Effects of the field modulation on the Hofstadter's spectrum

17 Pages·0.73 MB·English

by  Gi-Yeong Oh

Effects of the field modulation on the Hofstadter’s spectrum
∗
Gi-Yeong Oh
Department of Basic Science, Ansung National University,
Kyonggi-do 456-749, Korea
(February 7, 2008)
9
9 We study the effect of spatially modulated magnetic fields on the energy spectrum of a two-
9 dimensional (2D) Bloch electron. Taking into account four kinds of modulated fields and using
1
the method of direct diagonalization of the Hamiltonian matrix, we calculate energy spectra with

n varyingsystem parameters(i.e., thekindofthemodulation,therelativestrengthofthemodulated
a fieldtotheuniformbackgroundfield,andtheperiodofthemodulation)toelucidatethattheenergy
J band structuresensitively dependson such parameters: Inclusion of spatially modulated fields into

8 a uniform field leads occurrence of gap opening, gap closing, band crossing, and band broadening,
1 resulting distinctive energy band structure from the Hofstadter’s spectrum. We also discuss the
effect of the field modulation on the symmetries appeared in the Hofstadter’s spectrum in detail.
]
l
l PACS numbers: 71.28.+d, 71.25.-s, 73.20.Dx, 71.45.Gm
a
h
-
s I. INTRODUCTION cosine-modulated (1DCM) field on a 2D Bloch electron
e
toelucidatethatoccurrenceofgapclosingandgapopen-
m
The problem of a 2D Bloch electron under a uniform ingleadsdifferentenergyspectrumfromtheHofstadter’s
t. magnetic field has been intensively studied for several spectrum and that the symmetries appeared in the Hof-
a
m decades,1 and it is well known that the energy spectrum stadter’s spectrum except the dual property still remain
is characterized by the Hofstadter’s butterfly showing a despite the field modulation. In the meanwhile, Shi and
d- fractalnature.2Recentlytheproblemhasattractedtheo- Szeto26 studied the energy spectrum of a 2D Bloch elec-
n reticallyrenewedinterestinconnectionwithvariousphe- tron under a kind of 2D field modulation to argue that
o nomena such as the quantum Hall effect,3 the flux state there is no symmetry breaking in the energy spectrum
c modelforhigh-T superconductivity,4 andthemean-field andthatthe fractalstructure remainsirrespectiveofthe
c
[ transition temperature of superconducting networks or field modulation.
1 Josephson junction arrays.5 Besides, recent advance in Inthis paper we reexaminethe problemofa 2DBloch
v submicrontechnologythat makesitpossible to fabricate electronunderspatiallymodulatedmagneticfieldstoset-
7 any desired microstructures has stimulated experimen- tle the inconsistency discussed above. In doing this, we
5 tal study to find indication of the Hofstadter’s spectrum take into account four kinds of modulated fields (i.e.,
1 andits effectonthe transportandopticalproperties.6−9 1DSM, 1DCM, 2DSM, and 2DCM fields) in order to ob-
1
In parallel with this problem, the study on 2D elec- tain rather generic effects of the field modulation on the
0
9 tron systems under nonuniform (either disordered10 or Hofstadter’s spectrum. By means of direct diagonaliza-
9 periodic11−26) magnetic fields has been extensively per- tion of the Hamiltonian matrix, we calculate the energy
/ formed, and lots of interesting characteristics in the en- eigenvaluesandexaminehowthesystemparameterssuch
t
a ergy spectral and transport properties have been eluci- asthekindofthemodulation,therelativestrengthofthe
m dated. modulated field to the uniform field, and the period of
- Thoughtheproblemofa2DBlochelectronunderspa- the modulation influence on the energy band structure
d tiallymodulatedmagneticfieldshasattractedlessatten- and the symmetry of the Hofstadter’s spectrum. Intro-
n
tion compared with the problem of 2D electron gas un- duction of the field modulation is shown to change the
o
c der spatially modulated magnetic fields,11−22 it is still ~k dependence ofthe energyspectrumdrastically,leading
: an important problem not only in the view of theoreti- occurrence of gap opening, gap closing, band crossing,
v
calinterestbut also inthe view of experimentalinterest, and band broadening, which are the origin of distinctive
i
X andtherehavebeenattemptstosolvethisproblem.23−26 energy band structure from the Hofstadter’s spectrum.
