New magneto-polaron resonances in a monolayer of a transition metal dichalcogenide

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General formulation

For the description of the MP resonance we employed the Green’s function method that reveals the inherent symmetries of phonons in the TMD and their influence on the MP quasiparticles. Other existing methods (for example, the modified Wigner- Brillouin perturbation theory) allow one to consider the influence of the EPIs in the MP spectrum and the coupling between the Landau states⁴, rather than the direct evaluation of the life-time and the interconnection between these MP characteristics as a function of the external magnetic field.

We consider a 2D TMD in an external uniform magnetic field ${\textbf {B}}=B{\textbf {e}}_z$ applied along the z-direction perpendicular to the plane of the sample. For a description of the MP properties, we start from the Dyson equation for the full Green’s function G(E, N)⁴⁹ for a particle in a static B

$$\begin{aligned} G(E,N) = G^{(0)}(E,N)+ G^{(0)}(E,N)\sum _{M}S(E,N,M)G(E,M)\;, \end{aligned}$$

(3)

where $G^{(0)}(E,N)$ is the one-particle unperturbed Green’s function at $T=0$ K, S(E, N, M) is the self-energy considering the interaction of the electronic Landau levels with the optical phonons and the sum is taken over all Landau states. Assuming a free 2D electron gas with a parabolic dispersion and an isotropic effective mass m under a constant B and neglecting the spin degree of freedom, the electronic wave function can be cast as

$$\begin{aligned} \psi _{N,k_y}= & {} \frac{1}{\sqrt{L}}e^{ik_{y}y}\varphi _{N}(x-x_{0}(k_{y}))\;;\;\;\;\;\;\; (N=0,1,2,…)\;, \end{aligned}$$

(4)

where $k_{y}$ is the particle wave vector, $\varphi _{N}$ the harmonic oscillator functions, $x_0=-k_yl^2_c$ the center of the orbit, $l_c=(\hbar c/eB)^{1/2}$ the Landau magnetic length and L the normalization constant⁸. The corresponding energy, without EPI, is $\epsilon _{N}=\hbar \omega _{c}(N+1/2)$ with $\omega _{c}=eB/mc$. In the Landau representation, the zero-temperature bare electron Green’s function is $G^{(0)}(E,N)=[E-\epsilon _{N}-i\delta ]^{-1}$ with $\delta$ a residual life-time broadening⁵⁰. Solving the Eq. (3) for the dressed Green’s function G(E, N), we obtain the resonant MP energy. The contribution of the self-energy to the renormalized Green’s function is determined by the EPIs. In 2D TMDs, at the center of the Brillouin Zone (BZ), there exist in-plane optical phonons (degenerate LO and TO branches) and one out-of-plane mode (ZO-homopolar branch) belonging to the irreducible representations $E^{‘}(\text {LO})$ and $A_1$, respectively⁵¹. The symmetries of these phonons are responsible for the electron intra-valley transitions via the long-range PF interaction for the LO-phonons and the short-range DP mechanism for the $A_1$-homopolar mode. Hence, both interactions have to be summed up for a correct evaluation of the self-energy: $S(E,N,M)=S_{\text {PF}}(E,N,M)+S_{\text {DP}}{(E,N,M})$. It is possible to show that S(E, N, M) is diagonal in the Landau quantum numbers N, M, that is $S(E,N,M)=\delta _{N,M}S(E,N)$ (see below). Thus, the Dyson equation for the full Green’s function G(E, N) is reduced to

$$\begin{aligned} G(E,N) = \frac{1}{\left[ G^{(0)}(E,N)\right] ^{-1} – \left[ S_{\text {DP}}(E,N)+S_{\text {PF}}(E,N)\right] }\;. \end{aligned}$$

(5)

The complex energy is $E=\hat{\epsilon }+i\Gamma$, where ${\hat{\epsilon }}=$Re[E] and $\Gamma =$Im[E] are the MP energy and its life-time broadening, respectively. The pole of the dressed Green’s function determines the renormalized energy $\hat{\epsilon }(B)$, while the life-time broadening $\Gamma (B)$ is given by $\text {Im}[G(E,N)^{-1}]$^8,49. From Eq. (5), we derive

$$\begin{aligned} {\hat{\epsilon }} = \hbar \omega _c\left( N +1/2\right) + \text {Re}\left[ S_{\text {DP}}(E,N)+S_{\text {PF}}(E,N) \right] \;, \end{aligned}$$

(6)

and

$$\begin{aligned} \Gamma = \text {Im}\left[ S_{\text {DP}}(E,N)+S_{\text {PF}}(E,N)\right]. \end{aligned}$$

(7)

Equations (6) and (7) are a set of two coupled transcendental non-linear equations. Solving them simultaneously, we obtain in what follows the energy spectrum ${\hat{\epsilon }}(B)$ as well as the the life-time broadening $\Gamma (B)$.

