed here. Instead we shall discuss the density
of states in each subband.
=
=
=
Fig. 5. Spectrum of a tight-binding model on a honeycomb lattice in a perpendicular magnetic field. The eigenvalue energy a, ranging between - 3 and + 3, is the shown
vertical variable and 0/00 the reduced magnetic flux through
one lattice hexagonal cell, is the horizontal variable, ranging
from 0 to 1. Only rational values p/q of the field variable
0/00 are represented with q 20.
-
4.3 DENSITY
solutions of
of the
be
in a
obtained
equation (4.5)
since
manner
the
and
straight-forward
phases 01
02
appear in the constant term only. For this we follow
OF STATES. -
The
can
multiplicity
1352
the method proposed in references [7] and [17]. The
main idea is to transform the counting over an energy
range to a counting over a range of the polynomial
Pq(A), thus yielding a form of the density of states
appropriate for an arbitrary rational value of the field
parameter f Let Às(Ol’ (2) be a root of equation (4.13)
for a given 01 and O2 and a fixed value of the field. As
these phases cover their range the root will scan an
entire subband (in A). The phases have constant weight
since they are phase variables and the density of states,
in variable A is then given by the expression
The
sum
is
over
all subbands and the total number of
states has been normalized to one. The 5 function can
be integrated out by transforming the integral over 02
to an integral over A,, with the aid of equation (4.15).
In this way, P is treated as a variable, and the remaining integration can be performed in terms of complete
elliptic integrals. Using A = E2 - 3,
one
gets
coincides with argument
in equation (3.2).
k of the elliptic integral given
4.4 MEASURE OF THE SPECTRUM.
As a by-product
of the previous calculation, we have considered the
measure of the spectrum for rational f. Recently, the
total measure of the spectrum has been studied [18]
for different lattices. In particular, it was suggested
that for a given f p/q the total sum of bandwidths
(measure of the spectrum) has the asymptotic form
a/q for isotopic square and triangular lattices. Here a
denotes a constant of proportionality which asymptotically approaches 9.3300 on the square lattice, for
most values of the numerator p. This very accurate
result (one part in 105) suggests the vanishing of the
spectral measure, when q goes to infinity. The value 1
of the exponent of q was attributed to the self-similarity of the diagram of energy bands as a function
-
=
of f.
denote S(p/q) the sum of bandwidths for a
given f p/q. We have deduced the values of the
product qs(plq) from our numerical results. Although
the calculations were performed for q , 20 some
conclusions can be stated. A selection of the values of
qS(p/q) is shown in table I. For q a 10, qS(p/q) takes
its value inside the range from 7.2 to 8.6 independent
of the way in which p varies. It is clear that the general
form of the result is in agreement with what is expected, but it does not appear that data can be extrapolated in a reliable manner. The sequence p
2,
9, 11, 13, 15, 17, 19 suggests the limiting value
q
a/2 3.95, while the sequence q even, p q/2 - 1
is too short to show convergence. Other sequences
are scattered and do not follow an obvious pattern.
Therefore, it is likely that the sum of bandwidths is
tending to zero as fast as q - ’ is this case. However,
more extensive data are needed to extract the precise
value of the constant a.
Let
us
=
=
=
=
where
and
Application to superconducting networks.
The theory involved in the computation of the upper
critical field of a superconducting network (neglecting
fluctuations) consists of solving the linearized Ginzburg-Landau equation, which is essentially equivalent
to the Schrodinger equation. For superconducting
application, one is only interested in the lowest eigenvalue (edge of the spectrum). Note that the same equations are involved in the mean field theory of Josephsonjunction arrays [19]. The basic concepts for treating
superconducting networks near the second order
phase boundary have been worked in detail by various
authors. The linearized Ginzburg-Landau equations
5.
Here P is the value of the polynomial at A
s2 - 3
and K(x) is the complete elliptic integral of the first
kind. Within a subband, only this latter factor varies
=
significantly giving a A-shaped logarithmic singularity to the density of states. At subband edges, the
density of states drops discontinuously to zero as a
gap is encountered. Non-analytic singularities coming
from the touchings of neighbouring subbands appear
also as in zero field. In addition it is easy to recover the
expression of n(s) obtained in section 3 at f = 0. In
this case, P(A) = - À, v === !I s I1/2 and u =F[ e 11/2
Table I.
-
Values of qs(plq) for selected values ofplq
=
for the honeycomb lattice.