r However, unfortunately, some of the relevant works con- Our results indicate that there is no additional spider-
a
tain inconsistent results on the energy spectral proper- web structure in the energy spectrum contrary to the
ties: In Ref. 24, Gumbs et al. studied the effect of a result of Ref. 24 and that the field modulation generi-
one-dimensional sine-modulated (1DSM) field on a 2D cally breaks the symmetries and the fractal property of
Blochelectronto arguethatthe symmetries appearedin the Hofstadter’s spectrum.
the Hofstadter’s spectrum break down by the field mod- This paper is organizedas follows: In Sec. II we intro-
ulation. Most surprisingly, they also argued that the duce four kinds of magnetic fields and the tight-binding
field modulation leads an additional crisscross pattern model. And, in Sec. III we present numerical results on
like a spiderweb structure onto the Hofstadter’s spec- the energy band structure of a Bloch electron and the
trum. However, Oh et al.25 studied the effect of a 1D symmetries of the energy spectrum under these fields.
1
Section IV is devoted to a brief summary. 2πm
θ =2πmφ β γ φ cos ,
m 0 x x 0
− (cid:18) γ (cid:19)
x
2πn
II. MODULATED MAGNETIC FIELDS AND THE θn =βyγyφ0cos (8)
(cid:18) γ (cid:19)
TIGHT-BINDING MODEL y
We consideranelectronina 2Dsquarelattice under a for the SM field and
spatially modulated magnetic field
2πm
B~ =[B0+B1(x,y)]zˆ, (1) θm =2πmφ0+βxγxφ0sin(cid:18) γx (cid:19),
2πn
where B0 (B1) denotes the uniform (modulated) part of θn = βyγyφ0sin (9)
anappliedmagneticfield. Amongpossiblekindsofmod- − (cid:18) γy (cid:19)
ulatedfields,wepayattentiontotwokindsofmodulated
fields. The one is the SM field
for the CM field, respectively. Here β = B /B ,
x(y) x(y) 0
2πx 2πy γ =T /a, and φ =B a2, a being the lattice con-
x(y) x(y) 0 0
B (x,y)=B sin +B sin (2)
1 x (cid:18) T (cid:19) y (cid:18) T (cid:19) stant. Themagneticfluxperunitcell,inunitsoftheflux
x y
quantum hc/e, is given by φ=(1/2π) θ = A~ d~l =
ij
·
and the other is the CM field B~ dS~. P H
·
2πx 2πy R By means of Eqs. (6) and (7), the tight-binding equa-
B (x,y)=B cos +B cos . (3)
1 x y tion can be written as
(cid:18) T (cid:19) (cid:18) T (cid:19)
x y
Here Bx(y) is the strength of the modulated field and eiθnψm+1,n+e−iθnψm−1,n+λ eiθmψm,n+1
T is the period of the modulation along the x (y) di-
rexc(tyi)on. Under the Landau gauge, the vector potential +e−iθmψm,n−1 =Eψmn (cid:0) (10)
becomes (cid:1)
Ax = ByTy cos 2πy , twwheeernetλhe(≡xatnyd/tyx)diirsectthieonrsa,taiondofEhiosptphiengenienrtgeygrinalusnbites-
2π (cid:18) T (cid:19)
y
oft . Here,thewavefunctionisgivenby ψ = ψ j .
B T 2πx x | i j j| i
A =B x x x cos (4) Denoting R as the periodicity of θ , tPhe Bloch
y 0 − 2π (cid:18) T (cid:19) x(y) m(n)
x theorem can be written as
for the SM field and
B T 2πy ψm+Rx,n =eikxRxψmn, ψm,n+Ry =eikyRyψmn, (11)
y y
A = sin ,
x
− 2π (cid:18) T (cid:19)
y
andthefirstmagneticBrillouinzone(FMBZ)isgivenby
B T 2πx
x x
A =B x+ sin (5) k π/R . We calculate the energy eigenvalues
y 0 x(y) x(y)
2π (cid:18) T (cid:19) | | ≤
x for all the values of ~k in the FMBZ by directly diago-
for the CM field, respectively. nalizingtheHamiltonianmatrixobtainedfromEqs.(10)
The tight-binding Hamiltonian describing the motion and (11).
of an electron in a magnetic field is given by
H = tijeiθij i j , (6)
− | ih |
Xij III. NUMERICAL RESULTS AND DISCUSSION
where t is the hopping integral between the nearest-
ij
neighboring sites i and j, and θ (2πe/hc) jA~ d~l
ij ≡ i · In what follows, we assume the modulated field has a
is the magnetic phase factor. Under the vectoRr poten-
square-symmetry (i.e., β = β = β and γ = γ = γ)
tialsgivenbyEqs.(4)and(5),themagneticphasefactor x y x y
and consider only the case of the isotropic hopping in-
becomes
tegral (i.e., λ = 1) for the sake of simplicity. And we
′ ′
θ , (m ,n )=(m,n 1) payattentiontothe energydispersionsforq =2,3(with
m
θmn;m′n′ = ±θn , (m′,n′)=(m 1±,n) (7) setting p=1) and γ =2,3,4 because energy dispersions
 ±0 , otherwise ± forothervaluesof(q,γ)canbeobtainedinasimilarway.