Self-energy

In 2D TMDs, the LO and $A_1$(ZO)-modes are responsible for the particle intra-valley transitions at the K point. Thus, the long-range PF and short-range DP interactions must be taken into account for a correct evaluation of the self-energy. The general structure of the EPI Hamiltonian is cast as

$$\begin{aligned} {\hat{H}}_{j}= \sum _{{\textbf {q}}} \left[ C_{{\textbf {q}}}^{(j)}e^{i\small {{\textbf {q}}\cdot \varvec{\rho }}}{\hat{b}}_{{\textbf {q}}}+C_{\textbf {q}}^{{(j)}\dag }e^{-i\small {{\textbf {q}}\cdot \varvec{\rho }}}{\hat{b}}_{{\textbf {q}}}^\dag \right], \end{aligned}$$

(8)

where $C_{{\textbf {q}}}^{(j)}$ is the coupling constant, $j=$ PF and DP, $\varvec{\rho }$ the in-plane coordinate and ${\hat{b}}_{{\textbf {q}}}$ (${\hat{b}}_{{\textbf {q}}}^\dag$) the annihilation (creation) phonon operator with the phonon wave vector ${\textbf {q}}$. Employing Eqs. (4) and (8), the polaron effect is considered through the self-energy S(E, N, M) of the Green’s function of Eq. (5). Considering the electronic wave function of Eq. (4), the irreducible self-energy (obtained when keeping only the Feynman diagrams, which cannot be separated into two disconnected pieces^49,52) to the lowest order in the coupling constant $C_{{\textbf {q}}}^{(j)}$ can be written as

$$\begin{aligned} S_{j}(E,N,M)= & {} \sum _{N’,k^{\prime }_y,k^{\prime \prime }_y,{\textbf {q}}}G^{(0)}(E-\hbar \omega _0,N’) C_q^{(j)} \langle N’,k^\prime _{y}\vert e^{i\small {{\textbf {q}}\cdot \varvec{\rho }}}\vert N,k_{y}\rangle \arrowvert C_q^{*(j)}\langle M,k_{y}^{\prime \prime }\vert e^{i\small {{\textbf {q}}\cdot \varvec{\rho }}}\vert N’,k_{y}^{\prime }\rangle \end{aligned}$$

(9)

$$\begin{aligned}= & {} \sum _{N’,k^{\prime }_y,k^{\prime \prime }_y,{\textbf {q}}} G^{(0)}(E-\hbar \omega _0,N’) \arrowvert C_q^{(j)}\arrowvert ^2\mathscr {L}_{N’,N}({\textbf {q}})\delta _{k^{\prime }_{y},k_{y}-q_{y}} \mathscr {L}^{*}_{M,N’}({\textbf {q}})\delta _{k^{\prime \prime }_{y},k_{y}^{\prime }-q_{y}}. \end{aligned}$$

(10)

The matrix element $\mathscr {L}_{N’,N}({\textbf {q}})$ is

$$\begin{aligned} \mathscr {L}_{N’,N}({\textbf {q}})=\int _{-\infty }^{\infty }\varphi _{N’}(x-x_0(k^{\prime }_{y}) )\varphi _N(x-x_0(k_{y}))e^{-iq_{x}x} dx, \end{aligned}$$

(11)

and $k^{\prime }_{y}=k_{y}-q_y$. Integrating over the coordinate x, we obtain

$$\begin{aligned} \mathscr {L}_{N’,N}({\textbf {q}})= & {} e^{-\frac{i}{2} q_x(k_y+k^{\prime }_y)l_c^{2}}e^{-\frac{1}{4}l_c^2q_{\perp }^2}\times \nonumber \\{} & {} \left\{ \begin{array}{lcl} 2^{\frac{N-N’}{2}}\sqrt{\frac{N’!}{N!}}\left[ \frac{l_c}{2}(q_y-iq_x)\right] ^{N-N’} L_{N’}^{N-N’}\left( \frac{1}{2}l_c^2 q_{\perp }^2\right) \,, &{}\text {for}\,&{} N \geqslant N’,\\ 2^{\frac{N’-N}{2}}\sqrt{\frac{N!}{N’!}}\left[ -\frac{l_c}{2}(q_y+iq_x)\right] ^{N’-N} L_{N}^{N’-N}\left( \frac{1}{2}l_c^2 q_{\perp }^2\right) \,, &{} \text {for} &{} N<N’, \end{array} \right. \end{aligned}$$

(12)

with $L_{p}^{q}(z)$ the generalized Laguerre polynomials⁵³. Representing q in the cylindrical coordinates and integrating over the polar angle result in $S_{j}(E,N,M)=\delta _{N,M}S_{j}(E,N)$, where

$$\begin{aligned} S_{j}(E,N)&=l_c^{-2}&\sum _{N’} G^{(0)}(E-\hbar \omega _{0},N’) \frac{S}{2\pi }\int _0^{\infty }\arrowvert C_Q^{(j)} \arrowvert ^2T_{N’,N}(Q)dQ, \end{aligned}$$

