Landau level spectrum of Bloch electrons in a honeycomb lattice

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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�. �jpa-00210078�

1345

Landau level spectrum ^of Bloch electrons in a honeycomb ^lattice

R. Rammal

Centre de Recherches

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)

Résumé.

On analyse ^le spectre d’énergie ^du modèle des liaisons fortes,

^réseau

^nid d’abeilles,

présence

d’un champ magnétique uniforme. Le graphe ^du spectre ^est montré pour différents flux réduits rationnels à ^travers les cellules élémentaires. On montre que

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

propriété de frustration et sont comparées

celles des réseaux de Bravais. Nos résultats ont

une

implication directe pour les

récentes du champ critique supérieur d’un réseau supraconducteur

nid d’abeilles. Nous comparons aussi la ^structure du bord du spectre dans les trois réseaux : carré, triangulaire ^et

nid d’abeilles.

Abstract.

The energy spectrum ^of

tight-binding honeycomb lattice in the presence of

uniform magnetic

field is analysed. ^The graph ^{of the} spectrum

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

the square and triangular lattices. New features of the spectrum

also obtained. Specific properties (gaps, subbands, etc.)

^{shown to be}

direct consequence of frustration and

compared ^{with the} spectrum ^of Bravais lattices. Our results

shown to be relevant for the recent measurements of the upper critical field of

superconducting honeycomb network. A comparison ^{of the structure of the} edge ^{of the} spectrum

square,

triangular ^and honeycomb lattices is also outlined.

J. Physique ⁴⁶ (1985) ^{1345-1354 AOÛT} 1985,

Classification

Physics ^Abstracts

71.10

71.25

74.10

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 experi- mentally [5] by ^{a direct} measurement of the critical temperature ^{of a} regular ^two dimensional _super-

conducting ^{network. The observed} well-defined struc- tures 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 ^lat-

tice [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 non-}

Bravais 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 frustra- tion [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 incommen- surate 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>14>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

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 ^net-

works [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 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 a/, ^{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

hexagon ^formed by ^the appropriate perpendicular

bisectors. Particular points of relative importance, lying ^{at the} edge of the first Brillouin zone, ^are shown

on figure ^{lb :} F(2 n/3a, 0) ^and respectively.

As a model for the honeycomb ^{lattice, we consider}

an stight-binding model with nearest-neighbour ^over- lap only. ^If cp(r) ^{is used to} denote the wave function

amplitude ^{on site r, then the} Schrodinger equation

reduces to a finite-difference equation

where

denotes the associated eigenvalue, ^{r’ denotes}

nearest neighbours ^{of r and V the} strength ^{of the over-} lap integral. ^{In the} following, ^{we take V}

^{1 for} simplicity.

Fig. ^1.

(a) ^The honeycomb ^lattice made up of hexagonal

cells of side

A(x - a, y) ^and B(x, y) ^{denote two atoms in}

the unit cell. (b) ^{The first Brillouin}

^{in the} reciprocal space k

(kX’ ky). Particular points lying ^{at the} edge ^{of the}

are

shown : F(2 n/3a, 0) ^and K(2 nl3a, ± 2 7r/3 a -VI-3).

In the presence of ^a uniform magnetic ^{field H, per-}

pendicular ^{to the} planar lattice, ^{the above} equation ^is

modified by phase ^factors depending ^{on the} magnetic

field strength. ^Let _Yap ^{be the} phase factor between atoms

and fl :

A being ^{the vector} potential, equation (2. 1) ^{becomes :}

As shown in section 4, ^{the whole} spectrum ^of equa- tion (2. 3) depends only ^{on the} reduced flux f

0/00, giving ^the ratio of the magnetic ^flux through ^{an elemen-}

tary hexagonal cell I(o

⁶ Ha’,13-14) ^{to the flux}

quantum (00

hcle ^for electrons).

3. Spectrum ^{at zero and low} magnetic ^fields.

3.1 ZERO FIELD SPECTRUM.

In the absence of the

magnetic field, ^the dispersion relation is given by :

where k

(k, ky) ^{is the wave vector} associated to the

periodic ^solutions CPa

tfa exp(ikx. x ⁺ iky.Y) ^{of equa-}

tion (2.1), (a

A, B).

The spectrum (two bands) ^{is confined in the interval} [ - 3, + 3] ^{and the} edges ^are reached for k

0.

Close to this point ^{we have two} parabolic ^{bands and}

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

⁰ (point K) ^are

shown on figure ^2. Similarly ^the associated eigen-

states 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

Here is the

argument ^{of the} complete elliptic integral K(k) ^of

first kind.