Herepandqdenotethenumbers(primeeachother)given

with by φ =p/q.
0
2
A. Energy spectrum in a uniform magnetic field The lattice is called the stripped flux lattice23 and the
energy spectrum can be obtained by diagonalizing the
In a uniform field (β =0), θn becomes zero. Thus, by (Rx×Rx) Hamiltonian matrix.
means of the translational invariance along the y direc-
tion, Eq. (10) reduces to the Harper equation
1. 1DSM Field
ψm+1+ψm−1+2cos(θm+ky)ψm =Eψm, (12)
andthereareq eigenvaluesforagivenvalueofk . When When (q,γ) = (2,2), we have R = 2 and the energy
y x
q isodd,thefullenergyspectrumconsistsofq subbands. dispersion is given by
In the meanwhile, when q is even, the full energy spec-
trum consists of (q − 1) subbands because two central E(~k)=2sinβsinky 2 cos2kx+cos2βcos2ky. (16)
subbandstouchatzeroenergy. Herethefullenergyspec- ± q
trum means a set of energy eigenvalues obtained by tak-
inginto accountallthe valuesof~k inthe FMBZ.Energy The two subbands touch at (kx,ky) = (π/2,π/2) and
(π/2,3π/2), and the energy eigenvalues at the touching
dispersions for q =2 and 3 are given as follows:
points are given by E = 2sinβ. Since the touching
(a) q = 2: We have Rx = 2 and the energy dispersion pointsexistirrespectiveofβ±,thefullenergyspectrumex-
is given by
hibits a single subband structure as in the case of β =0.
In order to demonstrate the γ dependence of the en-
E(~k)= 2 cos2k +cos2k . (13)
± q x y ergyspectrumwithq =2,wecalculateenergyspectrafor
variousvaluesofγ andplotsomeoftheminFig.2. Even
Therearetwosubbandsandtheytouchatzeroenergyas
thoughtherearesixandfoursubbandsfor(q,γ)=(2,3)
shown in Fig. 1(a), resulting a single subband structure
and(2,4),thereoccursdirecttouchingbetweenthenear-
with E 2√2.
| |≤ est neighboring subbands, which leads a single subband
(b) q = 3: We have R = 3 and the energy dispersion
x structureofthefullenergyspectrum. Inthecalculations,
is given by the equation
we find that the energybandstructure is independent of
the values of β and γ while the total bandwidth slightly
E3 6E 2(cos3k +cos3k )=0. (14)
x y
− − changes with varying β.
There are three subbands as shown in Fig. 1(b). But, Figure 3 shows the ky dependence of the energy spec-
there is no touching point in the dispersion and the trum for (q,γ) = (3,2). In this case, we have Rx = 6
full energy spectrum exhibits a three subband struc- and there are six subbands. For small values of β, the
ture with (1+√3) E 2, E (√3 1), and upper(lower)twosubbandstouchatseveralpointsofky.
2 E (1−+√3). ≤ ≤ − | | ≤ − Besides,thereexistsindirectoverlapping(i.e.,crossingof
≤It is≤well known that the Hofstadter’s spectrum has subbands at different values of ky) between the central
subbands. Thus, the fullenergyspectrumstillexhibits a
the following symmetries; (i) the dual property between
threesubbandstructureascanbeseeninFig.3(a). How-
the k and k directions (let us denote it as S ), (ii)
x y D
ever, as β increases, there occurs gap opening between
the symmetry between E and E (S ), (iii) the sym-
E
− the subbands andthe fullenergyspectrumexhibits asix
metry between k andk (S ), and(iv)the symmetry
x x X
− subband structure as in Fig. 3(b). Further increase of β
between k and k (S ). It should be noted that S
y y Y E
holdsonly−whenallthevaluesof~kintheFMBZaretaken makes the second and the fourth gaps closed, leading a
foursubbandstructureasinFig.3(c). Figure4showsthe
into account; for a given value of~k, one can easily check
k dependence ofthe energy spectrum for (q,γ)=(3,3).
y
that S does not hold for odd q.
