Strain-induced quantum Hall phenomena of excitons in graphene

We study direct and indirect pseudomagnetoexcitons, formed by an electron and a hole in the layers of gapped graphene under strain-induced gauge pseudomagnetic field. Since the strain-induced pseudomagnetic field acts on electrons and holes the same way, it occurs that the properties of single pseudomagnetoexcitons, their collective effects and phase diagram are cardinally different from those of magnetoexcitons in a real magnetic field. We have derived wave functions and energy spectrum of direct in a monolayer and indirect pseudomagnetoexcitons in a double layer of gapped graphene. The quantum Hall effect for direct and indirect excitons was predicted in the monolayers and double layers of gapped graphene under strain-induced gauge pseudomagnetic field, correspondingly.


A. Eigenvalues and eigenfunctions
In expanded form the operatorP 2 is given bŷ In the center-of-mass reference system the Hamiltonian (6) from the main text can be written in the following form: . (5) and M and µ are the total and reduced exciton masses, respectively, given by Eqs.
Let us find the eigenfunctions of the operatorˆ H 0 defined aŝ where the vector A 0 is given by The eigenfunctions and eigenvalues of the Hamiltonian are considered for the both cases A 0 = 0 and A 0 ̸ = 0. If A 0 = 0, the eigenfunction of H 0 is given by n,m (R), which is the wavefunction for a free particle of unit charge in the effective PMF 2B in the cylindrical gauge in eigenvalue E n ofP 2 is defined as [5] where ω c = 2B/M is the cyclotron frequency for the motion of the center-of-mass of a non-interacting electron-hole pair. The quantum numbers n = 0, 1, 2, . . . and m = 0, 1, 2, . . . for ψ (0) = ψ (0) n,m (R) and in Eq. (9) are related to the motion of the center-of-mass of a non-interacting electron and hole.
If A 0 ̸ = 0, we define the scalar function f (R, r) so that A 0 ≡ ∇ R f (R, r) and we have f (R, r) = A 0 · R. The Schrödinger equation is invariant with respect to the translation of the coordinate R and simultaneous gauge transformation. In this case the eigenvalue ofP 2 is the same as the eigenvalue at A 0 = 0 given by E n = E (0) n = P 2 n /(2M ), and the eigenfunction ofP 2 denoted as ψ is given by We can see that By substituting Eq. (11) into Eq. (10), one obtains The eigenfunction ψ of the Hamiltonian H 0 is given by The function Φ(r) can be obtained from the solution of the following equation: E 0 = E n +Ẽ is the eigenvalue of the HamiltonianĤ 0 (12) from the main text.Ẽ and Φ(r) can be obtained from the solution of the following equation: Equation (16) can be rewritten as Equation (17) is invariant with respect to the translation of the coordinate r and simultaneous gauge transformation. In this case,Φ(R, r) is defined asΦ By substitutingΦ(R, r) from Eq. (18) into Eq. (17), one obtains for the relative motion of a non-interacting electron-hole pair. The corresponding eigenvalues readẼñ and for eigenfunctuions of the operatorĤ 0 given by Eq. (12) from the main text, we get whereφ (0) n,m (r) is the wavefunction for a free particle of unit charge the effective PMFB = ( m 2 e + m 2 h ) B/M 2 in the cylindrical gauge [1,2,5]: where l = √h /B is the pseudomagnetic length. In Eq. (22), L |m| n denotes Laguerre polynomials. Let us mention that l is measured in m, sinceB is measured in kg/s. Note that we consider a PME formed by an electron and a hole located in the same type of valley, e.g., in the point K (or K ′ ) of the Brillouin zone.
Combining Eqs. (14), (18), and (21), one can see that the wavefunction of the electron-hole pair in the strain-induced PMF, neglecting the electron-hole attraction, can be written as where γ is defined by Eq. (11) from the main text, ψ n,m (R) is the wavefunction for a free particle in the effective PMF 2B in the cylindrical gauge [1,2,5],φ

B. The energy of a PME
To find the energy of a direct and indirect exciton PME one should evaluate the following matrix elements and V (r) is the Coulomb or RK potential. In the case of an indirect PME in the potential V (r) the corresponding interparticle distance should be replaced by the expression √ r 2 + D 2 [6][7][8][9], where D is the interlayer separation.

C. The energy for direct PME's for the Coulomb and Rytova-Keldysh potentials
The energy of a direct PME in a monolayer of gapped graphene double layer can be calculated by substituting the Coulomb potential into Eq. (6) and one obtains In Eq. (30) E 0 is given by where l = √h /B is the pseudomagnetic length. The analytical expressions for the energy of a direct PME obtained using the Rytova-Keldysh (RK) potential [3,4] are the following: where γ is Euler constant, Erfi(x) is the imaginary error function, the Maijer G−function, Ei(x) is the exponential integral function, and 1 F 1 (a, b; x) is the Kummer confluent hypergeometric function.

D. The energy for indirect PME's for the Coulomb potential
where Erfc(x) is the complementary error function and E 0 is given by (31). These expressions partially concise with the expressions obtained in the case of uniform magnetic field [2].

Interactions in double layer
The potential energy of e − h attraction V (|r e − r h |) in Eq. (5) from the main text can be described by the Rytova-Keldysh [3,4] or Coulomb potentials. The corresponding expressions that describe the interaction between the electron and hole which are located in different parallel graphene monolayers are the following: for the RK potential [8,9], and for the Coulomb potential. In Eqs. eq:indkeld and eq:indcoul D is the separation between two graphene layers. The results of our calculations are compared with the calculation, employing the Coulomb potential.