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Relation between position and quasi-momentum operators in band theory
R. Balian
To cite this version:
R. Balian. Relation between position and quasi-momentum operators in band theory. Journal de Physique, 1989, 50 (18), pp.2629-2635. �10.1051/jphys:0198900500180262900�. �jpa-00211087�
Relation between position and quasi-momentum operators in band theory
R. Balian
Service de Physique Théorique(*) de Saclay, 91191 Gif-sur-Yvette Cedex, France
(Reçu le 21 mars 1989, accepté le 18 avril 1989)
Résumé. 2014 On utilise les fonctions de Wannier pour écrire une expression de la différence entre
l’opérateur de position et le conjugué de l’opérateur de quasi-moment. Ceci foumit une approche simple à la polarisation et à la dynamique des électrons dans les cristaux.
Abstract. 2014 The difference between the position operator and the conjugate of the quasi-momentum operator is expressed in terms of Wannier functions. This provides a simple approach to polarization
and to electron dynamics in crystals.
Classification
Physics Abstracts
71.55C - 77.20 - 03.65
The use of Wannier functions to describe electronic phenomena in crystals is often enlightening [1,2]. They are not only a convenient representation of Bloch waves especially suited to the tight- binding limit, but can also provide natural interpretations for many properties since they constitute
the extension to crystal structures of the concept of localized orbitals. The purpose of this note,
mainly tutorial, is to show how they can shed light on the partial conjugacy existing in band theory
between the position and the quasi-momentum and thus be useful in applications.
We recall that, for simple bands, Wannier functions wb(r) are defined as Fourier transforms of Bloch waves PbK.(r) with respect to K, :
The integral on the quasi-momentum x (which is defined within translations P of the reciprocal lattice) runs in the Brillouin zone, n denotes the volume of the primitive cell (we let h = 1). The
two sets of vectors cpbK.(r) == (rlbK) and wb(r - R) - r 6R) , labelled by the band index b and by
either the quasi-momentum x or the vector R of the Bravais lattice, constitute orthonormal bases
exhibiting the translational invariance of the crystal. Provided the phases of the Bloch functions
are suitably chosen, the Wannier function wb(r) defined by (1) is localized near the origin (usually
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180262900
2630
around either an atom or a bound of the crystal), and it decreases exponentially [3,2,4]. However,
for a multiple band with degeneracy d, these properties have been shown under rather general
conditions [4] to hold only for generalized Wannier functions which mix the d Bloch waves {JbOlK.
(where a = 1, ...d labels the branches of the band). These Bloch functions are then expressed as
in terms of d localized Wannier functions Wbp (p. = 1, .ud) related simply to one another by rotations
or symmetries of the crystal group. The d x d matrix Upa(bK,) is unitary for each value of x. For
simplicity, we write below mainly equations for simple bands.
Our main tool is a decomposition
of the position operator r as the sum o f two operators Il and p, analogous to the decomposition of the
vector r as the sum of a vector R of the Bravais lattice plus a vector p belonging to the primitive cell.
The splitting (4) comes out naturally when expressing r in the representation of Wannier functions
wb (r - R) = (r 1 bR) . By taking r = r’ - R as a variable and using the orthonormality of Wannier orbitals, we get
(The second term is equal to J dr Wb’ (r)rwb(r - R + R’).) Equation (5) is rewritten in the repre- sentation of Bloch waves cpb,,(r) == (rlb) by means of (2) and of
this b-function is meant modulo the vectors P of the reciprocal lattice. We obtain
where
The first term of either (5) or (7) defines the operator
which is nothing but the Bravais lattice vector R in the Wannier representation. The expression (9)
exhibits its eigenvalues and eigenvectors. The matrix elements (7) of R in the Bloch representation
are reminiscent of those (p’ Ifl p) = -ivb(p - p’) of r between plane waves of momenta p, p’.
Thus, the operator R, equal to 18/8x in the Bloch representation, appears as canonically conjugale
to the quasi-momentum operator in this representation [1]. It is identified with the smooth part of î-
at the scale of the lattice.
