Introduction

Since its invention of monolayer form1, graphene has attracted noticeable interest due to its extraordinary optical2, electrical3 and structural4 properties apart from its bulk form of graphite especially. Nowadays, another family of atomically thin materials, transition metal dichalcogenides (TMDs), have came into interest5.

As direct-gap semiconductors6, the monolayer members of TMD family (WS2, WSe2, MoS2, MoSe2 etc.) have a strong potential in optoelectronic applications7. In semiconductors, promoting of an electron from valance to conduction band leaving a hole behind creates an exciton by photo-excitation which especially finds applications in monolayer TMDs due to higher binding energies than in that of bulk semiconductors8,9 The spatial distance of these two particles is a result of screening effect of surrounding medium through Coulomb interaction10. These excitons can be used in variety of applications11,12. For example, extending exciton lifetimes in monolayer TMDs give possibility of energy storage in excitonic dark states even at room temperatures13.

Although there are much experimental studies focusing on measurement of exciton binding energies for monolayer TMDs14,15 and theoretical approaches for prediction of energy spectra of excitonic states16,17,18,19,20 the obtained results and predictions vary in the range of 0.3–1 eV9,14,15. Some research groups explain the differences by the substrate effects in which the monolayer TMD sample rely on it21. In addition, triangular lattice model was developed in order to explain non-hydrogen-like behavior of these quasi-particles22. In the traditional approach; the exciton is regarded as a single particle where the electron is attracted by the Coulomb potential of an hole where screening effect gives possibility of excitonic states.

On the other hand, exploring of relativistic dynamics of interacting particles has been very attractive in Quantum Field Theory (QFT). Although these topics have been studied for a long time, there are still much efforts to explain significant differences between experimental and theoretical results23. For example, relativistic two-body equation was written first by Eddington and Gaunt in 192924. In addition, a new relativistic Two Body Equation (TBE) was introduced short after25 which is relatively straightforward for interacting spin-1/2 particles since it includes approximate Darwin Lagrangian as an interaction term and 2-free Dirac Hamiltonian. This effective Hamiltonian fails either the velocity of particles or distance between particles is very high. Therefore, it covers actually weak coupling approximation.

Another research group obtained a different formulation of relativistic equation26 in the Quantum Field Theory (QFT) which satisfies the physics in25 up to the non-relativistic regime. Since they found negative solutions for binding energy, their equation was not valid as a bound-state solution of 2-interacting particles due to relative time approach.

Since we focus on the exciton interaction energy, bound-state equation of relativistic 2-Dirac particles is very important27,28. However, the interaction potential was written phenomenologically in these studies. A complete equation which is quite similar to27,28, starting from Quantum Electro-Dynamics (QED) and using Lagrange formalism assumed as a first principal in theoretical physics has been derived. This equation consists of most general electric and magnetic potentials29. Therefore, its solution gives us a well-known spectra similar to hydrogen-like atoms in all order (α6) even though it could not be solved completely30.

One can use equations in29 for solving of 2-body problem in (2 + 1) dimensions due to axial symmetry in Coulomb-like potentials and to obtain especially binding energy of electron-hole coupling (exciton) in monolayer materials. Although evaluating equations in (3 + 1) space-time is sometimes more complicated23,31,32,33, published results studied in (2 + 1) dimensions34,35 indicate that obtained results in (2 + 1) dimensions are very close to results obtained in (3 + 1) dimensions. Therefore, the direct solution of an exciton binding energy would be perfect regardless of surrounding medium. In this current work, exciton is regarded as 2 interacting oppositely charged fermions. Moreover, the solutions were obtained in 2 + 1 dimensions for a free-standing monolayer medium in order to eliminate any other external effects such as substrate, surrounding medium. For these purposes, two-body problem is exactly solved in 2 + 1 dimensions, firstly. Then, exciton binding energies and decay times are obtained in terms of spin and energy level of interacting particles as follows.

