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Formation of new fermi surfaces in 3D crystals at ultra-high magnetic field with different orientations
27 March 2000
Physics Letters A 267 Ž2000. 408–415 www.elsevier.nlrlocaterphysleta
Formation of new fermi surfaces in 3D crystals at ultra-high magnetic field with different orientations V.Ya. Demikhovskii ) , A.A. Perov, D.V. Khomitsky Nizhny NoÕgorod State UniÕersity, Gagarin AÕe., 23, 603600 Nizhny NoÕgorod, Russia Received 4 October 1999; received in revised form 1 February 2000; accepted 6 February 2000 Communicated by A.R. Bishop
Abstract In the tight-binding approximation the Harper like equation describing an electron in 3D crystal subject to a uniform magnetic field is obtained. It is supposed that the vector H can be oriented along several directions in the lattice. The Fermi surfaces relevant to a magnetic flux prq s 1r2 in a simple cubic lattice are built. The quantization rules in magnetic fields slightly distinguished from prq s 1r2 are investigated. q 2000 Elsevier Science B.V. All rights reserved. PACS: 71.18.q y; 71.20.-b Keywords: Fermi surfaces; Energy band structure; Ultra-high magnetic field
The effects of a high magnetic field on Bloch electrons are fascinating problem with very rich physics. This is not surprising since there are two fundamental periods in the problem – the period of the potential and the period of the phase of the wave function – and the interplay of these two periods gives a very interesting spectrum ŽHofstadter butterfly. and eigenstate structure. Starting from classical works of Azbel w1x, Hofstadter w2x, Wannier w3x Žsee also w4–7x. this problem attracts the increasing attention. During the last decade the one electron quantum states in 2D lateral superlattices in the presence of perpendicular magnetic field have been studied in ) Corresponding author. Mailing address: M. Gorky St., 156-3, 603006 Nizhny Novgorod, Russia. Fax: q7-8-312-658592. E-mail address: [email protected] ŽV.Y. Demikhovskii..
a series of theoretical w8–10x and experimental w11,12x works. The problem of flux states in 3D crystals were also the subject of investigations w13,14x. In these papers energy spectrum and density of states were calculated in the tight-binding approximation for several directions of the magnetic field in a simple cubic lattice. However, the translational properties of electron wave function and the geometry of Fermi surfaces for different orientations of the ultrahigh magnetic field have not yet been studied. It is well-known that the Fermi surface shape defines both single electron dynamics and kinetic properties of an electron gas. Nowadays when in VNIIEF ŽSarov. the magnetic field up to 28 MGs is reached w15x the idea of observation of details forecasted by the theory in the crystals with the lattice spacing of a s 0.6 nm and more does not seem to be unaccessible.
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 1 2 6 - 2
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
In this paper we shall consider electronic states in 3D crystals subjected to ultrahigh magnetic field. We suppose that the vector H is parallel to arbitrary translation vector of a lattice. In a tight-binding approximation the wave function satisfying the generalized Bloch–Peierls conditions is obtained and Harper like equation for different orientations of a magnetic field is derived. As an example, the Fermi surfaces for magnetic subbands relevant to rational value of a magnetic flux prq s 1r2 are built. The quantization rules for magnetic flux near this rational value of flux quanta are studied. First of all in the tight-binding approximation let us derive Harper’s equation for the simple cubic lattice subject to uniform magnetic field. To be specific let the basis vectors on the x, y plane be a1 and a 2 , and the magnetic field is applied along lattice basis vector a 3 I 0 z Ž< a1 < s < a 2 < s < a 3 < s a.. Let vector potential is be chosen in Landau gauge A s Ž0, H x,0.. Then electron wave function must satisfy the generalized Bloch conditions ŽPeierls conditions. w16x
c k Ž r . s c k Ž x q qa, y q a, z q a . =exp Ž yik x qa . exp Ž yik y a . =exp Ž yik z a . exp Ž y2p ipyra . ,
Ž 1.
due to the fact that the vector potential is not a periodic function of the coordinates. Here p and q are mutually prime integers which define the number of magnetic flux quanta per two-dimensional square elementary cell
F
p s q
s
F0
< e < H < w a1 = a 2 x < 2p "c
,
Ž 2.
where F 0 s hcr< e < is the magnetic flux quanta. In a presence of a magnetic field the electron quantum states must be classified in accordance with irreducible representations of magnetic translation group a mag s nqa1 q ma 2 q la 3 ,
Ž n,m,l are integer numbers. The quasimomentum k takes values in the magnetic Brillouin zone p p y FkxF , qa qa
p
p FkyF
y a
p y a
409
,
a
p FkzF
a
.
Ž 3.
In the tight-binding approximation the electron wave function which satisfies the conditions Ž1. can be written in the form w17x
ck Ž r . s
Ý
g n Ž k . exp Ž ika n .
n, m ,l
p Ž y y ma .
ž
=exp y2p i
q
a
n
/
=c 0 Ž x y na, y y ma, z y la .
Ž 4.
where c 0 Ž r y a n . is the atomic function. The pathdependent geometric phase 2p Ž p q.w Ž y y ma . an x of the wave function is known to play a fundamental role in the problem. By substituting of Ž4. into Schrodinger equation ¨ and evaluating the transfer integrals between neighbour sites we shall obtain a system of difference equations for coefficients g n . The magnetic field has a non-trivial influence on the transfer integrals between neighbour lattice sites in the x direction
ž
A s exp "2p i
H
p y y ma q
a
/
c 0 Ž r y a nX .
= Ž V Ž r . y U Ž r y a n . . c 0 Ž r y a n . dt ,
Ž 5.
where a nX s Ž Ž n " 1 . a,ma,la . , V Ž r . is the crystal scalar potential with periodicity a in three dimensions and U is the atomic potential. Calculating the transfer integrals in the mean-value approximation we substitute y s ma and obtain A s E0 where E0 is the transfer integral Ž5. in the absence of a magnetic field. In the y direction the transfer integral is
ž
B s exp "2p i
p q
n
/H
c 0 Ž r y a nX .