E In this case, we have R = 3 and there are three sub-
x
bands. For small values of β, the full energy spectrum
exhibits a three subband structure as in Fig. 4(a). How-
B. Energy spectrum in 1D modulated magnetic
ever, indirect overlapping between subbands occurs as β
fields
increases, and the full energy spectrum exhibits a single
subband structure as in Fig. 4(b). Occurrence of gap
When the fieldmodulationis alongthex direction,θ
n closing due to indirect overlapping is also found in the
isstillzeroandthetight-bindingequationisformallythe
case of (q,γ)=(3,4).
same as Eq. (12). However, β and γ are introduced in
As for the symmetry of the energy spectrum, Figs. 2
θ , and φ becomes periodic along the x direction with
m and3showthatS remainsunderthe1DSMfield,which
E
the period γ:
is contrary to the arguments of Ref. 24. The reason of
1+(βγ/2π)[cos(2π(m+1)/γ) this discrepancylies in the rangeof~k takeninto account
cos(2πm/γ)] for 1DSM in discussing the symmetry of the energy spectrum. In
φ/φ0 = 1−(β−γ/si2nπ()2[πsimn/(2γπ)](m+1)fo/rγ)1DCM. (15) cRoeuf.nt2,4wohnilley aallptahreticvuallauresvaolfu~ke ionf ~kthewaFsMtBakZenarienttoakaecn-
−

3
into account in this paper. Here we would like to stress comesnonuniformonlyundertheseconditions. However,
thatallthevaluesof~kintheFMBZshouldbeconsidered the authorsofRef. 24violatedthis condition;they chose
inordertodiscussthesymmetryoftheenergyspectrum. γ =1 in their calculations. But, when γ =1, φ becomes
This is because S of the Hofstadter’s spectrum holds uniformascanbe easilycheckedby Eq.(15)andthe en-
E
only when the all the values of~k in the FMBZ is consid- ergy spectrum should be identical with the Hofstadter’s
ered. WealsofindthatS stillremainsunderthe1DSM spectrumifallthevaluesof~kintheFMBZaretakeninto
X
field while whether S remains or not depends crucially account, which implies that the results of the cited pa-
Y
on q and γ. Note that S breaks down by introducing per, i.e., the appearance of additional crisscross pattern
D
the 1D field modulation. like a spiderweb structure and the symmetry breaking
of the energy spectrum, are erroneous ones coming from
mistakes in choosing the values of γ and in choosing the
2. 1DCM Field values of ~k. Note that the method of diagonalization of
the Hamiltonian matrix used in the present paper or in
the cited paper and the method of the transfer matrix
In orderto test how the energy spectrumis influenced
used in Ref. 25 are equivalent. Actually, Fig. 5(a) in the
on the kind of the field modulation, we also calculate
presentpaper is exactlythe same asFig.2(b)in Ref.25.
the energy spectrum of a 2D Bloch electron under the
Second,thetight-bindingmodelweareconsideringisba-
1DCM field. Figure 5 shows the k dependence of the
y sically a one-band model. Thus, we focus our attention
energy spectrum for (q,γ)=(2,3). In this case,we have
on energy spectra only for the values of β that are not
R =6andtherearesixsubbands. Forsmallvaluesofβ,
x large (i.e., β 1) since there might be interband mixing
the two central subbands directly touch at two points of ≤
between different Landau levels for large values of β.
k andtheupper(lower)twosubbandsindirectlyoverlap
y
witheachother,resultingasinglesubbandstructureasin
Fig. 5(a). However, as β increases, indirect overlapping
C. Energy spectrum in 2D modulated magnetic
between the upper (lower) two subbands disappears and
fields
there occurs gap opening between them. Thus the full
energyspectrumexhibitsathreesubbandstructureasin
When the field modulation is along both lateral di-
Fig.5(b). Furtherincreaseofβmakestheremainingindi-
rections, θ becomes nonzero and φ becomes periodic
rect overlapping disappear and the full energy spectrum n
in both lateral directions with the period γ. The lat-
exhibits a five subband structure as in Fig. 5(c). The β
tice is called the checkerboard flux lattice23 and the
dependence of the energy spectrum for (q,γ) = (2,4) is
energy spectrum can be obtained by diagonalizing the
quite different from (q,γ)=(2,3). In this case, we have
(R R R R ) Hamiltonian matrix.
R = 4 and there are four subbands. Each subband di- x y x y
x ×
rectlytoucheswithitsneighboringsubbandsandthefull
energy spectrum exhibits a single subband structure re-
gardlessof β; the only effect ofβ is to changethe energy 1. 2DSM field
bandwidth.