The second term of (5) or (7),(8) defines the operator
It is invariant under lattice translations R, and can thus be regarded as the part of r which acts within each cell. In the Bloch representation (10), p is diagonal in the quasi-momentum x (but it does not
commute with the momentum p, nor with R = 18/8x, and its three components do not commute with one another). It depends on x only through the Bloch function b It can induce interband transitions, contrary to R. The localization of Wannier functions shows that the matrix elements
( b’R/1 p IbR) given by (5) decrease exponentially with R - R’. The integrals (8) extend to the
ranges of z.vb or Wb’; if the band b’ is lower than b, the second form of (8) is more convenient than the first one since Wb’ is then more localized than wb (we shall use this remark to interpret Eq.(36) below).
In agreement with the translational invariance of p, the evaluation of the matrix elements of
h(f) is simple if h(r) is a periodic function, such as the band potential V(r), satisfying h(r) = h(r + R) for any R. Only the second term of (7) then contributes, and we get
For multiple bands, the calculation proceeds similarly, starting from (3) instead of (2). The decom- position (4) still holds, with the same expression for R, but with an operator p now given by
The decomposition (7) is currently written, but with Pblb(K,) being expressed in terms of the peri-
odic functions Ub,,,,(r) =- e-iK..rpbK.(r) and their derivative with respect to K, instead of (8). This decomposition is also reminiscent of the Zak representation [5], which relies on the existence of
a basis lkq) of states characterized by their quasi-momentum k and their quasi-coordinate q (de-
fined as r modulo R). More precisely, the state Ikq) is the Bloch wave of quasi-momentum k which
would be constructed as (2) from Wannier functions 6(r - q - R) localized at the point q and at
its translated points. Zak’s operators
look like the quasi-momentum operator
2632
and like the operator p defined by (10) or (12), respectively. Indeed, r-q is equal to iâ/âk in
the Zak representation, just as r-p is equal to ialam in the Bloch representation; they have the
same eigenvalues R, but different eigenfunctions. Moreover, k and q commute, just as k and p.
However, defining unambiguously an operator such as r modulo à requires a choice of basis. Zak’s basis retains the symmetry between position and momentum. In contrast, the splitting (4) of r is adapted to the band structure of the material. It depends not only on the Bravais lattice R but also
on the Hamiltonian, as obvious from (9) and (10) which involve the associated Bloch or Wannier functions.
Expressions analogous to (11) are obtained for the momentum operator p, or more generally
for any function g(p) of p, by starting from the Wannier representation. The result,
exhibits the commutation of p with Ê. Checking the commutation relations of r and p from (7),(8)
and (15) is straightforward but tedious, which indicates that the representation (7) of r is not trivial.
If g(p) is periodic and satisfies g(p) = g(p + P) for any P, (15) reduces to g(p) = g(î,.).
Dynamics in bands.
As a first illustrative example, let us give an elementary derivation of the equations of motion
of non-interacting electrons in a crystal structure under the effect of a constant electric field E. Al- though the existing derivations are well-known and included in most textbooks, they often contain
approximations which are not well-controlled or, when they are exact, they are not very simple.
The full Hamiltonian Ê’ is
where H is the band Hamiltonian with eigenvalues -(br.). Denoting as D(bK, bic’) the single- particle density matrix, we get from (16) then (7) the equation of motion
If the density matrix reduces to a unique band, the velocity reduces to the first term B7£ >=
f dicD(br., br.)Ve(br.). Otherwise, interband contributions
arise from the second term of (17). For a multiple band, (12) also provides contributions associated
with a 34 a’ and describing transitions from one branch of the band to another. Moreover, p always
contributes to the spreading d [ X2 > - x >2] /dt of the wave packet.
In contrast, the equation of motion for (14) yields the single term
in all cases. More generally, we get the exact equation
for any function g(r.) (in particular for periodic functions required to by-pass irrelevant difficulties
at the boundaries of the Brillouin zone).
