Binding energies and current density of heavy-hole trions of monolayer transition metal dichalcogenides: analytical perturbation treatment of Coulomb interaction with 2D H-like basis set

Abstract

Heavy-hole trions of transition metal dichalcogenides (TMDCs) have 2D counterpart $H^{-}$. Thus, 2D He-isoelectronic ions can be the ideal model for it even within Born–Oppenheimer (BO) approximation depending on versatile mass ratios. Schr$\ddot{o}$dinger equation of such systems is a long-standing problem because of secular divergence of Coulomb interactions. Analytical description of Coulomb (exchange) potential by algebraic-calculus Green’s function multipole expansion for ground-state energy (GSE) of 2D He-isoelectronic ions and binding energies of TMDCs has become one of indispensable methods. Employing associated Laguerre polynomial and Whittaker-M or Bessel functions in planar hydrogenic orbitals furnishes exact, terminable, simple and finitely summed integrals in terms of Lauricella functions for multipoles and also remedies the difficulties due to different scaling factors of higher-order perturbation calculations. Monopole and dipole factors are exploited upto third-order perturbation corrections of GSE with singly and doubly excited hydrogenic orbitals. The orbitals with circular and dumbbell symmetries exhibit GSEs differing 1.82% from reported results for 2D He atom. Binding energies of heavy-hole trions of TMDCs are observed to be in good agreement with the experimentally reported results.

Graphic abstract

Appendices

Appendix 1: 2D Coulomb interaction and Green$'$s function expansion (algebraic-calculus approach)

The free electrostatic Green$'$s function expansion of 2D Coulomb interaction reads as [41]:

$$\begin{aligned} \text {ln}\bigg (\frac{1}{r_i^2+r_j^2-2r_ir_jcos(\phi _i-\phi _j)}\bigg )^{1/2}&=\text {ln}\bigg (\frac{1}{r_{>}}\bigg )+\sum _{m=1}^{\infty }\frac{1}{m} \bigg (\frac{r_{<}}{r_{>}}\bigg )^mcos[m(\phi _i-\phi _j)]\\ &\Rightarrow \text {ln}\bigg (\frac{1}{r_i^2+r_j^2-2r_ir_jcos(\phi _i-\phi _j)}\bigg )^{1/2} \bigg /\bigg (\frac{1}{r_{>}}\bigg )=\sum _{m=1}^{\infty }\frac{1}{m} \bigg (\frac{r_{<}}{r_{>}}\bigg )^mcos[m(\phi _i-\phi _j)]\\ & \Rightarrow \bigg (\frac{1}{r_i^2+r_j^2-2r_ir_jcos(\phi _i-\phi _j)}\bigg )^{1/2}\bigg / \bigg (\frac{1}{r_{>}}\bigg )=\text {exp}\bigg (\sum _{m=1}^{\infty } \frac{1}{m}\bigg (\frac{r_{<}}{r_{>}}\bigg )^mcos[m(\phi _i-\phi _j)] \bigg )\\ &\quad=\text {exp}\bigg (\sum _{m=1}^{\infty }x^mB_m(\phi )\bigg ) \end{aligned}$$

(46)

where $x=\big (\frac{r_{<}}{r_{>}}\big )$, $B_m(\phi )=\frac{T_m(cos(\phi ))}{m}, \phi =\phi _i-\phi _j$ and $T_m(cos(\phi ))$ is the Chebyshev Polynomial.

$$\begin{aligned} \Rightarrow \text {exp}\bigg [\sum _{m=1}^{\infty }\frac{1}{m}\bigg (\frac{r_{<}}{r_{>}}\bigg )^mcos[m(\phi _i-\phi _j)]\bigg ]&=1+\sum _{m=1}^{\infty } \frac{1}{m}\bigg (\frac{r_{<}}{r_{>}}\bigg )^mcos[m(\phi _i -\phi _j)]+\frac{1}{2!}\bigg [\sum _{m=1}^{\infty }\frac{1}{m} \bigg (\frac{r_{<}}{r_{>}}\bigg )^mcos[m(\phi _i-\phi _j)]\bigg ]^2\\ &\quad+\frac{1}{3!}\bigg [\sum _{m=1}^{\infty }\frac{1}{m} \bigg (\frac{r_{<}}{r_{>}}\bigg )^mcos[m(\phi _i-\phi _j)]\bigg ]^3 +\cdots+f_k+\cdots+f_\infty \end{aligned}$$