Two-body Dirac Equation in (2 + 1) Dimensions

We first start from two-body Dirac equation in (3 + 1) dimensions29 and fully covariant separation of center of mass and relative variable for acquiring energy spectrum of 2-interacting particles.

$$\begin{array}{l}\{[{\gamma }_{\mu }^{(1)}i\hslash ({\partial }_{\mu }^{(1)}-{e}_{1}{A}_{\mu }^{(2)})-{m}_{1}c]\otimes {\gamma }_{0}^{(2)}+\,{\gamma }_{0}^{(1)}\otimes [{\gamma }_{\mu }^{(2)}i\hslash ({\partial }_{\mu }^{(2)}-{e}_{2}{A}_{\mu }^{(1)})-{m}_{2}c]\}\varphi ({{\bf{X}}}_{1},{{\bf{X}}}_{2})=0\end{array}$$
(1)

Since the Dirac matrices have been chosen as \({\gamma }_{0}={\sigma }^{3}\), \({\gamma }_{1}=i{\sigma }^{1}\) and \({\gamma }_{2}=i{\sigma }^{2}\) satisfying Dirac algebra and two-body Dirac spinor is defined by Eq. 2 as follows;

$$\varphi ({{\bf{X}}}_{1},{{\bf{X}}}_{2})=(\begin{array}{c}(\begin{array}{c}{D}_{1}\\ {D}_{2}\end{array})\\ (\begin{array}{c}{D}_{3}\\ {D}_{4}\end{array})\end{array})$$
(2)

Equation 1 is reduced to (2 + 1) dimensions. Thus, we could obtain a coupled equation set from Eq. 1 where \({D}_{i}={D}_{i}({\bf{r}},{\bf{R}},{R}_{0})\). The explicit form of Di is given by the following expression.

$${D}_{i}={{\rm{\Psi }}}_{i}(r){\rm{\Phi }}(R){e}^{-iw{R}_{0}},{\rm{\Phi }}(R)={e}^{-i{\bf{k}}.{\bf{R}}}$$

Here, Di’s where i = 1, 2, 3, 4 are the spinor components and we use relative (r) and center of mass (R) coordinates explicitly. Then, we introduce the (2 + 1)-dimensional center of mass and relative variables with the following expressions.

$$\begin{array}{rcl}R & = & \frac{1}{M}({m}_{1}{X}_{1}+{m}_{2}{X}_{2})\\ r & = & {X}_{1}-{X}_{2}\\ M & = & {m}_{1}+{m}_{2}\\ {{\bf{X}}}_{1} & = & ({x}^{(1)},{y}^{(1)},{t}^{(1)})\\ {{\bf{X}}}_{2} & = & ({x}^{(2)},{y}^{(2)},{t}^{(2)})\end{array}$$
(3)

Here, m1 and m2 are the masses of particles where X1 and X2 are the magnitudes of position vectors of corresponding particles in 2 + 1 dimensions, respectively. Writing positions of particles in terms of center of mass and relative coordinates

$$\begin{array}{rcl}{X}_{1} & = & \frac{{m}_{2}}{M}(r+\frac{M}{{m}_{2}}R)\\ {X}_{2} & = & \frac{{m}_{1}}{M}(-r+\frac{M}{m1}R)\end{array}$$
(4)

and partial derivative of X1 and X2 give us the following equations.

$$\begin{array}{rcl}\frac{{\partial }^{(1)}}{\partial {X}_{\mu }} & = & \frac{\partial }{\partial {r}_{\mu }}+\frac{{m}_{1}}{M}\frac{\partial }{\partial {R}_{\mu }}\\ \frac{{\partial }^{(2)}}{\partial {X}_{\mu }} & = & -\frac{\partial }{\partial {r}_{\mu }}+\frac{{m}_{2}}{M}\frac{\partial }{\partial {R}_{\mu }}\\ \frac{{\partial }^{(1)}}{\partial {X}_{\mu }}+\frac{{\partial }^{(2)}}{\partial {X}_{\mu }} & = & \frac{\partial }{\partial {R}_{\mu }}\\ \frac{\partial }{\partial {r}_{\mu }} & = & \frac{1}{M}({m}_{2}\frac{{\partial }^{(1)}}{\partial {X}_{\mu }}-{m}_{1}\frac{{\partial }^{(2)}}{\partial {X}_{\mu }})\end{array}$$
(5)