We also perform similar calculations for the energy When (q,γ) = (2,2), we have (R ,R ) = (2,2) and
x y
spectrum with q = 3. When (q,γ) = (3,3), we have the Hamiltonian matrix becomes
R = 3 and there are three subbands. For small values
x
0 a b 0
ofβ,thethreesubbandsseparateeachother,exhibitinga
∗
a 0 0 c
threesubbandstructureasinthecaseofβ =0. However,  ∗  (17)
b 0 0 d
as β increases, each subband overlaps indirectly with its
 0 c∗ d∗ 0 
neighboring subbands and the gaps between the sub-  
bands become closed, resulting a single subband struc-
ture. We also find similar phenomena for (q,γ)=(3,4). where a = eiβ e−i(β+2ky), b = e−iβ + ei(β−2kx),
In Table 1, we summarize the number of subbands of c = eiβ +e−i−(β+2k−x), and d = e−iβ +ei(β−2ky), respec-
the full energy spectra for the parameters (q,γ,β) we tively. By diagonalizing Eq.(17) we obtain the energy
took into account. As for the symmetry of the energy dispersion as
spectrum, we find that S , S , S remain, while S
E X Y D
breaks down under the 1DCM field. Note that SY de- E(~k)= 2 cosβ cos2kx+cos2ky
pends on the kind of the field modulation; it remains ± (cid:12) q
(cid:12)
(breaks) in the 1DCM (1DSM) field. (cid:12)
(cid:12)sinβ sin2k +sin2k , (18)
Before ending this subsection, we would like to men- ± q x y(cid:12)
(cid:12)
tion two comments. First, we assume γ to be an integer (cid:12)
(cid:12)
for the sake of simplicity even though it can be arbi- whichis plottedin Fig.6. Numbering the four subbands
trary real value. Besides, we consider only the values of in order of lowering energy, one can see that the first
γ 3 (2) inthe caseof the CM (SM) field becauseφ be- (second) and the third (fourth) subbands directly touch
≥
4
at the center of FMBZ regardless of β. There also ex- When(q,γ)=(3,3),thereareninesubbandsandthey
ists bandcrossingbetweenthe secondandthe thirdsub- compose a three subband structure for small values of
bands, which is absent in the case of the 1D field modu- β. However, gap closing occurs with increasing β, and
lation. Equation(18)indicates thatthe fullenergyspec- the three subband structure becomes a single subband
trum exhibits a single subband structure regardless of β structure. Further increase of β makes the full energy
and the total bandwidth changes with varying β. spectrum consist of seven subbands. For (q,γ) = (3,4),
When (q,γ) = (2,3), we have (R ,R ) = (6,3) and we show the β dependence of the energy spectrum in
x y
there are eighteen subbands. However, due to occur- Fig. 8, where concurrent occurrence of gap opening and
rence of direct touching and indirect overlapping be- gap closing can be seen. In this case, there are twenty
tweenneighboringsubbands,thefullenergyspectrumex- four subbands and they compose a three subband struc-
hibits a single subband structure regardless of β. When ture forsmallvaluesofβ. However,asβ increases,there
(q,γ) = (2,4), the situation becomes different. In this occurs band splitting and the full energy spectrum ex-
case, we have (R ,R ) = (4,4) and there are sixteen hibits a seven subband structure. Further increase of β
x y
subbands. For small values of β, due to direct touching makesthefullenergyspectrumconsistofthreesubbands.
between neighboring subbands, the full energy spectrum We summarizein Table2 the number ofsubbands ofthe
exhibits a single subband structure. However,gap open- fullenergyspectrafortheparameters(q,γ,β)takeninto
ingbetweensubbandsoccurswithincreasingβ,resulting account.
a five subband structure of the full energy spectrum. The symmetry of the energy spectrum under the 2D
When (q,γ) = (3,2), we have (R ,R ) = (6,2) and modulated fields is rather complicated than the cases of
x y
there are twelve subbands. In this case, even for small the1Dmodulatedfields. SE remainsforboththe 2DSM
values of β, the full energy spectrum is quite different and 2DCM fields as can be seen in Figs. 6–8. However,
fromthe caseofβ =0; the full energyspectrumexhibits SX andSY dependsensitivelyonthesystemparameters.