(47)

where

$$f_k=\frac{1}{k!}\bigg (\sum _{m=1}^{\infty }x^mB_m(\phi )\bigg )^k$$

Therefore, it is clearly seen from Eq. (47) that it can be expanded in polynomial expansion of x as:

$$\sum _{q=0}^{\infty }A_q(\phi )x^q=\sum _{k=0}^{\infty }f_k(x,\phi )$$

(48)

Differentiating Eq. (48) w.r.t. x n-times,

$$\sum _{q=n}^{\infty }{^{q}}P_{n}A_q(\phi )x^{q-n} =\sum _{k=0}^{\infty }f_k^{(n)}(x,\phi )$$

(49)

Now $f_k$ can be written as:

$$\begin{aligned} f_k&=\frac{1}{k!}\bigg (\mathop {\sum \limits _{r_1,r_2,\ldots ,r_\infty }}\nolimits ^{\!\!\!\!\!\!\!\!\!'} \frac{k!}{r_1!,r_2!,\ldots ,r_\infty !}\prod _{m=1}^{\infty }(x^mB_m(\phi ))^{r_m}\bigg )\\ f_k&=\frac{1}{k!}\bigg (\mathop {\sum \limits _{r_1,r_2,\ldots ,r_\infty }}\nolimits ^{\!\!\!\!\!\!\!\!\!'}\frac{k!}{\prod r_m!}\bigg (\prod _{m=1}^{\infty }(B_m(\phi ))^{r_m}\bigg )x^M\bigg )\end{aligned}$$

(50)

where $M=\sum _{m=1}^{\infty }mr_m$. The nth derivative of $f_k$ w.r.t. x can be written as:

$$f_k^{(n)}=\frac{1}{k!}\bigg (\mathop {\sum \limits _{r_1,r_2,\ldots ,r_\infty }}\nolimits ^{\!\!\!\!\!\!\!\!\!'}\frac{k!}{\prod r_m!}\bigg (\prod _{m=1}^{\infty }(B_m(\phi ))^{r_m}\bigg ) {^M}P_{n}x^{M-n}\bigg )$$

(51)

Substituting Eq. (51) in Eq. (49):

$$\sum _{q=n}^{\infty }{^{q}}P_{n}A_q(\phi )x^{q-n}=\sum _{k=0}^{\infty } \bigg (\mathop {\sum \limits _{r_1,r_2,\ldots ,r_\infty }}\nolimits ^{\!\!\!\!\!\!\!\!\!'}\frac{1}{\prod r_m!}\bigg (\prod _{m=1}^{\infty }(B_m(\phi ))^{r_m}\bigg ){^M}P_{n}x^{M-n}\bigg )$$

(52)

Letting $x=0$, terms with $q>n$ and $M>n$ on L.H.S. and R.H.S., respectively, vanish.

$$A_n(\phi )=\sum _{k=0}^{\infty }\bigg (\mathop {\sum \limits _{r_1,r_2,\ldots ,r_\infty }}\nolimits ^{\!\!\!\!\!\!\!\!\!'} \frac{1}{\prod r_m!}\bigg (\prod _{m=1}^{\infty }(B_m(\phi ))^{r_m}\bigg )\bigg )$$

(53)

From Eq. (53), it can easily be evaluated that for monopole ($n=0$)$\Rightarrow$ $A_0=1$, dipole ($n=1$)$\Rightarrow$ $A_1=cos(\phi )$, quadrupole ($n=2$)$\Rightarrow$ $A_2=\frac{cos^2(\phi )+cos(2\phi )}{2}=\frac{3cos^2(\phi )-1}{2}$, octupole ($n=3$)$\Rightarrow$ $A_3=\frac{cos^3(\phi )+3cos(\phi )cos(2\phi )+2cos(3\phi )}{6}=\frac{5cos^3(\phi ) -3cos(\phi )}{2}$ and so on. It is clear from Eq. (53) that $A_n(\phi ) = P_n(\cos (\phi ))$, Legendre Polynomial. This is unprecedentedly improvised explicit relation between Chebyshev and Legendre polynomials, best to our knowledge. We believe that applying principle of mathematical induction, 2D Coulomb potential can be conjectured as:

$$\frac{1}{|\overrightarrow{r_i}-\overrightarrow{r_j}|}=\sum _{l=0}^{\infty } \frac{r_{<}^l}{r_{>}^{l+1}}P_l(cos(\phi _i-\phi _j))$$