1353
lead in general to an eigenvalue problem which is
best expressed in terms of the order parameter values
at the nodes. If node a is linked to n nodes via strands
1 to n), the basic equation, at
of length L,,,# (#
node a may be written as
the continuum approximation (Eq. (3.4))
[2] the linear behaviour in temperature
reprodu-
ces
=
More generally, the variation of the lowest eigenvalue
function of f
41/410 (Fig. 6) determines the effect
of the frustration, induced by the magnetic field, on the
transition temperature Tc (or ground state energy) of
the superconducting transition, treated in mean field
theory. When the flux through one lattice cell is half a
flux quantum, f
1/2 the lattice is effectively fully
frustrated [10, 19]. The above study leads to the prediction of visible dips on the superconducting critical
line, actually observed experimentally [5], at rational
values of f. More pronounced depressions are at
f 1/3. Noticeable dips of smaller amplitudes appears
also at f
2/5 and 1/2.
We conclude by comparing the relative reduction
of the critical temperature Tr on different lattices. For
this we consider only the fully frustrated case f
1/2.
In this case, the total bandwidth is : 4,.,,/-2, 9 and 2 J6
respectively for square (S), triangular (T) and honeycomb (H). These numbers must be compared with the
corresponding widths at f 0 : 8, 9 and 6. A direct
comparison is given by the value of s at the edge for
f 1/2 :
and V6 for S, T and H. This leads to
cos (a/g) = V2/2, 1/2 and J6/2 respectively on these
three lattices. However AT, - l/ç;, therefore
ATe ,(T) &#x3E; AT,,(S) &#x3E;, AT,(H) for superconducting
networks made with the same material and same a.
The frustration is therefore more efficient on triangular lattice than on the two other lattices, as expected
from intuitive expectations. In comparing the square
to the honeycomb lattice, the large number of sides
(6 against 4) in the elementary loop shows that frusas
where A# is the value of the order parameter at nodes
and y,,,p is the circulation of the vector potential along
the strand (Eq. (2.2)) linking a and jS. Çs is the superconducting coherence length.
For a regular network, L,,,, = a is the same for all
strands and equation (5.1) reduces to
=
=
=
=
In equation (5.2), z denotes the coordination number
of the lattice. Thus the problem reduces to the Landau
level spectrum of a tight-binding model in the same
geometry. The energy of the corresponding Bloch elecz. cos (a/çs).
trons (Eq. (2. 3)) is given by : c
The upper critical field of the superconducting
network is given by the edge of the spectrum calcu3
lated in section 4. For a honeycomb lattice, z
and the results are given in table II.
The results obtained in the previous section have a
direct significance for the magnetic behaviour of the
superconducting network. In particular, at low field H,
=
=
Table II.
Edge (s) of the spectrum for rational values
reduced
of the
flux f( = 0/00) p/q with p and q inteto
each other (q , 20).
gers prime
-
=
=
=
2V2, 3
=
Fig.
0 %
’6.
Edge
0/00 1/2.
-
of the spectrum shown in
figure 5, for
1354
tration has less dramatic effect on the honeycomb
lattice. Further comparison between different superconducting networks as well as experimental results
will be discussed elsewhere [20].
6. Conclusion.
The results presented in this paper for an electron
described by a tight-binding honeycomb lattice in the
presence of a magnetic field complete those previously
obtained on square and triangular lattices. Our
results show that the touching of neighbouring subbands, already present in zero field is still present at
finite rational field. The recursive property is also
present. New features of the spectrum were also
obtained. Specific properties (gaps, subbands, ...)
are shown to be a direct consequence of frustration
and are compared with the other lattices. However
two important questions have not been discussed in
detail. The first concerns the gaps and the definition
of a gap index. The value of this index is defined by the
integrated density of states below a gap and is of
direct interest for the quantized Hall effect [21, 22].
The second is related to the nesting hypothesis and
the recursive property of the spectrum. Both of these
questions have been discussed in great detail in the
literature and the analysis can be carried over step by
step from known cases to ours.
Let us conclude by noting that the question of
irrationality is no longer artificial in real materials,
as was pointed out in reference [4] some years ago.