a eight subband structure as can be seen in Fig. 7(a). For the parameters considered, we find that SX and SY
And, as β increases, two more gaps are open to yield a remain for the cases of (q,γ) = (2,2),(2,4),(3,2) in the
ten subband structure as in Fig. 7(b). Further increase 2DSMfieldsand(q,γ)=(2,4),(3,3)inthe2DCMfields,
ofβ makesthefullenergyspectrumexhibitatwelvesub- while they break for the cases of (q,γ) = (2,3),(3,4) in
band structure as in Fig. 7(c). When (q,γ)=(3,3), the the 2DSM fields and (q,γ) = (2,3),(3,4) in the 2DCM
effectofβisquitedifferentfromthecaseof(q,γ)=(3,2). fields. In the meanwhile, the energy spectrum in the
Inthiscase,wehave(R ,R )=(3,3)andtherearenine 2DSM field with (q,γ) = (3,3) has a flipped symme-
x y
subbands. Forsmallvaluesofβ,thefullenergyspectrum try with respect to kx(y) = 0. Here the flipped sym-
consists of three subbands as in the case of β =0. How- metry means that the energy spectrum in the range of
ever, there occurs gap closing with increasing β and the 0 kx(y) π/Rx(y) is the same as that in the range
≤ ≤
energyspectrumexhibitsasinglesubbandstructure. We of π/Rx(y) kx(y) 0 when E is replaced by E.
− ≤ ≤ −
also find similar gap closing behavior for (q,γ)=(3,4). SD alsodepends sensitivelyonthe systemparameters;it
holds (breaks) for the values of (q,γ) that make S and
X
S remain (break).
Y
Beforeendingupthissubsection,wewouldliketomen-
2. 2DCM Field
tion a few comments. The first is that the above results
onS arenotfromatheoreticalanalysisbutfrom
E(X,Y,D)
In order to know how the energy spectrum is influ- anumericalstudy. Thus,furtherstudysuchasthegroup
enced by the kind of the 2D field modulation, we also theoretical analysis23,26,27 is required in order to deepen
calculate the energy spectra of a Bloch electron under the understanding of the symmetry breaking. The sec-
the 2DCM field with various values of (q,γ,β). And, we ond is that the k and k directions taken into account
x y
find that occurrence of gap closing, gap opening, band in discussing S are not the high symmetry direc-
X(Y,D)
crossing, and band broadening is a generic effect of the tions of the Hamiltonian under the 2D modulated fields.
field modulation. The reasonfor taking into accountthese directions, nev-
When (q,γ) = (2,3), there are eighteen subbands. ertheless, lies in testing how S appeared in the
X(Y,D)
For small values of β, due to direct touching and indi- Hofstadter’s spectrum is influenced by the field modu-
rectoverlappingbetweentheinnersixteensubbands,the lation. Since symmetry breaking is generally expected if
full energy spectrum exhibits a three subband structure. thesymmetryaxisisnotproperlychosen,ourresultthat
However, there occurs gap opening between the inner the breaking of S depends on the system param-
X(Y,D)
sixteen subbands with increasing β, resulting a five sub- eters is quite natural. Finally, it may be worthwhile to
band structure. We also find that the widths of gaps notetheresultofRef.26,whereShiandSzetoconsidered
broaden with the increase of β. For (q,γ)=(2,4), there a Bloch electron in the magnetic field
aresixteensubbandsandtheycomposeasinglesubband B~ =[B +( 1)m−nB ]zˆ (19)
structurefor smallvaluesofβ. However,with increasing 0 − 1
β,thereoccursgapopeningandthefullenergyspectrum and found that there is no symmetry breaking in the
exhibits a five subband structure. energy spectrum and that the fractal structure remains
5
eventhoughtheenergyspectrumbecomesdifferentfrom ∗ Electronic address: [email protected]
theHofstadter’sspectrum,whichseemstobecontraryto 1R. Peierls, Z. Phys. 80, 763 (1933); P. G. Harper, Proc.
ourresults. However,sincethe directionsusedinRef.26 Phys. Soc. London A 68, 874 (1955); W. Kohn, Phys.
are the highly symmetric (k k ) directions while the Rev. 115, 1460 (1959); M. Ya Azbel, Sov. Phys. JETP
x y
directionsusedinthispaperar±ethek andk directions, 19, 634 (1964); D. Langbein, Phys. Rev. 180, 633 (1969);
x y
itmaynotbe easytocomparebothresultsdirectly. The G. H. Wannier, Phys. Status Solidi B 88, 757 (1978); G.
only thing we can say is that the field modulation given H. Wannier, G. M. Obermaier, and R. Ray, ibid. 93, 337
by in this paper [i.e., Eqs. (1)-(3)] is more generalized (1979); F. H. Claro and G. H. Wannier, Phys. Rev. B 19,
6068 (1979); A. H. MacDonald, ibid. 28, 6713 (1983); J.
than that of Ref. 26 [i.e., Eq. (19)]; specifying the peri-
B. Sokoloff, Phys. Rep. 126, 189 (1985); M-C. Chang and
odic magnetic field by a magnetic unit cell, the lattice
Q. Niu, Phys. Rev. B 53, 7010 (1996); G. Gumbs and P.
with Eq. (19) becomes the simple checkerboard lattice
Fekete,ibid. 56, 3787 (1997).
