Exciton Wannier - Motta

	Exciton Wannier - Motta
Structure:	Quasiparticle
Classification:	Frenkel exciton
A family:	Boson
Group:	Exciton
Theoretically substantiated:	Frenkel in 1931
Spin :	Whole ħ

The Wannier – Mott exciton is an exciton whose radius significantly exceeds the characteristic period of the crystal lattice (in contrast to the Frenkel excitons ).

Wannier - Mott excitons exist in semiconductors due to the high dielectric constant of the latter. High dielectric constant leads to a weakening of the electrostatic attraction between the electron and the hole, which leads to a large radius of the exciton.

Content

1 On the origin of the term
2 Energy spectrum of an exciton
- 2.1 Three-dimensional case
- 2.2 Two-dimensional case
- 2.3 The effect of shielding
- 2.4 Manifestations of the exciton spectrum
3 Literature

About the origin of the term

The very concept of an exciton was proposed by Frenkel in 1931 . Frenkel expressed and substantiated the idea of the existence of such quasiparticles. The concept of a large-radius exciton, as one of the limiting cases of an exciton in general, is based on the theoretical work of Wannier , but is finally formulated in the works of Mott . Therefore, such a quasiparticle is called the Wannier – Mott exciton.

Exciton energy spectrum

3D Case

To calculate the energy spectrum of the Wannier – Mott exciton, we use the simplest model. Since the distance between the electron and the hole is large, we can use the effective mass method . We assume that the masses of the electron and the hole are isotropic, and the interaction between them is determined by the Coulomb law . Then the Schrödinger equation for such a system will have the form:

\left({\frac {{\hat {p}}_{e}^{2}}{2m_{e}}}+{\frac {{\hat {p}}_{h}^{2}}{2m_{h}}}-{\frac {e^{2}}{{\varepsilon }r}}\right)\Psi =E\Psi

{\ displaystyle \ left ({\ frac {{\ hat {p}} _ {e} ^ {2}} {2m_ {e}}} + {\ frac {{\ hat {p}} _ {h} ^ {2}} {2m_ {h}}} - {\ frac {e ^ {2}} {{\ varepsilon} r}} \ right) \ Psi = E \ Psi}

The change of variables, separating the translational motion of the center of mass and the rotational motion of particles around the center of mass, leads the equation to the form

\left({\frac {{\hat {p}}_{ex}^{2}}{2\mu }}-{\frac {e^{2}}{{\varepsilon }r}}\right)\Phi (\mathbf {r} )=\left(E-{\frac {\hbar ^{2}k_{ex}^{2}}{2M}}\right)\Phi (\mathbf {r} )

{\ displaystyle \ left ({\ frac {{\ hat {p}} _ {ex} ^ {2}} {2 \ mu}} - {\ frac {e ^ {2}} {{\ varepsilon} r} } \ right) \ Phi (\ mathbf {r}) = \ left (E - {\ frac {\ hbar ^ {2} k_ {ex} ^ {2}} {2M}} \ right) \ Phi (\ mathbf {r})}

Here $M=m_{e}+m_{h}$ ${\ displaystyle M = m_ {e} + m_ {h}}$ , $\mu =\left({\tfrac {1}{m_{e}}}+{\tfrac {1}{m_{h}}}\right)^{-1}$ ${\ displaystyle \ mu = \ left ({\ tfrac {1} {m_ {e}}} + {\ tfrac {1} {m_ {h}}} \ right) ^ {- 1}}$ - reduced mass $\mathbf {r} =\mathbf {r} _{e}-\mathbf {r} _{h}$ ${\ displaystyle \ mathbf {r} = \ mathbf {r} _ {e} - \ mathbf {r} _ {h}}$ .