(13)

and

$$\begin{aligned} T_{N’,N}(Q)=e^{-Q}{} & {} \left\{ \begin{array}{lcl} \frac{N’!}{N!}Q^{N-N’}\arrowvert L_{N’}^{N-N’}(Q)\arrowvert ^2 &{} \text {for}\,, &{} N \geqslant N’\;, \\ \frac{N!}{N’!}Q^{N’-N}\arrowvert L_{N}^{N’-N}(Q)\arrowvert ^2 &{} \text {for}\,, &{} N<N’. \end{array} \right. \end{aligned}$$

(14)

Table 1 Employed parameters for the calculation of the energy spectrum from the set of Eqs. (6) and (7). The value of $\delta /\hbar \omega _{A_1}=0.02$ is used. a is the lattice constant and $\mathbb {G}_{Ph}$ is the PF coupling constant of Eq. (19).

Intra-band deformation potential

The DP characterizes the changes of the band energy under mechanical deformations of the primitive unit cell due to the optical lattice oscillations of the out-of-plane $A_1$-homopolar branch. In first-order approximation, the EPI is independent of the phonon wave vector, and the DP coupling constant is⁴¹

$$\begin{aligned} C_q^{\text {DP}} = \left( \frac{\hbar }{2\rho _mA N_c\omega _{A_1}}\right) ^{1/2}D_c, \end{aligned}$$

(15)

where $\rho _m$ is the 2D reduced mass density associated with the two chalcogen atoms, $\omega _{A_1}$ the ZO-phonon frequency, $D_c$ the deformation potential constant, $A=\sqrt{3}a^2/2$ the area of the unit cell, a the lattice constant, and $N_c$ the number of cells. Taking advantage of the fact that $C_q^{\text {DP}}$ is independent of the phonon wave vector and employing the result $\int _0^{\infty }T_{N^{‘},N}(Q)dQ=1$⁵⁵, the self-energy acquires the form

$$\begin{aligned} S_{\text {DP}}(E,N) = \hbar \omega _{c}\hbar \omega _{A_1}\alpha _{\text {DP}}&\sum \limits _{N’}[E-\hbar \omega _{A_1}-\hbar \omega _{c}(N’+1/2)-i\delta ]^{-1}\; \end{aligned}$$

(16)

with

$$\begin{aligned} \alpha _{\text {DP}}=\frac{m}{4\pi \rho _m }\left( \frac{D_c}{\hbar \omega _{A_1}}\right) ^2\;. \end{aligned}$$

(17)

Pekar–Fröhlich interaction

The in-plane motion of the positive M-ion relative to the X$_{i}$-ions is responsible for the long-range interaction. The macroscopic electrostatic potential associated with the in-plane LO-vibrations acts on the electron, leading to the EPI valid for a ML of TMD with a coupling constant⁴¹

$$\begin{aligned} C_q^{\text {PF}}= & {} \frac{\mathbb {G}_{Ph}}{\sqrt{N{_c}}(1+r_0q)}, \end{aligned}$$

(18)

$$\begin{aligned} \mathbb {G}_{Ph}= & {} \left( \frac{2 \pi ^2 e^2\hbar \alpha ^2}{\rho _mA\omega _{\text {LO}}}\right) ^{1/2}, \end{aligned}$$

(19)

where $\omega _{\text {LO}}$ is the in-plane phonon-frequency at ${q}={0}$, $\rho _{m}$ the mass density with the reduced atomic mass $\mu =m_{_{\text {M}}}^{-1}+(2m_{_{\text {X}}})^{-1}$, $\alpha$ the coupling constant between the atomic displacement and the in-plane macroscopic electric field, and $r_0$ the screening parameter⁴¹. In this case, the contribution of the long-range interaction to the self-energy is

$$\begin{aligned} S_{\text {PF}}(E,N)=\hbar \omega _{c}\hbar \omega _{A_1}\alpha _{\text {PF}}\sum _{N’}\frac{f_{N’,N}(r_0/l_c)}{E-\hbar \omega _{\text {LO}}-\hbar \omega _c\left( N’+\frac{1}{2}\right) -i\delta }, \end{aligned}$$

(20)

where

$$\begin{aligned} \alpha _{\text {PF}}=\frac{A\mathbb {G}_{Ph}^2}{2\pi \hbar \omega _{A_1}}\frac{m}{\hbar ^2} \end{aligned}$$

(21)

and

$$\begin{aligned} f_{N’,N}(z)=\int \limits _0^\infty \frac{dQ}{\left( 1+z\sqrt{2Q}\right) ^2} T_{N’,N}(Q). \end{aligned}$$

From the self-energy as given by Eqs. (16) and (20), it follows immediately that a mixing effect for the MP energy is present. The summation over all Landau levels leads to a coupling between different states. For a given quantum number N, the relative contribution of the other states $N’$ to E depends on the coupling constants $\alpha _{\text {DP}}$ and $\alpha _{\text {PF}}$.

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