[see, for example, Fig. 2(a) of Ref. 23] while the lattice
2D.R. Hofstadter, Phys. Rev.B 14, 2239 (1976).
withEqs.(1)-(3)becomesageneralizedcheckerboardlat- 3D.J. Thouless, M. Kohmoto, P.Nightingale, and M. Den.
tice [see Fig. 2(b) of Ref. 23]. Thus we expect that the
Nijs, Phys. Rev. Lett. 49, 405 (1982); R. Rammal, G.
latter may presentmore generic effect of the field modu-
Toulouse, M. T. Jaekel, and B. I. Halperin, Phys. Rev.
lation than the former. Further theoretical study on the
B 27, 5142 (1983); M. Kohmoto, ibid. 39, 11943 (1989);
comparison between two cases is also required.
Y.Hatsugai and M. Kohmoto, ibid.42, 8282 (1990).
4P. W. Anderson, Phys. Scr. T 27, 60 (1989); P. Lederer,
D. Poilblanc, and T. M. Rice, Phys. Rev. Lett. 63, 1519
(1989).
IV. SUMMARY 5S. Alexander, Phys. Rev. B 27, 1541 (1983); R. Rammal,
T. C. Lubensky and G. Toulose, ibid. 27, 2820 (1983); Q.
Niu and F. Nori, ibid.39, 2134 (1989).
6B. Pannetier, J. Chaussy, R. Rammal, and J-C. Villegier,
In summary, we studied the effects of spatially mod- Phys.Rev.Lett. 53, 1845 (1984).
ulated magnetic fields on the Hofstadter’s spectrum. In 7D.Weiss, M.L.Roukes,A.Menshig, P.Grambow, K.von
order to obtain rather general properties of the effect of Klitzing,andG.Wieman,Phys.Rev.Lett.66,2790(1991).
the field modulation, we took into account four kinds of 8A. Lorke, J. P. Kotthaus, and K. Ploog, Phys. Rev. B 44,
modulated fields and calculated energy eigenvalues with 3447 (1991).
varyingq,β,andγ. Occurrenceofgapopening,gapclos- 9R.R.Gerhardts,D.Weiss,andU.Wulf,Phys.Rev.B43,
ing, band crossing, and band broadening was found to 5192(1991);R.R.GerhardtsandD.Pfannkuche,Surf.Sci.
be characteristic features of the field modulation, which 263, 324 (1992).
leads distinctive energy band structure from the Hofs- 10P.A.LeeandD.S.Fisher,Phys.Rev.Lett.47,882(1981);
tadter’s spectrum. It should be noted that these charac- H. Aoki and T. Ando, ibid. 54, 831 (1985); B. Huckestein
teristics are very important since the change of the en- andB.Kramer,ibid.64,1437(1990); Y.AvishiandR.M.
ergy dispersion may have crucial influence on the trans- Redheffer,Phys.Rev. B47, 2089 (1993).
port properties; it may allow an electron to change its 11D.YoshiokaandY.Iye,J.Phys.Soc.Jpn.56,448(1987).
12D.P. Xueand G. Xiao, Phys. Rev.B 45, 5986 (1992).
orbit, in view of the semiclassical language. There is
13F. M. Peeters and P. Vasilopoulos, Phys. Rev. B 47, 1466
also a possibility of quenching the integer quantum Hall
(1993).
behavior and exhibiting normal behavior when the sub-
14X.Wu and S. E. Ulloa, Phys. Rev.B 47, 10028 (1993).
gaps are smeared out. Sensitive dependence of the en-
15G. J. O. Schmidt,Phys. Rev.B 47, 13007 (1993).
ergy spectrum on the system parameters implies that it 16R.B. S.Oakeshott and A.MacKinnon, J. Phys.Condens.
may be difficult to detect direct indication of the Hofs-
Matter 5, 9355 (1993).
tadter’sspectruminexperimentsinceevenatinychange 17R.Yagi and Y. Iye,J. Phys.Soc. Jpn. 62, 1279 (1993).
ofthemagneticfieldleadsmuchcomplicatedenergyband 18P. Schmelcher and D. L. Shepelyansky, Phys. Rev. B 49,
structure. We also studied the effect of the field modu-
7418 (1994).