This equation is similar to the Schrödinger equation for a hydrogen atom . It follows that the dispersion dependence of the exciton energy has the form

E_{n}(k_{ex})=-{\frac {{\mu }e^{4}}{2\hbar ^{2}\varepsilon ^{2}n^{2}}}+{\frac {\hbar ^{2}k_{ex}^{2}}{2M}}=-{\frac {R_{ex}}{n^{2}}}+{\frac {\hbar ^{2}k_{ex}^{2}}{2M}}

{\ displaystyle E_ {n} (k_ {ex}) = - {\ frac {{\ mu} e ^ {4}} {2 \ hbar ^ {2} \ varepsilon ^ {2} n ^ {2}}} + {\ frac {\ hbar ^ {2} k_ {ex} ^ {2}} {2M}} = - {\ frac {R_ {ex}} {n ^ {2}}} + {\ frac {\ hbar ^ {2} k_ {ex} ^ {2}} {2M}}}

Value $R_{ex}={\tfrac {\mu e^{4}}{2\hbar ^{2}\varepsilon ^{2}}}$ ${\ displaystyle R_ {ex} = {\ tfrac {\ mu e ^ {4}} {2 \ hbar ^ {2} \ varepsilon ^ {2}}}}$ by analogy with Rydberg for a hydrogen atom is called exciton Rydberg .

Two-dimensional case

Shielding Impact

At high concentrations of charge carriers in the semiconductor, screening of the Coulomb interaction becomes significant and the destruction of Wannier – Mott excitons can occur. In the presence of free carriers, the potential of the Coulomb interaction has the form

V(r)={e^{2} \over \varepsilon r}e^{-r/r_{D}}

{\ displaystyle V (r) = {e ^ {2} \ over \ varepsilon r} e ^ {- r / r_ {D}}}

,

Where $r_{D}={\mathcal {E}}kT/4\pi e^{2}N$ {\ displaystyle r_ {D} = {\ mathcal {E}} kT / 4 \ pi e ^ {2} N} - Debye screening radius . Here $N$ ${\ displaystyle N}$ - concentration of free charge carriers.

If the radius of the first exciton state with $n=1$ ${\ displaystyle n = 1}$ $a_{ex}={\tfrac {\hbar ^{2}\varepsilon }{\mu e^{2}}}$ ${\ displaystyle a_ {ex} = {\ tfrac {\ hbar ^ {2} \ varepsilon} {\ mu e ^ {2}}}}$ ( Bohr radius of the Wannier – Mott exciton), then the condition for the disappearance of the exciton series due to screening: $a_{ex}>r_{D}$ ${\ displaystyle a_ {ex}> r_ {D}}$ . For Wannier – Mott exciton in crystals $G e$ ${\ displaystyle Ge}$ this condition is satisfied at a donor concentration of ~ 10 ¹⁷ cm ^{– 3} and T = 77 K. Thus, to observe weakly bound excitons in semiconductors, low temperatures and pure crystals are required.

Manifestations of the exciton spectrum

Absorption spectrum near the edge of the forbidden band in a direct-gap semiconductor with the participation of excitons (solid lines) and without exciton effects (dashed line).

Wannier - Mott excitons are clearly manifested in the absorption spectra of semiconductors in the form of narrow lines shifted by $E_{n}$ ${\ displaystyle E_ {n}}$ below the edge of the optical absorption . The hydrogen-like spectrum of Wannier – Mott excitons was first observed in the absorption spectrum of Cu ₂ O in 1952 by E. F. Gross and H. A. Karyev and independently by M. Hayasi and K. Katsuki, but an excitonic interpretation of it was absent in the work of Japanese authors. Excitons also appear in the luminescence spectra, in photoconductivity, in the Stark effect and the Zeeman effect .

Literature

Gross E.F. Exciton and its motion in the crystal lattice, "Advances in Physical Sciences", 1962, v. 76, c. 3;
Knox R. Theory of excitons, trans. from English, - M. , 1966;
Agranovich V. M. Theory of excitons. - M. , 1968;
Davydov A.S. Theory of molecular excitons. - M. , 1968;
Excitons in semiconductors, [Sat. articles], - M. , 1971;
Osipyan Yu. A. Solid state physics is at the forefront, - "Nature", 1975, No. 10.

Exciton Wannier - Motta
Structure:	Quasiparticle
Classification:	Frenkel exciton
A family:	Boson
Group:	Exciton
Theoretically substantiated:	Frenkel in 1931
Spin :	Whole ħ