= Ž V Ž r . y U Ž r y a n . . c 0 Ž r y a n . dt ,
Ž 6.
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
410
where a nX s Ž na, Ž m " 1 . a,la . and in the z direction the transfer integral is equal to E0 . As a result we obtain the well-known Harper’s equation w4x
one can obtain the following expressions for dispersion laws for q magnetic subbands: for prq s 1r2 Žtwo subbands.
exp Ž ik x a . g nq1 q exp Ž yik x a . g ny1
Ž1,2. ´H Ž k x ,k y . s "2 cos 2 k x a q cos 2 k y a ,
ž
q 2 g n cos 2p n
p q
(
Ž 8.
for prq s 1r3 Žthree subbands.
/
q k y a s ´ Ž k x ,k y ,k z . g n ,
Ž 7.
Ž1. ´H Ž k x ,k y . s y2'2 cos
where
ž
2p 3
/
q Ž arctan b . r3 ,
p
´ Ž k x ,k y ,k z . s ´ H Ž k x ,k y . y 2cos k z a
Ž 7’ .
is the dimensionless energy measured in terms of E0 . The spectrum ´ H Ž k x ,k y . in Eq. Ž7. consists of q bands and depend on one single parameter prq counting the number of magnetic flux quanta per unit cell ŽFig. 1.. If flux is irrational the spectrum is a singular-continuum – uncountable but measure zero set of points ŽCantor set.. Originally the Eq. Ž7. was derived by Harper w4x using Peierls substitution w16x. For some values of prq s 1r2, 1r3, 1r4 it is possible to solve the Eq. Ž7. analytically. As a result
Ž2. ´H Ž k x ,k y . s 2'2 cos
ž
3
/
q Ž arctan b . r3 ,
Ž3. ´H Ž k x ,k y . s y2'2 cos Ž Ž arctan b . r3. ,
Ž 9.
where b s '8 y a 2 ra , a s cos3k x a q cos3k y a; and for prq s 1r4 Žfour subbands. Ž1,2,3,4. ´H Ž k x ,k y .
( (
s " 4 " 2 4 y sin2 2 k x a y sin2 2 k y a .
Ž 10 .
Fig. 2 shows the Fermi surfaces in the lowest subband for prq s 1r2 obtained using Ž7’. and Ž8.
Fig. 1. Parts of the classical Hofstadter ‘butterfly’ near prq s 0, 1r3, 1r2.
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
411
Fig. 2. The Fermi surfaces for the lower Ža. and upper Žb. magnetic subbands of 3D crystals with simple cubic lattice at prq s 1r2, H I a 3 . Ža.: 1. ´ s y4.3; 2. ´ s y2.0; 3. ´ s 0.0; Žb.: 1. ´ s 4.3; 2. ´ s 2.0; 3. ´ s 0.0.
in the first Brillouin zone yprqaF k x F prqa,y prqa F k y F prqa,y pra F k z F pra. Three surfaces for different representative energies are plotted. For an ellipsoid-type surface such as Ž1. Ž< k < 0. the simple analytical expression for an energy spectrum ´ Žk ) can be easily obtained with the help of Ž7’. and Ž8.. The relevant effective masses are determined as
™
"2 m )x , y s
'2 E0 a2
"2 , m )z s
2 E0 a 2
.
Ž 11 .
™
The asymptotic spectrum near the point k x , y pr2 a ŽŽ2.-type surface. is ´ Ž k x ,k y ,k z . ; k 2x q k 2y y 2cos k z .
(
Now we shall discuss the magnetic quantization rules at values of a flux slightly distinguished from prq s 1r2: p
1 s
q
1 q
2
qX
,
qX 4 1.
Ž 12 .
This corresponds to the magnetic field H s H Ž1r2. q D H Ž1rqX .. As one can see from Fig. 1 near value of prq s 1r2 the spectrum represents a system of narrow subbands Žpractically discrete levels.. Here it is possible to see an equidistant spectrum, points of level accumulation and ranges where ´ H ; 'N Ž N is the level number.. The spectrum ´ H Ž k x ,k y . of Harper’s equation near one half of flux quanta can be studied analyti-
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
412
cally due to the self-similarity of its structure. In the case prq s 1r2 two subbands and two subsystems in the eigenvector distribution can be observed. It may be derived analytically from Harper’s equation that every eigenvector function from a subsystem has an argument step D n s 2. The magnetic flux Ž12. gives us the diagonal term in Harper’s equation of the form 2cosŽp n q 2p nrqX . g n s 2cosp ncosŽ2p nrqX . g n . Now qX is a new period of the equation. Since cosp n s Žy1. n our system of Eqs. Ž7. consists of two subsystems where only the sign varies. It is easy to construct an equation for each subsystem. After simple algebra we obtain Ž k x s k y s 0.: g nq2 q g ny2 q 2cos
4p qX
2 n gn s Ž ´ H y4 . g n .
Ž 13 .