(54)

Appendix 2: Dipole factor ($l=1$)

On substituting the values of $I_1$ and $I_2$ in Eq. (31) and taking $\xi (\alpha _1r_i)$ and $\xi (\alpha '_1r_i)$ in Whittaker-M function form, the integrals $I_s$ and $I_g$ become:

$$\begin{aligned} I_s&= \frac{BB'NN'}{(\alpha _1\alpha '_1)^{1/2}\,\alpha _2^3} \sum _{k_1=0}^{\mathscr {N}_2}\sum _{k_2=0}^{{\mathscr {N}}'_2}\frac{(-1)^{k_1+k_2}}{k_1!\,k_2!}\,(D+2)!\,w^{(|m'_2|+k_1)} g^{(D+3)}\left( {\begin{array}{c}n_2+|m_2|-1\\ 2|m_2|+k_1\end{array}}\right) \left( {\begin{array}{c}n'_2+|m'_2|-1\\ 2|m'_2|+k_2\end{array}}\right) \\ &\quad\times \bigg [\underbrace{\int _{0}^{\infty }r_i^{-2}M_{\kappa _1,\nu _1} (\alpha _1r_i)M_{\kappa '_1,\nu '_1}(\alpha '_1r_i)}_{I_{s_a}}-\sum _{s=0}^{(D+4)} \frac{1}{s!}\bigg (\frac{\alpha _2}{g}\bigg )^s \underbrace{\int _{0}^{\infty }r_i^{s-2}e^{-(\frac{w+1}{2}\alpha _2r_i)} M_{\kappa _1,\nu _1}(\alpha _1r_i)M_{\kappa '_1,\nu '_1}(\alpha '_1r_i)}_{I_{s_b}}\bigg ]\end{aligned}$$

(55)

$$\begin{aligned} I_g&= \frac{BB'NN'}{(\alpha _1\alpha '_1)^{1/2}\,} \sum _{k_1=0}^{\mathscr {N}_2}\sum _{k_2=0}^{{\mathscr {N}}'_2}\frac{(-1)^{k_1+k_2}}{k_1!\,k_2!} \,(D-1)!\,w^{(|m'_2|+k_1)}\,g^{(D)} \left( {\begin{array}{c}n_2+|m_2|-1\\ 2|m_2|+k_1\end{array}}\right) \left( {\begin{array}{c}n'_2+|m'_2|-1\\ 2|m'_2|+k_2\end{array}}\right) \bigg [\sum _{s=0}^{(D-1)}\frac{1}{s!}\bigg (\frac{\alpha _3}{g}\bigg )^s \\ &\quad\times \underbrace{\int _{0}^{\infty }r_i^{s+1}e^{-(\frac{w+1}{2}\alpha _2r_i)} M_{\kappa _1,\nu _1}(\alpha _1r_i)M_{\kappa '_1,\nu '_1}(\alpha '_1r_i)}_{I_{g_a}}\bigg ]\end{aligned}$$

(56)

Employing the standard integral (Eq. (64)) to solve $I_{s_a}, I_{s_b}$ and $I_{g_a}$ results in the following finite double summed forms.