where \(\mu =0,1,2\) and

$${\partial }_{0}={\partial }_{t}/c,\,{\partial }_{1}={\partial }_{x},\,{\partial }_{2}={\partial }_{y}.$$

For central interaction; the components of vector potential are taken as follows \({A}_{0}=V({{\bf{x}}}_{1}-{{\bf{x}}}_{2})\), \({A}_{1}=0\), \({A}_{2}=0\).

Derivation of Radial Equations

We can apply a method of separation of variables in terms of relative and center of mass coordinates provided that a central potential is taken into account; \(V\equiv V({\bf{r}})\).

In our calculations, we assume that the center of mass does not carry a momentum because we take k as k = 0. Therefore, one can write set of equation in terms of relative coordinates.

In the second part of calculations, we reorganized the coupled equation set by inserting Eq. 5 and definition of Di, we obtained the following equations.

$$(\frac{w}{c}-\frac{Mc}{\hslash }-i{V}_{t}(r)){{\rm{\Psi }}}_{1}+({\partial }_{{r}_{1}}-i{\partial }_{{r}_{2}}){{\rm{\Psi }}}_{2}-({\partial }_{{r}_{1}}-i{\partial }_{{r}_{2}}){{\rm{\Psi }}}_{3}=0,$$
(6)
$$(\,-\,\frac{w}{c}+\frac{{\rm{\Delta }}mc}{\hslash }+i{V}_{t}(r)){{\rm{\Psi }}}_{2}+({\partial }_{{r}_{1}}+i{\partial }_{{r}_{2}}){{\rm{\Psi }}}_{1}+({\partial }_{{r}_{1}}-i{\partial }_{{r}_{2}}){{\rm{\Psi }}}_{4}=0,$$
(7)
$$(\,-\,\frac{w}{c}-\frac{{\rm{\Delta }}mc}{\hslash }+i{V}_{t}(r)){{\rm{\Psi }}}_{3}-({\partial }_{{r}_{1}}-i{\partial }_{{r}_{2}}){{\rm{\Psi }}}_{4}-({\partial }_{{r}_{1}}+i{\partial }_{{r}_{2}}){{\rm{\Psi }}}_{1}=0,$$
(8)
$$(\frac{w}{c}+\frac{Mc}{\hslash }-i{V}_{t}(r)){{\rm{\Psi }}}_{4}-({\partial }_{{r}_{1}}+i{\partial }_{{r}_{2}}){{\rm{\Psi }}}_{3}+({\partial }_{{r}_{1}}+i{\partial }_{{r}_{2}}){{\rm{\Psi }}}_{2}=0$$
(9)

In order to write the coupled equation set in an explicit form, we use both

$$({\partial }_{{r}_{1}}-i{\partial }_{{r}_{2}})={e}^{-i{\rm{\Phi }}}(-\frac{i}{r}{\partial }_{{\rm{\Phi }}}+{\partial }_{r})$$
(10)

and

$$({\partial }_{{r}_{1}}+i{\partial }_{{r}_{2}})={e}^{i{\rm{\Phi }}}(\frac{i}{r}{\partial }_{{\rm{\Phi }}}+{\partial }_{r})$$
(11)

definitions and obtained the following explicit equation set in polar coordinates. In addition, we multiply the Eq. 6 by \({e}^{i{\rm{\Phi }}}\) and the Eq. 9 by \({e}^{-i{\rm{\Phi }}}\) on the left side. For the Eqs 7 and 8, the phase terms (\({e}^{i{\rm{\Phi }}}\) and \({e}^{-i{\rm{\Phi }}}\)) are required to be positioned in front of the operators paving the way that an additional term is included in front of the corresponding operator. By conducting these, we end up with the following equations;