2 Now ´ H y4 s ´ XH is the energy in Harper’s Eq. Ž7. and ´ H s " ´ XH q 4 describes the structure of the spectrum near one half of flux quanta and cos k z a s 0. Two signs at the square root correspond to two groups of energy subbands. The energy ´ H describes all parts of the spectrum near prq s 1r2 ŽFig. 1.. We can obtain from Ž13. the ‘square root’ dependence at ´ H s 0, the clustering points ´ H s "2 and Landau levels near ´ H s "2'2 . These peculiarities of a spectrum can be obtained with the help of Onsager–Lifshitz quasiclassical quantization rules on the Fermi surfaces ŽFig. 2a,b.. It is easy to see that magnetic quasiclassical quantization on the Ž1.-type surface results to the equidistant energy spectrum at the bottom of the lowest magnetic subband. Accordingly, the magnetic Onsager–Lifshitz quantization near k z 0 on Ž2.-type surface yields the ‘square root’ spectrum. At last, it is clear that in magnetic field H I 0 z there are self-crossing orbits on Ž3.-type surface. It gives the accumulation point in the spectrum ´ H near prq s 1r2 Žsee Fig. 1.. It can be stressed that level spacing obtained from Ž13. has the same value that was derived using quasiclassical quantization. In accordance with Harper’s Eq. Ž7. and Ž7’. the partial overlapping between q energy subbands is observed. When prq s 1r2 the overlapping takes place in the energy interval Žy2, 2.. It is clear that when the electron concentrations leading to ´ F g Žy2, 2. in the magnetic field H s H Ž1r2. q
(
™
D H Ž1rqX . the de Haas–van Alfen oscillations with different periods corresponding to the different energy level series ŽFig. 1. can be observed. Further it is necessary to make the following relevant note. The wave function Ž4. describes nonhomogeneous probability distribution on the lattice sites. This is due to the amplitudes g nŽ k . which give us non-uniform shapes. However, in real crystals due to Coloumb interaction the static electron density distribution must maintain its symmetry even in the presence of magnetic field. Fortunately, the spectrum of Harper’s equation is q-fold degenerate since ´ Ž k x ,k y q 2p jrqa,k z . s ´ Ž k x ,k y ,k z ., j s 0,1,2, . . . ,q y 1 and a wave function with homogenuous electron density distribution can be written as a combination of Ž4.. Let us introduce qy1
c kX Ž r . s
Ý C j Ž k . Kˆ j Tˆja c k Ž r . ,
Ž 14 .
js0
where c k Ž r . is from Ž4., C j Ž k . s expŽyik x ja., Tˆja c s c Ž x q ja, y, z .expŽ iefr"c . Ž f s yHyja., the operator Kˆ j transforms c k with k s Ž k x ,k y ,k z . into c k X with kX s Ž k x ,k y y 2p pjrqa,k z .. After this procedure we can write Ž14. in the form
c kX Ž r . s D Ž k . Ý exp y2p i n
ž
p Ž y y ma . q
=exp Ž ika n . c 0 Ž r y a n . ,
a
n
/ Ž 15 .
1 Ž . where DŽ k . s Ý qy js0 g nqj k does not depend on n due to the periodicity of the system. Now Ž15. describes homogeneous electron density distribution on the lattice sites and corresponds to the same energy as c k . Let us consider now the case of a simple cubic lattice subjected to a uniform magnetic field oriented along the diagonal of a square in the plane Ž x, y .. The magnetic field with amplitude H has now the following Cartesian coordinates:
Hs
H
'2 Ž 1,1,0 . .
Ž 16 .
It is convenient to choose a new coordinate system making a rotation about the old z axis by pr4 and
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
413
Fig. 3. a., b. The parts of Fermi surfaces bounded by planes k 3 s "p'2 r3a Ž l s 2. at prq s 1r2, H I a1 q a 2 for the upper subband; c. the energy spectrum ´ H Ž k 1 ,k 2 ,k 3 s p'2 r3a. at prq s 31r60.
defining the magnetic field orientation as x 3 . This transformation is written as following: x 3 y x1 x 3 q x1 xs , ys '2 '2 , z s x 2 . We choose a new elementary cell: it is a rectangular parallelepiped based on the new coordinate system vectors: a square with the side '2 a at the base in the plane Ž x 1 , x 3 . and the height a. This cell is non-primitive: it consists of two atoms Žadditional sites are located at the centers of top and bottom faces.. In this coordinate system the magnetic field is oriented along one of the elementary cell basic vectors Žnamely, a 3 .. According to the general principles in this case the classification of electron states is possible w18,19x. Now the first Brillouin zone has an area two times less with respect to the initial value. The magnetic field in this coordinate system is written: H s H Ž 0,0,1 .
Ž 17 .
and it is convenient to choose a vector potential using Landau gauge: A s Ž yH x 2 ,0,0 . .
Ž 18 .
The wavefunction in the tight-binding approximation can be written as following:
½ ž
c k Ž r . s Ý g n Ž k . exp 2p i n
p x 1 y m'2 a q
ar'2
n
/
=exp Ž ika n . c 0 Ž r y a n .
ž
qexp 2p i
p x 1 y Ž m q 1r2 . '2 a q
ar'2
n
/
5
=exp Ž ik Ž a n q d . . c 0 Ž r y Ž a n q d . . .
Ž 19 .
414
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
Here ka n s k 1'2 am q k 2 an q k 3'2 al, c 0 is an isolated atomic function in the presence of the magnetic field, the amplitudes g nŽ k . are the subject of further research and g nq q s g n . The indexes Ž m,n,l . are taking all integer values independently, n numerates sites along x 2 axis, perpendicular to the magnetic field. The sum is taken over all lattice sites and two addendums in Ž22. correspond to the two existing crystalline subsystems. Each of them is a simple tetragonal lattice with the elementary cell being a rectangular parallelepiped with square of the side '2 a in its base and the height a. These subsystems are displaced on the vector d with new coordinates Ž ar '2 ,0,ar '2 .. The magnetic translation group is constructed by the q - time multiplication of the translation period along x 2 axis: x 2 x 2 q qa. The other translations are not changed. Their periods in the new coordinate system are '2 a which is the diagonal of a square with the side a. The classifica-
™
tion of wavefunctions is possible when the number of flux quanta FrF 0 through the area a 2r '2 Žbeing the minimal inter-site area in the transversal to magnetic field projection. is a rational number: FrF 0 s prq where p and q are mutually prime integers. The substitution of Ž19. into Schrodinger equation ¨ is performed and the transfer integrals are calculated. We will have the equation for the energy ´ and the amplitudes g nŽ k .: e i k 2 a g nq 1 q eyi k 2 a g ny1 q 4cos
ž
= cos 2p
p q
nq
k1 a
'2
/
k3 a
ž' / 2
gn s ´ gn .