For ${{\varvec{I}}}_{{{\varvec{s}}}_{{\varvec{a}}}}, \varrho = -1, c=0, a_1 = \alpha _1, a_2 = \alpha '_1, A = \frac{1}{2}(\alpha _1+\alpha _2), c+A = \frac{\alpha _1+\alpha '_1}{2}, \gamma _1 = |m_1|+\frac{1}{2}, \gamma _2 = |m'_1|+\frac{1}{2}, M = \gamma _1+\gamma _2 = |m_1|+|m'_1|+1, \kappa _1=n_1-\frac{1}{2}, \kappa '_1=n'_1-\frac{1}{2}$:

$$I_{s_a}=\alpha _1^{(|m_1|+\frac{1}{2})}(\alpha '_1)^{(|m_1|+\frac{1}{2})}\bigg (\frac{\alpha _1+\alpha _2}{2}\bigg )^{-(|m_1|+|m'_1|)}\Gamma \left( |m_1|+|m'_1|\right) \sum _{p=0}^{{\mathscr {N}}_1}\sum _{q=0}^{{\mathscr {N}}'_1}\frac{(|m_1|+|m'_1|)_{(p+q)}(-{\mathscr {N}}_1)_p(-{\mathscr {N}}'_1)_q}{(2|m_1|+1)_p\,(2|m'_1|+1)_q \,p!\,q!}\bigg (\frac{2\alpha _1}{\alpha _1+\alpha '_1}\bigg )^p\bigg (\frac{2\alpha '_1}{\alpha _1+\alpha '_1}\bigg )^q$$

(57)

For ${{\varvec{I}}}_{{{\varvec{s}}}_{{\varvec{b}}}}, \varrho = s-1, c = \frac{(w+1)\alpha _2}{2}, a_1 = \alpha _1, a_2 = \alpha _2, A = \frac{1}{2}(\alpha _1+\alpha _2), c+A = \frac{(w+1)\alpha _2+\alpha _1+\alpha '_1}{2}, \gamma _1 = |m_1|+\frac{1}{2}, \gamma _2 = |m'_1|+\frac{1}{2}, M = \gamma _1+\gamma _2 = |m_1|+|m'_1|+1, \kappa _1=n_1-\frac{1}{2}, \kappa '_1=n'_1-\frac{1}{2}$:

$$\begin{aligned} I_{s_b}&=\alpha _1^{(|m_1|+\frac{1}{2})}(\alpha '_1)^{(|m_1|+\frac{1}{2})}\bigg (\frac{(w+1)\alpha _2+\alpha _1+\alpha _2}{2}\bigg )^{-(s+|m_1|+|m'_1|)}\Gamma \left( s+|m_1|+|m'_1|\right) \sum _{p=0}^{{\mathscr {N}}_1}\sum _{q=0}^{{\mathscr {N}}'_1}\frac{(s+|m_1|+|m'_1|)_{(p+q)}}{(2|m_1|+1)_p\,(2|m'_1|+1)_q}\\ &\quad\times \frac{(-{\mathscr {N}}_1)_p(-{\mathscr {N}}'_1)_q}{p!\,q!}\bigg (\frac{2\alpha _1}{(w+1)\alpha _2+\alpha _1+\alpha '_1}\bigg )^p\bigg (\frac{2\alpha '_1}{(w+1)\alpha _2+\alpha _1+\alpha '_1}\bigg )^q\end{aligned}$$

(58)

Similarly for ${{\varvec{I}}}_{{{\varvec{g}}}_{{\varvec{a}}}}, \varrho = s+2, c = \frac{(w+1)\alpha _2}{2}, a_1 = \alpha _1, a_2 = \alpha _2, A = \frac{1}{2}(\alpha _1+\alpha _2), c+A = \frac{(w+1)\alpha _2+\alpha _1+\alpha '_1}{2}, \gamma _1 = |m_1|+\frac{1}{2}, \gamma _2 = |m'_1|+\frac{1}{2}, M = \gamma _1+\gamma _2 = |m_1|+|m'_1|+1, \kappa _1=n_1-\frac{1}{2}, \kappa '_1=n'_1-\frac{1}{2}$:

$$\begin{aligned} I_{g_a}&=\alpha _1^{(|m_1|+\frac{1}{2})}(\alpha '_1)^{(|m_1|+\frac{1}{2})}\bigg (\frac{\alpha _1+\alpha _2}{2}\bigg )^{-(s+|m_1|+|m'_1|+3)}\Gamma \left( s+|m_1|+|m'_1|+3\right) \sum _{p=0}^{{\mathscr {N}}_1}\sum _{q=0}^{{\mathscr {N}}'_1}\frac{(s+|m_1|+|m'_1|+3)_{(p+q)}}{(2|m_1|+1)_p\,(2|m'_1|+1)_q \,}\\ &\quad\times \frac{(-{\mathscr {N}}_1)_p(-{\mathscr {N}}'_1)_q}{p!\,q!}\bigg (\frac{2\alpha _1}{(w+1)\alpha _2+\alpha _1+\alpha '_1}\bigg )^p\bigg (\frac{2\alpha '_1}{(w+1)\alpha _2+\alpha _1+\alpha '_1}\bigg )^q \end{aligned}$$