Note that in addition to the Van Hove’s singulari-

ties (at

± 1, ± 3), ^{which are} typical ^{of two}

dimensional lattices, ^the density ^{of states} n(E) ^vanishes

at

0 where a non-analytical singularity appears

The origin ^{of this} singularity ^at

0 may be traced 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

semiclassical theory (Onsager quantization ^scheme [13]) ^{or a continuum} approxi-

mation [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 singu- larities, ^but it will be shown to break down near the

non-analytic singularity ^{s = 0.}

3.2.1 Onsager’s quantization ^scheme.

^{Near the}

band edges

^{± 3, the} quantization ^rule gives ^the following Landau levels :

where _y

2 noloo ^{and n is a} positive integer (n > 0).

As shown in figure 4, equation (3.4) reproduces quantitatively ^the Landau level structure near the band edges.

Fig. ^2.

Degenerate eigenmodes associated with the energy level

0 at

magnetic field. Here

and b denote two

arbitrary numbers and w is the complex conjugate ^of

m ⁼

exp(2 in/3).

Fig. ^3.

Eigenmodes associated with

+ 1 and - 1 at

zero

magnetic ^field.

Similarly, ^{near the} singular point ^{s = 0, one obtains}

This result, implying ^a square-root departure ^of

^{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 + 1 2 ^{must be} replaced by

When this correction is taken into account, equa- tion (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 gauge A

H(O, x, 0).

Let

consider first the ^case of the band edge

^3.

Taking ^{into account the} translation symmetry ⁱⁿ

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

Close to

3, ^{one can choose} ky

^{0 and look at} solutions of equation (3.6) ^where t/J A

t/JB

0. Putting

u ⁼ x -

a/2, ^{a more} 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 :

Fig. ^4.

Energy spectrum ^{for broadened} Landau levels.

Only low index levels

shown and the fine structure corres-

ponds ^to just

few rational values of the reduced flux 0/00.

Solid lines correspond ^{to Landau levels} starting ^at

^field

from

3 and

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

0, ^{we use} equation (3.6),

written in variable

al2 ^{and for} ky

corresponding ^{to the} point (kx

0, ky) of the first Brillouin zone. In the limit *y ^{1, we} expand ^{the sys-}

tem as above and this yields

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

^{0. This is a}

consequence of the lattice structure 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 para- meter 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 ^beha-

viour. Other such touchings ^{of bands can be observed}

on the central horizontal axis. The case of the triangu-

lar 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.

1349

As was shown in section 2, ^{a band} touching ^occurs already ^{in the} honeycomb lattice in the absence of ^a

magnetic ^{field, at}

^{0. A natural} question ^{arises :}

to what extent do the above properties ^of band touch-

ings ^{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.

4.1 REDUCTION TO ONE DIMENSION PROBLEM.

As

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 tight- binding equations (Eq. (2. 3)) ^{for sites} B(x;y), A(x - a, y)

and A(x ^{+ a/2, y ± a} v’3/2), ^{one obtains}

^system

of four eigenvalue equations. Eliminating ^{the A sublat-}

tice sites, ^the following equation ^{for B site is obtained}

In equation (4.1), Fi(l ⁱ 6) ^denote phase ^factors, resulting ^{from the} elimination of A sites. For the choosen gauge, they ^are given by ^the following expressions

It is reasonable to assume plane ^wave behaviour in the y direction, ^{since the} coefficients in the above equation only ^involve

^{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

where t/lm

OB(x) ^at

^{m(3 a/2), A}

^{B2 - 3 and}

ky a,J3/2. ^{Here m denotes an} integer, and y

2n4>/4>o

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

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

-

6 and + 6, ^{i.e. - 3 6 .- + 3. A close} inspection

of equation (4.5) shows that the spectrum ^is unchanged

under reflections

for a given ^{value of y. In} addition, the spectrum is invariant under translations :

f - f ^{+ n, n} integer and exhibits a reflection _symme- try about half integer ^{values of} f. ^{All these} symmetry

properties ^{are also} present ^on the square lattice spectrum, reflecting translation and gauge inva- riance.

4.2 RATIONAL FIELDS.

Because of the symmetries

in the spectrum, ^we shall limit ^our discussion to

0 f 1/2. ^For rational values p/q of f (p, q ^inte-

gers 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),

1, 2,

q.

where 01

^{2 x and} y

2 nplq, Indeed, ^{one can} easily verify ^that g(u) ^and g(q ⁺ u) obey ^{the same} equation.