Ž 20 .
™
It should be mentioned that this equation may be obtained using Peierls substitution k kˆ y eA "c with the initial spectrum at null magnetic field. In old
Fig. 4. a., b. The Fermi surfaces in the first Brillouin zone at prq s 1r2 and l s 0 Ž H I a1 q a 2 . for the upper subband; c. the energy spectrum ´ H Ž k 1 ,k 2 ,k 3 s pra'2 . at prq s 31r60.
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
coordinates it was written as ´ s 2Žcos k x a q cos k y a q cos k z a ., and in new coordinates Ž x 1 , x 2 , x 3 . it is
´ s 2cos k 2 a q 4cos
k3 a
'2
cos
k1 a
'2
.
Eq. Ž20. is a generalization of Harper’s equation for the case of magnetic field parallel to a diagonal of a square which is a face of the elementary cell cube. Keeping a three-diagonal structure Ž20. has some remarkable features with respect to standard Harper’s equation. First, Ž20. describes total Žnot ‘transversal’ ´ H Ž k 1 ,k 3 .. energy as a function of quasimomentum k: ´ s ´ Ž k 1 ,k 2 ,k 3 .. Now k 3 s const. corresponds to the plane perpendicular to H and k 1 s const. describes the plane parallel to magnetic field. Second, Ž20. corresponds to anisotropic Harper’s equation with anisotropy ratio 4cos
k3 a
ž' / 2
sl
415
conditions and obtained Harper’s equation from initial principles of the tight-binding model. The Fermi surfaces are built for different orientations of the magnetic field. In the following paper we plan to study the electron states and Fermi surfaces at ultrahigh magnetic field in 3D crystals with more complicated lattices of real crystals. The properties of de Haas – van Alfen effect in this problem will be studied.
Acknowledgements One of us ŽV.Ya.Demikhovskii. wants to thank V.D. Selemir for constant support and Prof. Michael von Ortenberg for numerous discussions of this problem. This research was made possible due to financial support from the Russian Foundation for Basic Research ŽGrant No. 98-02-16412..
Ž 21 .
depending on the cross-section transversal to the magnetic field which is fixed by choosing the value of k 3 . It is well-known that Harper’s equation in the case l / 2 describes 2D square lattice with anisotropic transfer integrals w5,6x. According to Ž21. the persistent interval for l is y4 F l F 4. In the quasiclassical limit it describes the situation where open-type trajectories are available. The contribution of open trajectories to the whole topological structure varies with the respect to a cross-section k 3 s const. being chosen. For example, the cross-section k 3 s p'2 r3a corresponds to l s 2 which is the case of isotropic Harper’s equation and only closetype orbits are available. This is illustrated in Fig. 3 where Fermi surfaces ŽFig. 3a,b. and spectrum ŽFig. 3c. are shown for prq s 31r60. Then, the crosssection k 3 s pr '2 a gives us l s 0 which is a ‘full anisotropic’ limit for Harper’s equation. Here as one can see in Fig. 4a,b only opened-type orbits are available. It gives a continuous energy spectrum ŽFig. 4c.. For remaining values of l both opened and closed orbits are present. In the last case the spectrum ´ H consists of a continuous and quasi discrete Žnarrow subbands. parts. As a conclusion, we suggested the expression for electron wave function which satisfies to Peierls
References w1x M.Ya. Azbel’, Zh. Exper. i Teor. Fiz. 46 Ž1964. 929 win Russianx. w2x D.R. Hofstadter, Phys. Rev. B 14 Ž1976. 2239. w3x F.H. Claro, G.H. Wannier, Phys. Rev. B 19 Ž1979. 6068. w4x P.G. Harper, Proc. Phys. Soc. ŽLondon. A 68 Ž1955. 874. w5x H. Hiramoto, M. Kohmoto, Int. J. Mod. Phys. 6 Ž1992. 281. w6x M. Kohmoto, Y. Hatsugai, Phys. Rev. B 41 Ž1990. 9527. w7x A. Barelli, R. Fleckinger, Phys. Rev. B 46 Ž1992. 11559. w8x H. Silberbauer, J. Phys.: Condens. Matter 4 Ž1992. 7355. w9x V.Ya. Demikhovskii, A.A. Perov, JETP 87 Ž1998. 973. w10x V.Ya. Demikhovskii, A.A. Perov, Phys. Low-Dim. Structures 7r8 Ž1998. 135. w11x D. Weiss, M.L. Roukes, A. Menschig et al., Phys. Rev. Lett. 66 Ž1991. 27. w12x T. Schlosser, K. Ensslin, J.P. Kotthaus et al., Semicond. Sci. ¨ Technol. 11 Ž1996. 1582. w13x H. Hasegawa, J. Phys. Soc. Jpn. 59 Ž1990. 4384. w14x Z. Kunszt, A. Zee, Phys. Rev. B 44 Ž1991. 6842. w15x B.A. Boyko, A.I. Bykov, M.I. Dolotenko et al., book of abstracts, VIIIth Int. Conference on Megagauss Magnetic Field Generation and Related Topics, p. 149, Tallahassee, USA. w16x R.E. Peierls, Z. Phys. 80 Ž1933. 763. w17x V.Ya. Demikhovskii, A.A. Perov, D.V. Khomitsky, Proc. VIIIth International Conference on Megagauss Magnetic Field Generation and Related Topics ŽMEGAGAUSS-VIII., Tallahassee, USA Ž1998., in press. w18x J. Zak, Phys. Rev. A 134 Ž1964. 1602. w19x J. Zak, Phys. Rev. A 134 Ž1964. 1607.
