# Exciton energy in quantum wells¶

This tutorial aims to reproduce figures 6.4 (p. 196) and 6.5 (p. 197) of Paul Harrison’s excellent book
*Quantum Wells, Wires and Dots* (Section 6.5 *The two-dimensional and three-dimensional limits*”) ([HarrisonQWWD2005]),
thus the following description is based on the explanations made therein.

*We are grateful that the book comes along with a CD so that we were able to look up the relevant material parameters and to check the results for consistency.*

The following input file was used:

`1DExcitonCdTe_QW.in`

(input files for next**nano**³ software)CdTe quantum well with infinite barriers

In order to correlate the calculated optical transition energies of a 1D quantum well to experimental data,
one has to include *exciton* (electron-hole pair) corrections.
In this tutorial we study the exciton correction of the electron ground state to the heavy hole ground state (e1-hh1).

## Bulk¶

The 3D bulk exciton binding energy can be calculated analytically

\(E_\text{ex,b} = - \frac{ \mu e^4 }{ 32 \pi^2 \hbar^2 e_\text{r}^2 e_0^2 } = - \frac{\mu}{m_0 e_\text{r}^2} \cdot 13.61 \text{ eV}\),

where \(\mu\) is the reduced mass of the electron–hole pair, \(1/\mu = 1/m_\text{e} + 1/m_\text{h}\).

GaAs: \(1/\mu = 1 / 0.067 + 1 / 0.5\)

`==>`

\(\mu = 0.0591\)CdTe: \(1/\mu = 1 / 0.096 + 1 / 0.6\)

`==>`

\(\mu = 0.0828\)

\(\hbar\) is Planck’s constant divided by \(2\pi\)

\(e\) is the electron charge

\(e_\text{r}\) is the dielectric constant (GaAs: 12.93, CdTe: 10.6)

\(e_0\) is the vacuum permittivity

\(m_0\) is the rest mass of the electron and

13.61 eV is the Rydberg energy.

In GaAs, the *3D bulk exciton binding energy* is equal to -4.8 meV with a Bohr radius of \(\lambda = 11.6 \text{ nm}\).
In CdTe it is equal to -10.0 meV with a Bohr radius of \(\lambda = 6.8 \text{ nm}\).
Thus the energy of the exciton, i.e. the band gap transition, reads:

GaAs: \(E_\text{ex} = E_\text{gap} + E_\text{ex,b} = 1.519\text{ eV} - 0.005\text{ eV} = 1.514\text{ eV}\)

CdTe: \(E_\text{ex} = E_\text{gap} + E_\text{ex,b} = 1.606\text{ eV} - 0.010\text{ eV} = 1.596\text{ eV}\)

## Quantum well (type-I)¶

A 1D quantum well for a type-I structure has two exciton limits for the ground state transition (e1-hh1):

infinitely thin quantum well (2D limit)

\(E_\text{ex,QW} = 4 E_\text{ex}\), \(\lambda_{\text{ex,QW}} = \lambda_{\text{ex}} / 2\)

infinitely thick quantum well (3D bulk exciton limit)

\(E_\text{ex,QW} = E_\text{ex}\), \(\lambda_{\text{ex,QW}} = \lambda_{\text{ex}}\)

Between these limits, the exciton correction which depends on the well width has to be calculated numerically, not only for the ground state but also for excited states (e.g. e2-hh2, e1-lh1).

## CdTe quantum well with infinite barriers¶

In this tutorial we study the exciton binding energy of CdTe quantum wells (with infinite barriers) as a function of well width.

The material parameters used are the following ($binary-zb-default):

!---------------------------------------------------------! ! Here we are overwriting the database entries for CdTe. ! !---------------------------------------------------------! $binary-zb-default ! binary-type = CdTe-zb-default ! apply-to-material-numbers = 2 ! conduction-band-masses = 0.096 0.096 0.096 ! Gamma [m0] ... ! ! valence-band-masses = 0.6 0.6 0.6 ! heavy hole [m0] ... ! static-dielectric-constants = 10.6 10.6 10.6 ! []

We chose infinite barriers, in order to be able to compare the next**nano** calculations with standard textbook results,
originally published by [BastardPRB1982],
namely the exciton binding energy of a type-I quantum well (in units of the 3D bulk exciton energy \(E_{\text{ex}}\),
also called *effective Rydberg energy*) as a function of well width (in units of the 3D bulk exciton Bohr radius \(\lambda_{\text{ex}}\)).

### Template¶

The following screenshot shows how to use the *Template* feature of next**nano**mat in order to calculate the exciton binding energy as a function of the quantum well width.

Open input file in Template tab.

Select

*List of values*, select variable`QuantumWellWidth`

. The corresponding list of values are loaded from the template input file.Click on

*Create input files*to create an input file for each quantum well width.Switch to

*Simulation*tab and start the batch list of jobs.

### Results¶

The following figure shows the exciton binding energy in an infinitely deep quantum well as a function of well width. Both quantities are given in terms of the effective Rydberg energy and the Bohr radius for a 3D exciton in the same material.

Our numerical approach is the following:

The exciton binding energy is minimized with respect to the variational parameter \(\lambda\). We use a separable wave function:

\(\psi(r) = \sqrt{\frac{2}{\pi}} \frac{1}{\lambda} \exp(- r / \lambda)\)

see e.g. p. 562, Eq. (13.4.27), Section 13.4.3 *Variational Method for Exciton Problem* in [ChuangOpto1995] or [BastardPRB1982].

Thus the 3D limit is not reproduced correctly in our approach (not shown in the figure). To obtain the 3D limit, a nonseparable wave function has to be used, \(\psi(r,z_\text{e},z_\text{h})\).

The following figure shows the exciton binding energy in an infinitely deep CdTe quantum well as a function of well width.
The next**nano**³ results are in nice agreement with the Fig. 6.4 of [HarrisonQWWD2005] although we use a simpler trial wave function with only one variational parameter.

In order to calculate the exciton correction, the following flags have to be used:

$numeric-control simulation-dimension = 1 calculate-exciton = yes ! to switch on exciton correction exciton-electron-state-number = 1 ! electron ground state exciton-hole-state-number = 1 ! hole ground state

The output of the exciton binding energies can be found in this file:
`Schroedinger_1band/exciton_energy1D.dat`

The output for the 5 nm CdTe QW looks like this:

Exciton correction for 1D quantum wells (type-I structures) =========================================================== static dielectric constant = 10.6000000000 [] effective mass electron = 0.0960000000 [m0] effective mass hole = 0.6000000000 [m0] reduced mass = 0.0827586207 [m0] Bulk Bohr radius of 3D exciton = 6.7778780735 [nm] Bulk 3D exciton energy = -10.0212560410 [meV] lambda [nm] exciton energy [meV] exciton energy [Rydberg] 0.338893904E+001 -0.158496790E+002 0.158160603E+001 ... 0.421888329E+001 -0.215591082E+002 0.215133793E+001 ... 0.553296169E+001 -0.232757580E+002 0.232263879E+001 ----------------------------------------------------------------- ----------------------------------------------------------------- Calculated lambda and exciton energy: 0.546379967E+001 -0.232817837E+002 0.232324009E+001 -----------------------------------------------------------------

The last iteration yields -23.28 meV for the exciton binding energy.
`lambda`

is the variational parameter \(\lambda\) which is equivalent to the exciton Bohr radius in units of `[nm]`

.