$$\begin{array}{l}(\frac{w}{c}-\frac{Mc}{\hslash }-i{V}_{t}(r)){{\rm{\Psi }}}_{1}(r,{\rm{\Phi }}){e}^{i{\rm{\Phi }}}+(-\frac{i}{r}{\partial }_{{\rm{\Phi }}}+{\partial }_{r}){{\rm{\Psi }}}_{2}(r,{\rm{\Phi }})\\ \,-\,(-\frac{i}{r}{\partial }_{{\rm{\Phi }}}+{\partial }_{r}){{\rm{\Psi }}}_{3}(r,{\rm{\Phi }})=0\end{array}$$
(12)
$$\begin{array}{l}(-\frac{w}{c}+\frac{{\rm{\Delta }}mc}{\hslash }+i{V}_{t}(r)){{\rm{\Psi }}}_{2}(r,{\rm{\Phi }})+(\frac{i}{r}{\partial }_{{\rm{\Phi }}}+\frac{1}{r}+{\partial }_{r}){{\rm{\Psi }}}_{1}(r,{\rm{\Phi }}){e}^{i{\rm{\Phi }}}\\ \,+\,(-\frac{i}{r}{\partial }_{{\rm{\Phi }}}+\frac{1}{r}+{\partial }_{r}){{\rm{\Psi }}}_{4}(r,{\rm{\Phi }}){e}^{-i{\rm{\Phi }}}=0\end{array}$$
(13)
$$\begin{array}{l}(-\frac{w}{c}-\frac{{\rm{\Delta }}mc}{\hslash }+i{V}_{t}(r)){{\rm{\Psi }}}_{3}(r,{\rm{\Phi }})-(-\frac{i}{r}{\partial }_{{\rm{\Phi }}}+\frac{1}{r}+{\partial }_{r}){{\rm{\Psi }}}_{4}(r,{\rm{\Phi }}){e}^{-i{\rm{\Phi }}}\\ \,-\,(\frac{i}{r}{\partial }_{{\rm{\Phi }}}+\frac{1}{r}+{\partial }_{r}){{\rm{\Psi }}}_{1}(r,{\rm{\Phi }}){e}^{i{\rm{\Phi }}}=0\end{array}$$
(14)
$$\begin{array}{l}(\frac{w}{c}+\frac{Mc}{\hslash }-i{V}_{t}(r)){{\rm{\Psi }}}_{4}(r,{\rm{\Phi }}){e}^{-i{\rm{\Phi }}}-(\frac{i}{r}{\partial }_{{\rm{\Phi }}}+{\partial }_{r}){{\rm{\Psi }}}_{3}(r,{\rm{\Phi }})\\ \,+\,(\frac{i}{r}{\partial }_{{\rm{\Phi }}}+{\partial }_{r}){{\rm{\Psi }}}_{2}(r,{\rm{\Phi }})=0\end{array}$$
(15)

where \({{\rm{\Psi }}}_{g}(r,{\rm{\Phi }})={{\rm{\Psi }}}_{g}(r){e}^{i{j}_{g}{\rm{\Phi }}}\) \((g=1,2,3,4)\).

Since the \({\rm{\Phi }}\) is a cyclic coordinate, we derive a coupled first order 4-differential equations by using this property. Then, multiplying the Eq. 12 by \((\frac{w}{c}+\frac{Mc}{\hslash }-i{V}_{t}(r))\), the Eq. 15 by \((\frac{w}{c}-\frac{Mc}{\hslash }-i{V}_{t}(r))\), the Eq. 13 by \((-\frac{w}{c}-\frac{{\rm{\Delta }}mc}{\hslash }+i{V}_{t}(r))\) and the Eq. 14 by \((-\frac{w}{c}+\frac{{\rm{\Delta }}mc}{\hslash }+i{V}_{t}(r))\) on the left side of equations cause in a new definition set indicated as follows.

