Electrons in graphene populate vicinities of two nonequivalent Dirac points in the Brillouin zone, or two valleys K and K'. We do not consider intervalley scattering and neglect valley splitting, thus it is sufficient to consider electrons in only one valley and treat existence of the other valley as additional twofold degeneracy of electron states.
We consider magnetoexciton as an electron-hole pair, and we will denote all electron and hole variables by the indices 1 and 2 respectively. In the valley K, Hamiltonian of free electrons in graphene in the basis {A1,B1} of sublattices takes a form [7]:
(1)
where are the cyclic components of electron momentum and v
F
≈ 106m/s is the Fermi velocity of electrons.
For external magnetic field H, parallel to the z axis, we take the symmetrical gauge, when . Introducing the magnetic field as substitution of the momentum p1 → p1 + (e/c)A(r1) in (1) (we treat the electron charge as -e), we get the Hamiltonian of the form:
(2)
Here the operators and (where ) obey bosonic commutation relation .
Using this relation, by means of successive action of the raising operator we can construct Landau levels for electron [18] with energies
(3)
and wave functions
(4)
Here k = 0,1, 2,... is the index of guiding center, which enumerates electron states on the n th Landau level (n = -∞,...,+∞), having macroscopically large degeneracy , equal to a number of magnetic flux quanta penetrating the system of the area S. Eigenfunctions ϕ
nk
(r) of a 2D harmonic oscillator have the explicit form:
(5)
s
n
= sign(n) and are associated Laguerre polynomials.
Consider now the hole states. A hole wave function is a complex conjugated electron wave function, and the hole charge is +e. Thus, we can obtain Hamiltonian of the hole in magnetic field from the electron Hamiltonian (2) by complex conjugation and reversal of the sign of the vector potential A(r2). In the representation of sublattices {A2,B2} it is
(6)
where the operators and commute with and obey the commutation relation . Energies of the hole Landau levels are the same as these of electron Landau levels (3), but have an opposite sign.
Hamiltonian of electron-hole pair without taking into account Landau level mixing is just the sum of (2) and (6), and can be represented in the combined basis of electron and hole sublattices {A1A2,A1B2,B1A2,B1B2} as
(7)
It is known [41] that for electron-hole pair in magnetic field there exists a conserving 2D vector of magnetic momentum, equal in our gauge to
(8)
and playing the role of a center-of-mass momentum. The magnetic momentum is a generator of simultaneous translation in space and gauge transformation, preserving invariance of Hamiltonian of charged particles in magnetic field [42].
The magnetic momentum commutes with both the noninteracting Hamiltonian (7) and electron-hole Coulomb interaction V(r1-r2). Therefore, we can find a wave function of magnetoexciton as an eigenfunction of (8):
(9)
Here R = (r1 + r2)/ 2, r = r1 - r2, e
z
is a unit vector in the direction of the z axis. The wave function of relative motion of electron and hole is shifted on the vector . This shift can be attributed to electric field, appearing in the moving reference frame of magnetoexciton and pulling apart electron and hole.
Transformation (9) from Ψ to Φ can be considered as a unitary transformation Φ = U Ψ, corresponding to a switching from the laboratory reference frame to the magnetoexciton rest frame. Accordingly, we should transform operators as A → UAU+. Transforming the operators in (7), we get: . Here the operators contain only the relative electron-hole coordinate and momentum and obey commutation relations (all other commutators vanish).
Thus, the Hamiltonian (7) of electron-hole pair in its center-of-mass reference frame takes the form
(10)
A four-component wave function of electron-hole relative motion , being an eigenfunction of (10), can be constructed by successive action of the raising operators and (see also [20, 21]):
(11)
The bare energy of magnetoexciton in this state is a difference between energies (3) of electron and hole Landau levels:
(12)
Here we label the state of relative motion by numbers of Landau levels n1 and n2 of electron and hole, respectively. The whole wave function of magnetoexciton (9) is additionally labeled by the magnetic momentum P. In the case of integer filling, when all Landau levels up to ν th one are completely filled by electrons and all upper levels are empty, magnetoexciton states with n1 > ν, n2 ≤ ν are possible. For simplicity, we neglect Zeeman and valley splittings of electron states, leading to appearance of additional spin-flip and intervalley excitations [20, 21, 24].