(59)

Substituting Eqs. (57–59) in Eq. (31), we get:

$$\begin{aligned} I_s&=\frac{BB'NN'\alpha _1^{(|m_1|)}(\alpha '_1)^{(|m'_1|)}}{\alpha _2^3} \sum _{k_1=0}^{\mathscr {N}_2}\sum _{k_2=0}^{{\mathscr {N}}'_2} \frac{(-1)^{k_1+k_2}}{k_1!\,k_2!}\,(D+2)!\, w^{|m'_2|+k_1}\,g^{(D+3)}\left( {\begin{array}{c}n_2+|m_2|-1\\ 2|m_2|+k_1\end{array}}\right) \left( {\begin{array}{c}n'_2+|m'_2|-1\\ 2|m'_2|+k_2\end{array}}\right) \\ &\quad\times \bigg [\bigg (\frac{\alpha _1+\alpha _2}{2} \bigg )^{-(|m_1|+|m'_1|)}\Gamma \left( |m_1|+|m'_1|\right) \sum _{p=0}^{{\mathscr {N}}_1}\sum _{q=0}^{{\mathscr {N}}'_1} \frac{(|m_1|+|m'_1|)_{(p+q)}(-{\mathscr {N}}_1)_p(-{\mathscr {N}}'_1)_q}{(2|m_1|+1)_p\, (2|m'_1|+1)_q \,p!\,q!}\bigg (\frac{2\alpha _1}{\alpha _1+\alpha '_1}\bigg )^p \bigg (\frac{2\alpha '_1}{\alpha _1+\alpha '_1}\bigg )^q \\ &\sum _{s=0}^{(A-1)}\bigg (\frac{\alpha _1+\alpha _2}{2}\bigg )^{-(s+|m_1|+|m'_1|)} \Gamma \left( s+|m_1|+|m'_1|\right) \sum _{p=0}^{{\mathscr {N}}_1} \sum _{q=0}^{{\mathscr {N}}'_1}\frac{(s+|m_1|+|m'_1|)_{(p+q)(-{\mathscr {N}}_1)_p(-{\mathscr {N}}'_1)_q}}{(2|m_1|+1)_p\,(2|m'_1|+1)_q \,p!\,q!} \\ &\quad\times \bigg (\frac{2\alpha _1}{(w+1)\alpha _2+\alpha _1+\alpha '_1} \bigg )^p\bigg (\frac{2\alpha '_1}{(w+1)\alpha _2+\alpha _1+\alpha '_1}\bigg )^q\bigg ] \end{aligned}$$

(60)

$$\begin{aligned}I_g&=NN'BB'\alpha _1^{|m_1|}\alpha _1^{|m_1'|}\sum _{k_1=0}^{{\mathscr {N}}_2} \sum _{k_1=0}^{{\mathscr {N}}_2^{'}}\frac{(-1)^{k_1+k_2}}{k_1!\,k_2!}w^{(|m_2|+k_2)} g^{(D)}\left( {\begin{array}{c}n_2+|m_2|-1\\ 2|m_2|+k_1\end{array}}\right) \left( {\begin{array}{c}n_2+|m_2|-1\\ 2|m_2'|+k_2\end{array}}\right) \\ &\qquad\bigg [\sum _{s=0}^{(D-1)}\frac{1}{s!}\bigg (\frac{\alpha _2}{g} \bigg )^s\Gamma \left( s+|m_2|+|m_2^{'}|+3\right) \bigg (\frac{(w+1)\alpha _3+\alpha _1 +\alpha _2}{2}\bigg )^{-(s+|m_2|+|m_2'|+3)} \\ &\quad \times \sum _{p=0}^{{\mathscr {N}}_1} \sum _{q=0}^{{\mathscr {N}}_2}\frac{(s+|m_2|+|m_2'|+3)_{p+q} (-{\mathscr {N}}_1)_p(-{\mathscr {N}}_2)_q}{(2|m_1|+1)_p(2|m_1'|+1)_q\,p!\,q!} \bigg (\frac{2\alpha _1}{(w+1)\alpha _2+\alpha _1+\alpha _1'}\bigg )^p\bigg (\frac{2\alpha _1'}{(w+1)\alpha _2+\alpha _1+\alpha _1'}\bigg )^q\bigg ]\end{aligned}$$