The observation of the predicted features on figures 5
and 6 requires, of course, an enormous magnetic field
of about 106 kOe to let f - 1 for a typical lattice
spacing of the order of 2 A. Nowadays ,experimental
limitations do not allow for magnetic fields higher than
103 kOe, which implies that the influence of the
magnetic field on the motion of the electrons is very
small and the rich spectrum is not seen experimentally
in this regime. As shown in this paper and in references [5 and 20], the superconducting networks
provide a powerful and unique example, where some
of the main features of the spectrum can be observed
in realistic conditions. Furthermore, in these systems,
the frustration induced by the magnetic field enters
alone, and in a controllable way. Continuous variation
of the applied field allows for a fine tuning of the
frustration, that is not easy to achieve otherwise.
To our knowledge, this is the first available system
where a small modification of an external parameter
(y 0/00) may lead to a qualitatively different state
in a predictable and adjustable way.
=
Acknowledgments.
I would like to thank Dr G. Toulouse for useful and
friendly conversations. I would also like to express
my dept to my collaborator Dr J. C. Angles D’Auriac
for valuable help in various ways. Finally, I would
like to thank Professor T. C. Lubensky at the University of Pennsylvania for the hospitality in the Physics
Department. This work was supported in part by the
NSF under grand number DMR 82-19216.
References
[1] ALEXANDER, S., Phys. Rev. B 27 (1983) 1541.
ALEXANDER, S. and HALEVI, E., J. Physique 44 (1983)
805.
[2] RAMMAL, R., LUBENSKY, T. C. and TOULOUSE, G.,
Phys. Rev. B 27 (1983) 2820, J. Physique Lett. 44
(1983) L-65.
[3] For a recent review, see SIMON, B., Adv. Appl. Math.
3 (1982) 463.
[4] HOFSTADTER, D. R., Phys. Rev. B 14 (1976) 2239.
References to prior works will be found in references [6-8] and [14-17] below.
[5] PANNETIER, B., CHAUSSY, J. and RAMMAL, R., J. Physique Lett. 44 (1983) L-853.
PANNETIER, B., CHAUSSY, J., RAMMAL, R. and VILLEGIER, J., Phys. Rev. Lett. 53 (1984) 1845.
[6] CLARO, F., Phys. Status Solidi (b) 104 (1981) K31.
[7] CLARO, F. and WANNIER, G. H., Phys. Rev. B 19 (1979)
6068.
[8] WANNIER, G. H., Phys. Status Solidi (b) 88 (1978) 757.
[9] TOULOUSE, G., Commun. Phys. 2 (1977) 115.
[10] TOULOUSE, G., Proc. of the 1983 Geilo Institute, Frustration, Landau Levels and superconducting diamagnetism on networks, Geilo Inst. Proceeding (Plenum Press), to appear.
V., J. Phys. Chem. Solids 24 (1964) 1495.
HALPERN,
[11]
[12] HORIGUCHI, J. Math. Phys. 13 (1972) 1411. See also
HOBSON, J. P. and NIERENBERG, W. A., Phys. Rev.
(1953) 662.
[13] ONSAGER, L., Philos. Mag. 43 (1952) 1006; see also
WILKINSON, M., Proc. Soc. London A 391 (1984)
89
3051.
[14] ZILBERMAN, G. E., Sov. Phys. JETP 3 (1957) 385.
FISCHBECK, H. J., Phys. Status Solidi 38 (1970) 11.
[15] Hsu, W. Y. and FALICOV, L. M., Phys. Rev. B 13
(1976) 1595.
KAPO, P. S. and BROWN, E., Phys. Rev. B 7 (1973) 3429.
[16] LANGBEIN, D., Phys. Rev. 180 (1969) 633, and references
cited therein.
[17] WANNIER, G. H., OBERMAIR, G. M. and RAY, R.,
Phys. Status Solidi (b) 93 (1979) 337.
[18] THOULESS, D. J., Phys. Rev. B 28 (1983) 4272.
[19] TEITEL, S. and JAYAPRAKASH, C., Phys. Rev. B 27
(1983) 598.
SHIH, W. Y. and STROUD, D., Phys. Rev. B 28 (1983)
6575.
[20] PANNETIER, B., CHAUSSY,
J. and RAMMAL,
R.,
to be
published.
D. J., KOHMOTO, M., NIGHTINGALE, M. P.
and DEN NIJS, M., Phys. Rev. Lett. 49 (1982) 405.
RAMMAL, R., TOULOUSE, G., JACKEL, M. T. and HALPERIN, B. I., Phys. Rev. B 27 (1983) 5142.
[21] THOULESS,
[22]