# Exciton energy in quantum wells - Tutorial¶

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 nextnano³ 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 nextnano 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 nextnanomat in order to calculate the exciton binding energy as a function of the quantum well width.

Figure 12 Parameter sweep: QuantumWellWidth

• 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.

Figure 13 Exciton energy as a function of quantum well width

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, $$\psi(r,z_\text{e},z_\text{h})$$.

Figure 14 Exciton binding energy in an infinitely deep quantum well

The following figure shows the exciton binding energy in an infinitely deep CdTe quantum well as a function of well width. The nextnano³ 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.

Figure 15 Exciton Bohr radius energy in an infinitely deep quantum well

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].