Physics Letters A 267 Ž2000. 408–415 www.elsevier.nlrlocaterphysleta
Formation of new fermi surfaces in 3D crystals at ultra-high magnetic field with different orientations V.Ya. Demikhovskii ) , A.A. Perov, D.V. Khomitsky Nizhny NoÕgorod State UniÕersity, Gagarin AÕe., 23, 603600 Nizhny NoÕgorod, Russia Received 4 October 1999; received in revised form 1 February 2000; accepted 6 February 2000 Communicated by A.R. Bishop
Abstract In the tight-binding approximation the Harper like equation describing an electron in 3D crystal subject to a uniform magnetic field is obtained. It is supposed that the vector H can be oriented along several directions in the lattice. The Fermi surfaces relevant to a magnetic flux prq s 1r2 in a simple cubic lattice are built. The quantization rules in magnetic fields slightly distinguished from prq s 1r2 are investigated. q 2000 Elsevier Science B.V. All rights reserved. PACS: 71.18.q y; 71.20.-b Keywords: Fermi surfaces; Energy band structure; Ultra-high magnetic field
The effects of a high magnetic field on Bloch electrons are fascinating problem with very rich physics. This is not surprising since there are two fundamental periods in the problem – the period of the potential and the period of the phase of the wave function – and the interplay of these two periods gives a very interesting spectrum ŽHofstadter butterfly. and eigenstate structure. Starting from classical works of Azbel w1x, Hofstadter w2x, Wannier w3x Žsee also w4–7x. this problem attracts the increasing attention. During the last decade the one electron quantum states in 2D lateral superlattices in the presence of perpendicular magnetic field have been studied in ) Corresponding author. Mailing address: M. Gorky St., 156-3, 603006 Nizhny Novgorod, Russia. Fax: q7-8-312-658592. E-mail address: [email protected] ŽV.Y. Demikhovskii..
a series of theoretical w8–10x and experimental w11,12x works. The problem of flux states in 3D crystals were also the subject of investigations w13,14x. In these papers energy spectrum and density of states were calculated in the tight-binding approximation for several directions of the magnetic field in a simple cubic lattice. However, the translational properties of electron wave function and the geometry of Fermi surfaces for different orientations of the ultrahigh magnetic field have not yet been studied. It is well-known that the Fermi surface shape defines both single electron dynamics and kinetic properties of an electron gas. Nowadays when in VNIIEF ŽSarov. the magnetic field up to 28 MGs is reached w15x the idea of observation of details forecasted by the theory in the crystals with the lattice spacing of a s 0.6 nm and more does not seem to be unaccessible.
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 1 2 6 - 2
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
In this paper we shall consider electronic states in 3D crystals subjected to ultrahigh magnetic field. We suppose that the vector H is parallel to arbitrary translation vector of a lattice. In a tight-binding approximation the wave function satisfying the generalized Bloch–Peierls conditions is obtained and Harper like equation for different orientations of a magnetic field is derived. As an example, the Fermi surfaces for magnetic subbands relevant to rational value of a magnetic flux prq s 1r2 are built. The quantization rules for magnetic flux near this rational value of flux quanta are studied. First of all in the tight-binding approximation let us derive Harper’s equation for the simple cubic lattice subject to uniform magnetic field. To be specific let the basis vectors on the x, y plane be a1 and a 2 , and the magnetic field is applied along lattice basis vector a 3 I 0 z Ž< a1 < s < a 2 < s < a 3 < s a.. Let vector potential is be chosen in Landau gauge A s Ž0, H x,0.. Then electron wave function must satisfy the generalized Bloch conditions ŽPeierls conditions. w16x
c k Ž r . s c k Ž x q qa, y q a, z q a . =exp Ž yik x qa . exp Ž yik y a . =exp Ž yik z a . exp Ž y2p ipyra . ,
Ž 1.
due to the fact that the vector potential is not a periodic function of the coordinates. Here p and q are mutually prime integers which define the number of magnetic flux quanta per two-dimensional square elementary cell
F
p s q
s
F0
< e < H < w a1 = a 2 x < 2p "c
,
Ž 2.
where F 0 s hcr< e < is the magnetic flux quanta. In a presence of a magnetic field the electron quantum states must be classified in accordance with irreducible representations of magnetic translation group a mag s nqa1 q ma 2 q la 3 ,
Ž n,m,l are integer numbers. The quasimomentum k takes values in the magnetic Brillouin zone p p y FkxF , qa qa
p
p FkyF
y a
p y a
409
,
a
p FkzF
a
.
Ž 3.
In the tight-binding approximation the electron wave function which satisfies the conditions Ž1. can be written in the form w17x
ck Ž r . s
Ý
g n Ž k . exp Ž ika n .
n, m ,l
p Ž y y ma .
ž
=exp y2p i
q
a
n
/
=c 0 Ž x y na, y y ma, z y la .
Ž 4.
where c 0 Ž r y a n . is the atomic function. The pathdependent geometric phase 2p Ž p q.w Ž y y ma . an x of the wave function is known to play a fundamental role in the problem. By substituting of Ž4. into Schrodinger equation ¨ and evaluating the transfer integrals between neighbour sites we shall obtain a system of difference equations for coefficients g n . The magnetic field has a non-trivial influence on the transfer integrals between neighbour lattice sites in the x direction
ž
A s exp "2p i
H
p y y ma q
a
/
c 0 Ž r y a nX .
= Ž V Ž r . y U Ž r y a n . . c 0 Ž r y a n . dt ,
Ž 5.
where a nX s Ž Ž n " 1 . a,ma,la . , V Ž r . is the crystal scalar potential with periodicity a in three dimensions and U is the atomic potential. Calculating the transfer integrals in the mean-value approximation we substitute y s ma and obtain A s E0 where E0 is the transfer integral Ž5. in the absence of a magnetic field. In the y direction the transfer integral is
ž
B s exp "2p i
p q
n
/H
c 0 Ž r y a nX .