(61)

Adding Eqs. (60) and (61) gives the exact value of the integral (Eq. (31)).

Appendix 3: Standard equations and integrals

Associated Laguerre Polynomial [46]

$$L^{\mu }_{k}(x)=\sum _{m=0}^{k}(-1)^m\frac{(\mu +k)!}{(k-m)!(\mu +m)!m!}x^m \text {where}\quad \mu >-1$$

(62)

Lower and Upper Incomplete Gamma function[46]

$$\begin{aligned}&\gamma (a,x)=\int _{0}^{x}e^{-t}t^{a-1}dt=(a-1)!\bigg (1-e^{-x} \sum _{s=0}^{a-1}\frac{x^s}{s!}\bigg ) \\ &\Gamma (a,x)=\int _{x}^{\infty }e^{-t}t^{a-1}dt=(a-1)!\bigg (e^{-x} \sum _{s=0}^{a-1}\frac{x^s}{s!}\bigg )\quad \text {where} \quad Re(a) \ge 0 \end{aligned}$$

(63)

Standard Integral-I (Erdélyi’s Integral) [60]

$$\begin{aligned} &\int _0^{\infty }x^{\left( \varrho -1\right) } e^{-cx}M_{\kappa _1,\gamma _1- \frac{1}{2}}(a_1x)M_{\kappa _2, \gamma _2-\frac{1}{2}}(a_2x) \mathrm{d}x =a_1^{\gamma _1}a_2^{\gamma _2}(c+A)^{-\varrho -M}\Gamma (\varrho +M) \\ & \quad F_2\left( \varrho +M;\gamma _1-\kappa _1,\gamma _2-\kappa _2;2\gamma _1,2 \gamma _2;\frac{a_1}{c+A},\frac{a_2}{c+A}\right) \end{aligned}$$

(64)

where $Re(\varrho +M)>$0, $Re\left( c\pm \frac{1}{2}a_1\pm \frac{1}{2}a_2\right) >0$ and $M = \gamma _1+\gamma _2$

$F_2$ to $F_1$ relation [61]

$$F_{2}\left( \begin{array}{c} a,b,b' \\ c,c'\end{array}; {d,d'}\right) =\sum \limits _{k=0}^{\infty }\frac{(a)_k(b)_k \mathrm{d}^k}{(c)_k k!}{_{2}}F_{1}\left( \begin{array}{c} b',a+k \\ {c'}\end{array}; d' \right)$$

(65)

Chu-Vandermonde identity [62]

$${_{2}}F_{1}\left( \begin{array}{c} -b',a+k\\ {c'}\end{array}; {d'}\right) =\frac{(c'-(a+k))_{b'}}{(c')_{b'}} \quad \text {where}\quad Re(c') \ge 0\hbox { and }d'=1$$

(66)

Standard Integral-II (Erd$\acute{e}$lyi’s Integral) [63]

$$\int _0^{\infty } x^{\left( \sigma -1\right) }\,e^{-bx}\,L_{k_1}^{\mu _1}(a_1x)\,L_{k_2}^{\mu _2} (a_2x)\,\mathrm{d}x=(-1)^{(k_1+k_2)}\left( {\begin{array}{c}-(\mu _1+1)\\ k_1\end{array}}\right) \left( {\begin{array}{c}-(\mu _2+1)\\ k_2\end{array}}\right) \, b^{-\sigma }\,\Gamma (\sigma ) F_2\left( \sigma ;-k_1,-k_2;\mu _1+1,\mu _2+1;\frac{a_1}{b},\frac{a_2}{b}\right)$$

(67)

where $Re(\sigma )>0, Re(b) >0$