Derivation of Complete 2nd Order Equation

By pursuing above mathematical procedures and substituting \({{\rm{\Psi }}}_{g}(r,{\rm{\Phi }})\), we obtained the following equations for general central potentials.

$$(U{(r)}^{2}-{M}_{1}^{2}){{\rm{\Psi }}}^{+}(r)+2({M}_{1}\frac{j}{r}+U(r)\frac{d}{dr}){{\rm{\Psi }}}_{0}(r)=0$$
(16)
$$(U{(r)}^{2}-{M}_{1}^{2}){{\rm{\Psi }}}^{-}(r)+2(U(r)\frac{j}{r}+{M}_{1}\frac{d}{dr}){{\rm{\Psi }}}_{0}(r)=0$$
(17)
$$(U{(r)}^{2}-{M}_{2}^{2}){{\rm{\Psi }}}_{0}(r)+2({M}_{2}\frac{j}{r}){{\rm{\Psi }}}^{-}(r)-2{M}_{2}\frac{d}{dr}{{\rm{\Psi }}}^{+}(r)=0$$
(18)
$$(U{(r)}^{2}-{M}_{2}^{2}){{\rm{\Psi }}}_{0}(r)+2(U(r)\frac{j}{r}){{\rm{\Psi }}}^{-}(r)-2U(r)\frac{d}{dr}{{\rm{\Psi }}}^{+}(r)=0$$
(19)

where \({M}_{1}=\frac{Mc}{\hslash }\), \({M}_{2}=\frac{{\rm{\Delta }}mc}{\hslash }\), and \(U(r)=(\frac{w}{c}-i{V}_{t}(r))\). From Eqs 16 and 17, we rewrite \({{\rm{\Psi }}}^{+}(r)\) and \({{\rm{\Psi }}}^{-}(r)\) in terms of \({{\rm{\Psi }}}_{0}(r)\).

$$\begin{array}{rcl}{{\rm{\Psi }}}^{+}(r) & = & -\frac{2}{(U{(r)}^{2}-{M}_{1}^{2})}({M}_{1}\frac{j}{r}+U(r)\frac{d}{dr}){{\rm{\Psi }}}_{0}(r),\\ {{\rm{\Psi }}}^{-}(r) & = & -\frac{2}{(U{(r)}^{2}-{M}_{1}^{2})}(U(r)\frac{j}{r}r+{M}_{1}\frac{d}{dr}){{\rm{\Psi }}}_{0}(r)\end{array}$$

Inserting these expressions into the Eq. 19, we obtain a radial second order differential equation which contains general central potential.

Solution for a Coulomb Interaction

Since the exciton is composed of an electron and hole, the interaction between them can be written in terms of Coulomb potential. In addition, these particles (an electron and a hole) posses very small masses. Therefore, we can neglect the \({M}_{1}^{2}\) and \({M}_{2}^{2}\) in our equations.

$$\begin{array}{l}\frac{{d}^{2}}{d{r}^{2}}{{\rm{\Psi }}}_{0}(r)-[\frac{U^{\prime} (r)}{U(r)}]\frac{d}{dr}{{\rm{\Psi }}}_{0}(r)\,+\,[\frac{U{(r)}^{2}}{4}-\frac{{j}^{2}}{{r}^{2}}-\frac{2jU^{\prime} (r){M}_{1}}{U{(r)}^{2}}-\frac{j{M}_{1}}{U(r){r}^{2}}]{{\rm{\Psi }}}_{0}(r)=0\end{array}$$
(20)

