We consider the envelope tight-binding Hamiltonian of monolayer graphene as follows:

${H}_{\text{ex}}=\u220a\left({\mathbf{p}}_{e}\right)+\u220a\left({\mathbf{p}}_{h}\right)+V({\mathbf{r}}_{e}-{\mathbf{r}}_{h}),$

(1)

where

$\u220a\left(\mathbf{p}\right)={\gamma}_{0}\sqrt{1+4\text{cos}\frac{a{p}_{x}}{2}\text{cos}\frac{\sqrt{3}a{p}_{y}}{2}+4\stackrel{2}{cos}\frac{a{p}_{x}}{2}},$

(2)

is the single-electron energy, *a* = 0.246 nm is the lattice constant,$\hslash =1$, *V*(**r** )= −*e*^{2}/(*χr*) is the potential energy of the electron-hole interaction. The electron spectrum has conic points *ν* **K**,*ν* = ±1, **K** = (4*Π*/3*a*,0), where *∊*(**p**)≈*s*|**p**−*ν* **K**|,$s={\gamma}_{0}a\sqrt{3}/2$ is the electron velocity in the conic approximation.

The electron and hole momenta **p**_{e,h}can be expressed via pair **q**=**p**_{
e
} + **p**_{
h
}and relative **p**=**p**_{
e
}−**p**_{
h
} momenta. The momenta **p**_{e,h} can be situated near the same (*q* → *k* ≪ 2*K*) or near the opposite conic points (**q** = 2**K** + **k** ,*k* ≪ 2*K*).

We assumed that graphene is embedded into the insulator with a relatively large dielectric constant *χ* so that the effective dimensionless constant of interaction$g={e}^{2}/\left(\mathrm{s\chi \hslash}\right)\sim 2/\chi \ll 1$ and the many-body complications are inessential. In the conic approximation, the classical electron and hole with the same direction of momentum have the same velocities *s*. The interaction changes their momenta, but not their velocities. The two-particle Hamiltonian contains no terms quadratic in the component of the relative momentum **p** along **k**. In a quantum language, such attraction does not result in binding. Thus, the problem of binding demands accounting for the corrections to the conic spectrum.

Two kinds of excitons are potentially allowed in graphene: a direct exciton with

*k* ≪ 1/

*a*(when the pair belongs to the same extremum) and an indirect exciton with

**q** = 2

**K** +

**k**. Assuming

*p*≪

*k* (this results from the smallness of

*g*), we get to the quadratic Hamiltonian

${H}_{\text{ex}}=\text{sk}+\frac{{p}_{1}^{2}}{2{m}_{1}}+\frac{{p}_{2}^{2}}{2{m}_{2}}-\frac{{e}^{2}}{\text{\chi r}},$

(3)

where the coordinate system with the basis vectors

**e**_{1}≡

**k**/

*k* and

**e**_{2}⊥

**e**_{1} is chosen,

**r** = (

*x*_{1},

*x*_{2}). In the conic approximation, we have

*m*_{2} =

*k*/

*s*,

*m*_{1} =

*∞*. Thus, this approximation is not sufficient to find

*m*_{1}. Beyond the conic approximation (but near the conic point), we should expand the spectrum (2) with respect to

**k** up to the square terms, which results in the trigonal spectrum warping. As a result, we have for the indirect exciton,

$\begin{array}{l}\frac{1}{{m}_{1}}=\nu \frac{\text{sa}}{4\sqrt{3}}cos3{\varphi}_{\mathbf{k}},\end{array}$

(4)

where *ϕ*_{
k
} is an angle between **k** and **K**.

The effective mass

*m*_{1} ≫

*m*_{2}is directly determined by the trigonal spectrum warping, and the large value of

*m*_{1} follows from the warping smallness. The sign of

*m*_{1}is determined by

*ν* cos3

*ϕ*_{
k
}. If

*ν* cos3

*ϕ*_{
k
}> 0, electrons and holes tend to bind, or else to run away from each other. Thus, the binding of an indirect pair is permitted for

*ν* cos3

*ϕ*_{
k
}>0. Apart from the conic point, this condition transforms to

$\begin{array}{l}(1+u+{v}_{-})<0\wedge (1+u+{v}_{+})<0\vee \phantom{\rule{2em}{0ex}}\\ (1+u+{v}_{-})<0\wedge (1+{v}_{-}+{v}_{+})<0\vee \phantom{\rule{2em}{0ex}}\\ (1+u+{v}_{+})<0\wedge (1+{v}_{-}+{v}_{+})<0,\phantom{\rule{2em}{0ex}}\end{array}$

(5)

where$u=\text{cos}a{k}_{x},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{v}_{\pm}\text{=}cos\left(\right({k}_{x}\pm \sqrt{3}{k}_{y})a/2).$

To find the indirect exciton states analytically, we solved the Schrödinger equation with the Hamiltonian (3) using the large ratio of effective masses. This parameter can be utilized by the adiabatic approximation similar with the problem of molecular levels. Coordinates 1 and 2 play a role of heavy ‘ion’ and ‘electron’ coordinates. At the first stage, the ion term in the Hamiltonian is omitted, and the Schrödinger equation is solved with respect to the electron wave function at a fixed ion position. The resulting electron terms then are used to solve the ion equation. This gives the approximate ground level of exciton *ε*(**k**)=*sk*−*ε*_{
ex
}(**k**), where the binding energy of the exciton is *ε*_{
ex
}(**k**) = *Π*^{−1}*sk* *g*^{2} log^{2}(*m*_{1}/*m*_{2}) (the coefficient 1/*Π* here is found by a variational method).

A similar reasoning for the direct exciton gives negative mass *m*_{1}=−32/(*ks* *a*^{2}(7−cos6*ϕ*_{
k
})). As a result, the direct exciton kinetic energy of the electron-hole relative motion is not positively determined and that means the impossibility of binding of electrons with holes from the same cone point.