= Ž V Ž r . y U Ž r y a n . . c 0 Ž r y a n . dt ,
Ž 6.
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
410
where a nX s Ž na, Ž m " 1 . a,la . and in the z direction the transfer integral is equal to E0 . As a result we obtain the well-known Harper’s equation w4x
one can obtain the following expressions for dispersion laws for q magnetic subbands: for prq s 1r2 Žtwo subbands.
exp Ž ik x a . g nq1 q exp Ž yik x a . g ny1
Ž1,2. ´H Ž k x ,k y . s "2 cos 2 k x a q cos 2 k y a ,
ž
q 2 g n cos 2p n
p q
(
Ž 8.
for prq s 1r3 Žthree subbands.
/
q k y a s ´ Ž k x ,k y ,k z . g n ,
Ž 7.
Ž1. ´H Ž k x ,k y . s y2'2 cos
where
ž
2p 3
/
q Ž arctan b . r3 ,
p
´ Ž k x ,k y ,k z . s ´ H Ž k x ,k y . y 2cos k z a
Ž 7’ .
is the dimensionless energy measured in terms of E0 . The spectrum ´ H Ž k x ,k y . in Eq. Ž7. consists of q bands and depend on one single parameter prq counting the number of magnetic flux quanta per unit cell ŽFig. 1.. If flux is irrational the spectrum is a singular-continuum – uncountable but measure zero set of points ŽCantor set.. Originally the Eq. Ž7. was derived by Harper w4x using Peierls substitution w16x. For some values of prq s 1r2, 1r3, 1r4 it is possible to solve the Eq. Ž7. analytically. As a result
Ž2. ´H Ž k x ,k y . s 2'2 cos
ž
3
/
q Ž arctan b . r3 ,
Ž3. ´H Ž k x ,k y . s y2'2 cos Ž Ž arctan b . r3. ,
Ž 9.
where b s '8 y a 2 ra , a s cos3k x a q cos3k y a; and for prq s 1r4 Žfour subbands. Ž1,2,3,4. ´H Ž k x ,k y .
( (
s " 4 " 2 4 y sin2 2 k x a y sin2 2 k y a .
Ž 10 .
Fig. 2 shows the Fermi surfaces in the lowest subband for prq s 1r2 obtained using Ž7’. and Ž8.
Fig. 1. Parts of the classical Hofstadter ‘butterfly’ near prq s 0, 1r3, 1r2.
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
411
Fig. 2. The Fermi surfaces for the lower Ža. and upper Žb. magnetic subbands of 3D crystals with simple cubic lattice at prq s 1r2, H I a 3 . Ža.: 1. ´ s y4.3; 2. ´ s y2.0; 3. ´ s 0.0; Žb.: 1. ´ s 4.3; 2. ´ s 2.0; 3. ´ s 0.0.
in the first Brillouin zone yprqaF k x F prqa,y prqa F k y F prqa,y pra F k z F pra. Three surfaces for different representative energies are plotted. For an ellipsoid-type surface such as Ž1. Ž< k < 0. the simple analytical expression for an energy spectrum ´ Žk ) can be easily obtained with the help of Ž7’. and Ž8.. The relevant effective masses are determined as
™
"2 m )x , y s
'2 E0 a2
"2 , m )z s
2 E0 a 2
.
Ž 11 .
™
The asymptotic spectrum near the point k x , y pr2 a ŽŽ2.-type surface. is ´ Ž k x ,k y ,k z . ; k 2x q k 2y y 2cos k z .
(
Now we shall discuss the magnetic quantization rules at values of a flux slightly distinguished from prq s 1r2: p
1 s
q
1 q
2
qX
,
qX 4 1.
Ž 12 .
This corresponds to the magnetic field H s H Ž1r2. q D H Ž1rqX .. As one can see from Fig. 1 near value of prq s 1r2 the spectrum represents a system of narrow subbands Žpractically discrete levels.. Here it is possible to see an equidistant spectrum, points of level accumulation and ranges where ´ H ; 'N Ž N is the level number.. The spectrum ´ H Ž k x ,k y . of Harper’s equation near one half of flux quanta can be studied analyti-
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
412
cally due to the self-similarity of its structure. In the case prq s 1r2 two subbands and two subsystems in the eigenvector distribution can be observed. It may be derived analytically from Harper’s equation that every eigenvector function from a subsystem has an argument step D n s 2. The magnetic flux Ž12. gives us the diagonal term in Harper’s equation of the form 2cosŽp n q 2p nrqX . g n s 2cosp ncosŽ2p nrqX . g n . Now qX is a new period of the equation. Since cosp n s Žy1. n our system of Eqs. Ž7. consists of two subsystems where only the sign varies. It is easy to construct an equation for each subsystem. After simple algebra we obtain Ž k x s k y s 0.: g nq2 q g ny2 q 2cos
4p qX
2 n gn s Ž ´ H y4 . g n .
Ž 13 .