If

use Floquet’s ^{theorem to construct these two functions} through the relation

a 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

Expanding this secular determinant in terms of

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

Similarly, ^{the term} involving (Jl only ^is given by

Therefore the equation for A that results has the

following ^form

1351

where Pq(À) ^{is a} polynomial ^of degree q ^{in Å not} containing 01 ^and 02- W(01, 82) ^is given by

When 01 ^and 02 ^{cover their} range, W «(J l’ 02) ^varies

between W 1

^{3 and} W2

^{+ 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

of the secular equation

Furthermore, ^A

^{3 is a zero for this} equation ^for

all _p and _q. This solution corresponds ^{to a band} touching ^at

^{0 in the} spectrum. ^More generally, the q ^{bands in} variable A transform into 2 q ^bands

with a reflection symmetry ^about

^0.

Some examples ^of polynomials P,(A) ^are given ⁱⁿ

what follows, ^for simple ^values of f.

Intercepts ^of P,,(A) ^{with W 1 and W2 give} the subband

edges. ^More precisely ^the edges ^{are fixed} by ^{the fol-} lowing ^{two conditions}

and

For instance, ^if p

1 and q

2, equation (4.1b) yields ^the following ^four symmetrical ^{bands :}

,/3 ^{8 ! [} /6, ^{and 0 s [ %} /3 touching ^at

s ⁼

0 and

+..,/3-. ^For small values of _q, the cal- culations can be achieved analytically up to q

^5.

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

calculations at q

²⁰ because the band widths became

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 8=3 and

0 are clearly recognizable, ^{their field} 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 dis- cussion of the nesting hypothesis (recursiveness ^pro- perty) 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

tight-binding ^model

honey-

comb lattice in

perpendicular magnetic ^{field. The} eigen-

value _{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

represented with q ^20.

4.3 DENSITY OF STATES. - The multiplicity ^{of the}

solutions of equation (4.5) ^{can be obtained in}

straight-forward ^{manner since the} phases 01 ^and 02

appear in the constant term only. ^{For this we follow}

the method proposed in references [7] ^and [17]. ^The

main idea is to transform the counting

^an energy range ^to

counting ^{over a} range of the polynomial

Pq(A), ^{thus yielding}

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

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 remain-

ing integration ^{can be} performed ^{in terms of} complete elliptic integrals. Using A = E2 - 3, ^one gets

where

and

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 singula- rity ^{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

I1/2 ^and u =F [

11/2

coincides with argument k ^{of the} elliptic integral given

in equation (3.2).

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

denotes a constant of proportionality ^which asympto- tically 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-simi-

larity ^{of the} diagram of energy bands ^{as a} function

of f.

Let us 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 expec-

ted, but it does not appear that data can be extrapo- lated in a reliable manner. The sequence p

2,

q

9, 11, 13, 15, 17, ¹⁹ suggests ^the limiting ^value 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

5. 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 Ginz-

burg-Landau equation, ^{which is} essentially equivalent

to the Schrodinger equation. ^For superconducting application, ^{one is} only interested in the lowest eigen-

value (edge ^{of the} spectrum). Note that the same equa- tions are involved in the mean field theory of Josephson- junction 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

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

is linked to

nodes via strands of length L,,,# ^(#

^{1 to n),} the basic equation, ^at

node

may be written as

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

^and jS. Çs ^{is the} super-

conducting ^coherence length.

For a regular ^network, L,,,,

^{is the same for all}

strands and equation (5.1) ^{reduces to}

In equation (5.2),

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 elec- trons (Eq. (2. 3)) ^is given by :

^{z. cos} (a/çs).

The upper critical field of the superconducting

network is given by ^the edge ^{of the} spectrum ^calcu- lated in section 4. For ^a honeycomb lattice,

³

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

of the ^reduced flux f( = 0/00)

p/q ^with p and _{q inte-} gers prime ^to each other (q , 20).

the continuum approximation (Eq. (3.4)) reprodu-

ces [2] ^the linear behaviour in temperature

More generally, ^the variation of the lowest eigenvalue

as 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 pre- diction 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

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 honey-

comb (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

at the edge ^for f

1/2 : 2V2, 3 ^{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) > AT,,(S) >, AT,(H) ^for superconducting

networks made with the same material and same

The frustration is therefore more efficient on triangu-

lar 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 frus-}

Fig. ^’6.

Edge ^{of the} spectrum ^{shown in} figure 5, ^for

0 % 0/00 1/2.

tration has less dramatic effect ^on the honeycomb

lattice. Further comparison between different _super-

conducting ^{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 ^sub- bands, 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 refe-

rences [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 Univer-

sity ^of Pennsylvania ^{for the} hospitality ^{in the} Physics Department. ^{This work was} supported ⁱⁿ part by ^the

NSF under grand number DMR 82-19216.

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