The use of Wannier functions also provides useful hints on the dynamics in a uniform magnetic field [1]. The solution of this problem is known to be complicated [2, 5-9], in spite of the simplicity of
its result for a weak field, namely the replacement of the band energy by an effective Hamiltonian
Let us make à few comments on this question. Note first that the translational invariance is best
implemented in the Wannier representation. In the absence of fields, the band Hamiltonian has the form
which exhibits its commutation with the translation operators e-ip.R. The magnetic Hamiltonian
commutes with the magnetic translations
which include a gauge transformation, and which should be used to deduce the magnetic Wannier
functions from one of them as
1 --
Expressing the translational invariance 1 results in the general expression
more complicated than (22) because the translations (25) do not commute with one another.
An exact identification of (24) with (27) would require bR) and ê to depend on the field B. Let
us check, however, that neglecting these dependences is sufficient for an approximate identification.
The relation (26) then requires for consistency that r be replaced by its smooth part R in (25).
Expanding the exponential in (27) yields to first order the operator
2634
where we have used the decomposition (4), expressed the velocity i[H, f] as p/m, neglected p x p compared to R x p, and noted from (10), (22) that [H, pj vanishes within a band. The higher order
terms in B of (27), evaluated by means of the same approximations (one-band terms, replacement
of r by R), provide
readily identified with (24).
The expected relation (21) is also recovered from (27) by letting H" act on a Bloch wave and using (2), (9). The resulting equation
gives a precise meaning to the operator (21).
Polarization.
As a second application, let us write a general formal expression for the dielectric constant of an insulator, in the Hartree approximation. The electronic density n(r) at equilibrium, for a temperature f3-1 and a chemical potential y, is given in terms of the Hamiltonian H by
where
is the Fermi factor. A variation bfi of the Hamiltonian induces a change
of the operator f (H), and hence a change bn(r) of the density (30). Let us calculate the effect of a
small constant electric field E including both an external field Eexi and the self-consistent Hartree field Esc due to the density change Sn(r). Writing (30) in the basis of Bloch waves (for simple bands), integrating over u, and accounting for the spin, we get
In a good insulator, the Fermi factor f(bK,) is either 0 or 1, depending on the sign of e(bK) - y;
hence one of the bands b and b’ in (33) is empty, the other is full. Since these bands are different, only the part p of r contributes to (33), which reduces to
We have used (7) and the symmetry of (8) to constrain the band b to lie below the Fermi surface,
the band b’ above. The expression (34) is obviously periodic in r. Its average over one cell c, which defines the macroscopic charge density -(e/n) f drôn(r) in this cell, vanishes (as a result of the
orthogonality of Pbl,, and C{)bK. after the integral over r is extended to the whole space).
The polarization P produced in the cell c by the total electric field E is then
and its contribution to the Hartree field is Esc = -P/eo. Since f dr ôn(r) vanishes in any cell, P
is the average over the whole space of -erôn(r), and its calculation from (34) again introduces the matrix element (7),(8). We find thus the suscepti6ilily tensor X (defined by P = xE) as
and hence the dielectric constant tensor e = eo + x.
The expression (36) is quite suggestive. The matrix element Pb’ b, as expressed by the first form
of (8), describes the excitation of an electron by the electric field, from a filled localized Wannier
orbital b to an empty Bloch wave b’. Only the second term of (4) contributes, expressing that the polarization (a local phenomenon in each cell of the lattice) is produced only by the part r modulo R of the electric potential. The crystal structure enters through the deformations of the Wannier orbitals with respect to molecular orbitals, and of the Bloch waves with respect to plane waves, and also through the band energies. The discrete translational invariance is reflected by the conser-
vation of the quasi-momentum x in the energy denominator. The inclusion of multiple bands by
means of (12) is straightforward.
References
[1] WANNIER G.H., Rev. Mod. Phys. 34 (1962) 645.
[2] BLOUNT E.I., Solid State Phys. 13 (1962) 305.
[3] KOHN W., Phys. Rev. 115 (1959) 809.
[4] DES CLOIZEAUX J., Phys. Rev. 129 (1963) 554; 135 (1964) A685, A698.
[5] ZAK J., Solid State Phys. 27 (1972) 1.
[6] PEIERLS R., Z. Phys. 80 (1933) 763.
[7] ONSAGER L., Philos. Mag. 43 (1952) 1006.
[8] KOHN W., Phys. Rev. 115 (1959) 1460.
[9] ROTH L.M., J. Phys. Chem. Solids 23 (1962) 433.