2 Now ´ H y4 s ´ XH is the energy in Harper’s Eq. Ž7. and ´ H s " ´ XH q 4 describes the structure of the spectrum near one half of flux quanta and cos k z a s 0. Two signs at the square root correspond to two groups of energy subbands. The energy ´ H describes all parts of the spectrum near prq s 1r2 ŽFig. 1.. We can obtain from Ž13. the ‘square root’ dependence at ´ H s 0, the clustering points ´ H s "2 and Landau levels near ´ H s "2'2 . These peculiarities of a spectrum can be obtained with the help of Onsager–Lifshitz quasiclassical quantization rules on the Fermi surfaces ŽFig. 2a,b.. It is easy to see that magnetic quasiclassical quantization on the Ž1.-type surface results to the equidistant energy spectrum at the bottom of the lowest magnetic subband. Accordingly, the magnetic Onsager–Lifshitz quantization near k z 0 on Ž2.-type surface yields the ‘square root’ spectrum. At last, it is clear that in magnetic field H I 0 z there are self-crossing orbits on Ž3.-type surface. It gives the accumulation point in the spectrum ´ H near prq s 1r2 Žsee Fig. 1.. It can be stressed that level spacing obtained from Ž13. has the same value that was derived using quasiclassical quantization. In accordance with Harper’s Eq. Ž7. and Ž7’. the partial overlapping between q energy subbands is observed. When prq s 1r2 the overlapping takes place in the energy interval Žy2, 2.. It is clear that when the electron concentrations leading to ´ F g Žy2, 2. in the magnetic field H s H Ž1r2. q
(
™
D H Ž1rqX . the de Haas–van Alfen oscillations with different periods corresponding to the different energy level series ŽFig. 1. can be observed. Further it is necessary to make the following relevant note. The wave function Ž4. describes nonhomogeneous probability distribution on the lattice sites. This is due to the amplitudes g nŽ k . which give us non-uniform shapes. However, in real crystals due to Coloumb interaction the static electron density distribution must maintain its symmetry even in the presence of magnetic field. Fortunately, the spectrum of Harper’s equation is q-fold degenerate since ´ Ž k x ,k y q 2p jrqa,k z . s ´ Ž k x ,k y ,k z ., j s 0,1,2, . . . ,q y 1 and a wave function with homogenuous electron density distribution can be written as a combination of Ž4.. Let us introduce qy1
c kX Ž r . s
Ý C j Ž k . Kˆ j Tˆja c k Ž r . ,
Ž 14 .
js0
where c k Ž r . is from Ž4., C j Ž k . s expŽyik x ja., Tˆja c s c Ž x q ja, y, z .expŽ iefr"c . Ž f s yHyja., the operator Kˆ j transforms c k with k s Ž k x ,k y ,k z . into c k X with kX s Ž k x ,k y y 2p pjrqa,k z .. After this procedure we can write Ž14. in the form
c kX Ž r . s D Ž k . Ý exp y2p i n
ž
p Ž y y ma . q
=exp Ž ika n . c 0 Ž r y a n . ,
a
n
/ Ž 15 .
1 Ž . where DŽ k . s Ý qy js0 g nqj k does not depend on n due to the periodicity of the system. Now Ž15. describes homogeneous electron density distribution on the lattice sites and corresponds to the same energy as c k . Let us consider now the case of a simple cubic lattice subjected to a uniform magnetic field oriented along the diagonal of a square in the plane Ž x, y .. The magnetic field with amplitude H has now the following Cartesian coordinates:
Hs
H
'2 Ž 1,1,0 . .
Ž 16 .
It is convenient to choose a new coordinate system making a rotation about the old z axis by pr4 and
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
413
Fig. 3. a., b. The parts of Fermi surfaces bounded by planes k 3 s "p'2 r3a Ž l s 2. at prq s 1r2, H I a1 q a 2 for the upper subband; c. the energy spectrum ´ H Ž k 1 ,k 2 ,k 3 s p'2 r3a. at prq s 31r60.
defining the magnetic field orientation as x 3 . This transformation is written as following: x 3 y x1 x 3 q x1 xs , ys '2 '2 , z s x 2 . We choose a new elementary cell: it is a rectangular parallelepiped based on the new coordinate system vectors: a square with the side '2 a at the base in the plane Ž x 1 , x 3 . and the height a. This cell is non-primitive: it consists of two atoms Žadditional sites are located at the centers of top and bottom faces.. In this coordinate system the magnetic field is oriented along one of the elementary cell basic vectors Žnamely, a 3 .. According to the general principles in this case the classification of electron states is possible w18,19x. Now the first Brillouin zone has an area two times less with respect to the initial value. The magnetic field in this coordinate system is written: H s H Ž 0,0,1 .
Ž 17 .
and it is convenient to choose a vector potential using Landau gauge: A s Ž yH x 2 ,0,0 . .
Ž 18 .
The wavefunction in the tight-binding approximation can be written as following:
½ ž
c k Ž r . s Ý g n Ž k . exp 2p i n
p x 1 y m'2 a q
ar'2
n
/
=exp Ž ika n . c 0 Ž r y a n .
ž
qexp 2p i
p x 1 y Ž m q 1r2 . '2 a q
ar'2
n
/
5
=exp Ž ik Ž a n q d . . c 0 Ž r y Ž a n q d . . .
Ž 19 .
414
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
Here ka n s k 1'2 am q k 2 an q k 3'2 al, c 0 is an isolated atomic function in the presence of the magnetic field, the amplitudes g nŽ k . are the subject of further research and g nq q s g n . The indexes Ž m,n,l . are taking all integer values independently, n numerates sites along x 2 axis, perpendicular to the magnetic field. The sum is taken over all lattice sites and two addendums in Ž22. correspond to the two existing crystalline subsystems. Each of them is a simple tetragonal lattice with the elementary cell being a rectangular parallelepiped with square of the side '2 a in its base and the height a. These subsystems are displaced on the vector d with new coordinates Ž ar '2 ,0,ar '2 .. The magnetic translation group is constructed by the q - time multiplication of the translation period along x 2 axis: x 2 x 2 q qa. The other translations are not changed. Their periods in the new coordinate system are '2 a which is the diagonal of a square with the side a. The classifica-
™
tion of wavefunctions is possible when the number of flux quanta FrF 0 through the area a 2r '2 Žbeing the minimal inter-site area in the transversal to magnetic field projection. is a rational number: FrF 0 s prq where p and q are mutually prime integers. The substitution of Ž19. into Schrodinger equation ¨ is performed and the transfer integrals are calculated. We will have the equation for the energy ´ and the amplitudes g nŽ k .: e i k 2 a g nq 1 q eyi k 2 a g ny1 q 4cos
ž
= cos 2p
p q
nq
k1 a
'2
/
k3 a
ž' / 2
gn s ´ gn .