Here we defined a new dimensionless variable; \(z=\frac{iwr}{2\alpha c}\) for \({V}_{t}(r)=-\,2\frac{\alpha }{r}\). Then, the solution of simplified Eq. 20 becomes a HeunC\(({\alpha }_{1},\beta ,\gamma ,\delta ,\eta ,{\rm{z}})\) function. In this function, frequency relation is found by using \(\delta =-\,(n+\frac{\beta +\gamma +2}{2}){\alpha }_{1}\) expression. The following parameters;

$$\begin{array}{rcl}{\alpha }_{1} & = & 2\alpha ,\\ \beta & = & -\,2\sqrt{{\alpha }^{2}+{j}^{2}},\\ \gamma & = & \frac{2}{(w)}\sqrt{-\,4iN{c}^{2}j\alpha +{(w)}^{2}},\\ \delta & = & 2{\alpha }^{2}\\ 2{\alpha }^{2} & + & 2(n-\sqrt{{\alpha }^{2}-{j}^{2}}+\tfrac{\sqrt{-\,4iN\alpha {c}^{2}j+{w}^{2}}}{w})\alpha =0\end{array}$$

yields

$$\begin{array}{rcl}{w}_{(n,j)} & = & \pm 2\sqrt{iN{c}^{2}\alpha j}[2\sqrt{{\alpha }^{2}-{j}^{2}}\alpha +2\sqrt{{\alpha }^{2}-{j}^{2}}n-2{\alpha }^{2}\\ & & {-2\alpha n-{j}^{2}-{n}^{2}+2\sqrt{{\alpha }^{2}-{j}^{2}}-2\alpha -2n]}^{-1/2}\end{array}$$
(21)

where \(N=\frac{Mc}{\hslash }\), \(M={m}_{1}+{m}_{2}\).

The explicit expression of w(n,j) allows us to obtain interaction energy of an exciton. This expression covers actually all the energy levels (n) where the particles might be and spin value of interacting particles. For clarity and quantification, a detailed table (Table 1) is composed of these variables.

Table 1 Interaction energies for an exciton.
Full size table

Discussion

Our calculations are very close to results found in literature15. Since our model exclude any other environmental effects such as substrate or crystal structure, we found a little bit lower value for an exciton binding energy. The experimental measurements of an exciton binding energy are based on the photo-excitation of an electron from a valance to a conduction band leaving a hole behind. While the interaction between an electron and a hole is totally attractive, the screening effect allows an exciton to be created. Since we only use the masses of the electron and the hole in our calculations, the calculated value for exciton energy (0.214 eV) is valid in all 2-dimensional systems. Moreover, we presume that the differences between exciton binding energies in the literature (even for the same samples-WS214,15) stem from surrounding environment such as substrate or crystal structure. Moreover, we think that the origin of the differences in experimental results of exciton binding energy is due to screening effect depending on the substrate or the crystal structure. Therefore, experimental measurements include this screening energy. Another interesting result of our calculations is that the interaction energy of 2-fermions possesses imaginary part. By using this imaginary part, one can calculate the decay time (\(\tau \)) of an exciton. Based on the definition in \(\tau \propto \frac{1}{{w}_{(n,j)}}\), we found relaxation time as 0.01 ps for n = 1 condition.

Conclusion

In this work, we found a general definition of interaction energy for 2-interacting fermions. First, we had examined two Dirac particles interacting with their central potential before we derived two-body Dirac equation in (2 + 1) space-time geometry. For solution of this equation, we separated center of mass and relative coordinates by using explicit form of the equation. Then, we obtained 1st order coupled radial differential equation set. Since the masses of an electron and a hole are very small (m1 and m2), we neglected \({M}_{1}^{2}\) and \({M}_{2}^{2}\) values in our equations. The solution of 2-interacting fermions in our model indicates that one can obtain general definition for an interaction energy including quantum numbers. We apply our model to an exciton and found its binding energy which is very close to results found in literature. Moreover, since we work in 2 + 1 dimensions, our results do not include any other effects for a monolayer medium. Finally, we calculated the decay time of an exciton (~ps) directly because the interaction energy in our calculations possesses imaginary part.