lation on the symmetries appeared in the Hofstadter’s 19S. M. Stewart and C. Zhong, J. Phys.Condens. Matter 8,
spectrum andillustrated that the symmetries sensitively 6019 (1996).
depend onthe systemparameters. Inthis work,we have 20S.Izawa,S.Katsamoto,A.Endo,andY.Iye,J.Phys.Soc.
paidattentiontotheenergyspectralpropertiesonlywith Jpn.64, 706 (1995).
a rational flux and the isotropic hopping integral. Since 21H. A. Carmona, A. K. Geim, A. Nogaret, P. C. Main, T.
an introduction of hopping anisotropy is known to lead I. Foster, M. Henini, S. P. Beaumont, and M. G. Blamire,
interestingphenomenalikebandbroadening,28itmaybe Phys.Rev.Lett. 74, 3009 (1995).
interesting to study the effect of hopping anisotropy on 22P.D.Ye,D.Weiss,R.R.Gerhardts,M.Seeger,K.vonKl-
the energy spectrum of a 2D Bloch electron under the itzing, K. Eberl, and H. Nickel, Phys. Rev. Lett. 74, 3013
modulated magnetic fields. (1995).
6
23A. Barelli, J. Bellissard, and R. Rammal, J. Phys. (Paris) Fig. 3 Plot of E versus ky in the 1DSM field with
51, 2167 (1990). (q,γ) = (3,2), where (a) β = 0.3, (b) β = 0.6, and
24G. Gumbs, D. Miessein, and D. Huang, Phys. Rev. B 52, (c) β =0.9.
14755 (1995).
25G. Y. Oh, J. Jang, and M. H. Lee, J. Korean Phys. Soc. Fig. 4 Plot of E versus ky in the 1DSM field with
29, 79 (1996); G. Y. Oh and M. H. Lee, Phys. Rev.B 53, (q,γ)=(3,3), where (a) β =0.3 and (b) β =0.9.
1225 (1996).
26Q.W.ShiandK.Y.Szeto,Phys.Rev.B56, 9251 (1997). Fig. 5 Plot of E versus ky in the 1DCM field with
27M.C.ChangandQ.Niu,Phys.Rev.B50,10843(1994);Y. (q,γ) = (2,3), where (a) β = 0.3, (b) β = 0.6, and
(c) β =0.9.
Hatsugai,M.Kohmoto,andY-S.Wu,ibid.53,9697(1996);
M. C. Chang and M. F. Yang,ibid. 57, 13002 (1998). Fig. 6Energydispersioninthe2DSMfieldwith(q,γ)=
28S.N.SunandJ.P.Ralston,Phys.Rev.B44,13603(1991);
(2,2), where (a) β =0.3 and (b) β =0.9.
G.Y.Oh,J.Jang,andM.H.Lee,J.KoreanPhy.Soc.29,
261 (1996). Fig. 7 Plot of E versus k in the 2DSM field with
y
(q,γ) = (3,2), where (a) β = 0.3, (b) β = 0.6, and
(c) β =0.9.
Fig. 8 Plot of E versus k in the 2DCM field with
y
(q,γ)=(3,4), where (a) β =0.3 and (b) β =0.6.
FIGURE CAPTIONS
Fig. 1 Energy dispersion in the uniform magnetic field TABLE CAPTIONS
with (a) q = 2 and (b) q = 3. The horizontal plane is
drawn in units of π.
Table 1Numberofsubbandsofthefullenergyspectrum
under the 1D modulated fields.
Fig.2PlotofEversusk inthe1DSMfieldwithβ =0.3,
y
where (a) (q,γ)=(2,3) and (b) (q,γ)=(2,4). The hor- Table 2Numberofsubbandsofthefullenergyspectrum
izontal axis is drawn in units of π. under the 2D modulated fields.
7
4 (a)
E
-4
0
kx
1 2
0 ky
4 (b)
E
-4
0
kx
1
2
0 ky
Fig.1 : Gi-Yeong Oh
kind 1DSM 1DCM
q
2 3 2 3
(cid:13)
2 3 4 2 3 4 3 4 3 4
0.3 1 1 1 3 3 3 1 1 3 3
(cid:12) 0.6 1 1 1 6 1 1 3 1 1 1
0.9 1 1 1 4 1 1 5 1 1 1
Table 1 : Gi-Yeong Oh
4
(a)
E
-4
4
(b)
E
-4
0 2
ky
Fig.2 : Gi-Yeong Oh