Ž 20 .
™
It should be mentioned that this equation may be obtained using Peierls substitution k kˆ y eA "c with the initial spectrum at null magnetic field. In old
Fig. 4. a., b. The Fermi surfaces in the first Brillouin zone at prq s 1r2 and l s 0 Ž H I a1 q a 2 . for the upper subband; c. the energy spectrum ´ H Ž k 1 ,k 2 ,k 3 s pra'2 . at prq s 31r60.
V.Ya. DemikhoÕskii et al.r Physics Letters A 267 (2000) 408–415
coordinates it was written as ´ s 2Žcos k x a q cos k y a q cos k z a ., and in new coordinates Ž x 1 , x 2 , x 3 . it is
´ s 2cos k 2 a q 4cos
k3 a
'2
cos
k1 a
'2
.
Eq. Ž20. is a generalization of Harper’s equation for the case of magnetic field parallel to a diagonal of a square which is a face of the elementary cell cube. Keeping a three-diagonal structure Ž20. has some remarkable features with respect to standard Harper’s equation. First, Ž20. describes total Žnot ‘transversal’ ´ H Ž k 1 ,k 3 .. energy as a function of quasimomentum k: ´ s ´ Ž k 1 ,k 2 ,k 3 .. Now k 3 s const. corresponds to the plane perpendicular to H and k 1 s const. describes the plane parallel to magnetic field. Second, Ž20. corresponds to anisotropic Harper’s equation with anisotropy ratio 4cos
k3 a
ž' / 2
sl
415
conditions and obtained Harper’s equation from initial principles of the tight-binding model. The Fermi surfaces are built for different orientations of the magnetic field. In the following paper we plan to study the electron states and Fermi surfaces at ultrahigh magnetic field in 3D crystals with more complicated lattices of real crystals. The properties of de Haas – van Alfen effect in this problem will be studied.
Acknowledgements One of us ŽV.Ya.Demikhovskii. wants to thank V.D. Selemir for constant support and Prof. Michael von Ortenberg for numerous discussions of this problem. This research was made possible due to financial support from the Russian Foundation for Basic Research ŽGrant No. 98-02-16412..
Ž 21 .
depending on the cross-section transversal to the magnetic field which is fixed by choosing the value of k 3 . It is well-known that Harper’s equation in the case l / 2 describes 2D square lattice with anisotropic transfer integrals w5,6x. According to Ž21. the persistent interval for l is y4 F l F 4. In the quasiclassical limit it describes the situation where open-type trajectories are available. The contribution of open trajectories to the whole topological structure varies with the respect to a cross-section k 3 s const. being chosen. For example, the cross-section k 3 s p'2 r3a corresponds to l s 2 which is the case of isotropic Harper’s equation and only closetype orbits are available. This is illustrated in Fig. 3 where Fermi surfaces ŽFig. 3a,b. and spectrum ŽFig. 3c. are shown for prq s 31r60. Then, the crosssection k 3 s pr '2 a gives us l s 0 which is a ‘full anisotropic’ limit for Harper’s equation. Here as one can see in Fig. 4a,b only opened-type orbits are available. It gives a continuous energy spectrum ŽFig. 4c.. For remaining values of l both opened and closed orbits are present. In the last case the spectrum ´ H consists of a continuous and quasi discrete Žnarrow subbands. parts. As a conclusion, we suggested the expression for electron wave function which satisfies to Peierls
References w1x M.Ya. Azbel’, Zh. Exper. i Teor. Fiz. 46 Ž1964. 929 win Russianx. w2x D.R. Hofstadter, Phys. Rev. B 14 Ž1976. 2239. w3x F.H. Claro, G.H. Wannier, Phys. Rev. B 19 Ž1979. 6068. w4x P.G. Harper, Proc. Phys. Soc. ŽLondon. A 68 Ž1955. 874. w5x H. Hiramoto, M. Kohmoto, Int. J. Mod. Phys. 6 Ž1992. 281. w6x M. Kohmoto, Y. Hatsugai, Phys. Rev. B 41 Ž1990. 9527. w7x A. Barelli, R. Fleckinger, Phys. Rev. B 46 Ž1992. 11559. w8x H. Silberbauer, J. Phys.: Condens. Matter 4 Ž1992. 7355. w9x V.Ya. Demikhovskii, A.A. Perov, JETP 87 Ž1998. 973. w10x V.Ya. Demikhovskii, A.A. Perov, Phys. Low-Dim. Structures 7r8 Ž1998. 135. w11x D. Weiss, M.L. Roukes, A. Menschig et al., Phys. Rev. Lett. 66 Ž1991. 27. w12x T. Schlosser, K. Ensslin, J.P. Kotthaus et al., Semicond. Sci. ¨ Technol. 11 Ž1996. 1582. w13x H. Hasegawa, J. Phys. Soc. Jpn. 59 Ž1990. 4384. w14x Z. Kunszt, A. Zee, Phys. Rev. B 44 Ž1991. 6842. w15x B.A. Boyko, A.I. Bykov, M.I. Dolotenko et al., book of abstracts, VIIIth Int. Conference on Megagauss Magnetic Field Generation and Related Topics, p. 149, Tallahassee, USA. w16x R.E. Peierls, Z. Phys. 80 Ž1933. 763. w17x V.Ya. Demikhovskii, A.A. Perov, D.V. Khomitsky, Proc. VIIIth International Conference on Megagauss Magnetic Field Generation and Related Topics ŽMEGAGAUSS-VIII., Tallahassee, USA Ž1998., in press. w18x J. Zak, Phys. Rev. A 134 Ž1964. 1602. w19x J. Zak, Phys. Rev. A 134 Ž1964. 1607.