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## Transport in low dimensional structures

The case of resonant tunneling in a double barrier semiconductor is a textbook problem that is theoretically treated for instance in [7]. It was rst observed experimentally by Chang et al [8]. It consists in a tunneling through two successive barriers where the distance between the two barriers is small compared to the electron De Broglie wavelength (that is to say a few nanometers). In that case, the region between the barrier is a quantum well with conned states, and a peculiar behaviour is observed.

Let consider a case where a small band gap material A is sandwiched between two large band gap barriers in material B, the left and right reservoirs being material A (for instance, GaAs/AlGaAs/GaAs/AlGaAs/GaAs) which is respresented in gure 2.2.

The current is calculated using the Laudauer formalism described above. Equation on both sides of the system are Fermi-Dirac distributions at temperature T and Fermi energies 1 and 2 and the voltage across the double barrier is de- ned as V = (1 2)=q. The function (E) is simply determined by solving the Schrödinger equation in such geometry, so the current can be calculated.

At low voltage, the current increases exponentially with applied voltage. However, the position of the conned state is shifted (see gure 2.2). When this conned state is lower than the bottom of the conduction band in the left lead, resonant tunneling becomes impossible, and the current drops sharply. At higher voltage, the carriers can be transmitted by jumping above the barrier. A resonant behaviour is observed with a local maximum of current at a voltage Vr that correspond to a minimum of resistance. Immediately after the peak, the current decreases which can be seen as a negative dierential resistance eect.

Such architecture could be used as a possible selective contact. However, the transmission range is rather large (a few hundreds of meV, large compared to kT), and carriers outside of the transmission range are not blocked eciently. Another approch may provide better selectivity.

### State of the art of hot carrier solar cell models

Previous models of hot carrier solar cells were reminded in chapter 1. They are based on a detailed balance model introduced by Shockley and Queisser [13], where the current is dened as the dierence between the photogenerated carrier ux and the radiatively recombined carrier ux. Hot carrier solar cells though dier from this picture because the carrier temperature in the absorber TH is unknown and the quasi Fermi level splitting H is not the output voltage qV . Three variables are involved (TH, H and V ), so additional equations are necessary. An energy balance equation is considered, where the power delivered is the dierence between the absorbed power and the power lost by radiative recombinations [14, 1]. It was generally assumed that no other losses occur. In particular, thermalization losses due to the carrier cooling were neglected. Finally, ideal selective contacts were considered, which enables to express the output voltage V as a simple function of the other variables of the system. qV = (1 TC TH )Eext + H TC TH.

**Validation of the model**

Before investigating the inuence of the contact selectivity and conductivity on the cell eciency, it is necessary to check that the model developed here is in agreement with the previous Ross-Nozik model for ideal hot carrier solar cells with perfectly selective contacts. By setting the contact energy range to a value that is small compared to the kinetic energy of carriers (E kTH), for instance 1 meV, and with a contact degeneracy that is large enough to ensure carrier extraction without resistance losses, the two models should give the same results, at least for large band gaps (Eg > 0:5 eV). In the case of smaller band gaps, the Boltzman approximation made above is not valid because the quasi Fermi level splitting is below the band gap by less than kBTH. This model thus gives inaccurate results which might diverge from the Ross-Nozik model.

The eciency and the dierent losses for the two models are presented in gure 2.6 as a function of the band gap, in full concentration condition. A very good agreement is observed over the whole band gap range, which shows the validity of the model in the limit of an ideal hot carrier solar cell. A validation in the limit of a fully thermalized solar cell will be proposed in section 4.3.1.

**Criticality of the transmission range**

If perfectly selective contacts were shown to give optimal conversion eciency, provided that their conductance can be high enough, a quantitative analysis of the losses in contacts due to their non ideality is missing so far. A rst attempt was proposed in [17] but only in the case of a 0 eV band gap absorber. The case of a more realistic 1 eV band gap is treated here. Two cases will be considered: rst, a xed extraction energy equal to the average energy of absorbed photons, second, with an optimization on the extraction energy in each conguration to have the maximal achievable eciency. The rst situation is interesting to see the evolution with only one varying parameter (the contact transmission range). However, in order to determine the ultimate achievable eciency, it is more relevant to investigate the hot carrier solar cell behaviour in its optimal operating conditions. The contact conductance is taken optimal to enable maximal eciency. The cell eciency, heat losses in the contact and radiative losses are plotted as functions of the contact transmission range in gure 2.7, either with a xed extraction energy equal to the absorption energy (dashed line), or with an optimal extraction energy depending on the transmission range (solid line). The value of the optimal extraction energy as a function of the transmission range can be seen in gure 2.13. For E < 10 meV , the eciency is that of an ideal hot carrier solar cell. Both situations give the same eciency, because the absorption energy is the optimal extraction energy with highly selective contacts. In that case, E kBT so the contacts can be considered as a discrete level compared to the energy distribution of carriers.

**Table of contents :**

Remerciements

Résumé

Abstract

List of gures

List of tables

Introduction

Bibliography

**1 The hot carrier solar cell concept **

1.1 Principles of the hot carrier solar cell

1.2 Detailed balance models and limit of eciency

1.3 The mechanisms of carrier thermalization

1.3.1 Carrier-carrier scattering

1.3.2 Electron/hole-phonon interaction

1.3.3 Electron-hole plasma dynamic

1.4 Energy selective contacts for carrier extraction

1.4.1 Role of selective contacts

1.4.2 Technological feasibility

Bibliography

**2 Heat losses in the selective contacts **

2.1 Models of non-ideal contacts

2.1.1 Landauer formalism

2.1.2 Transport in low dimensional structures

2.2 Analytical model

2.2.1 State of the art of hot carrier solar cell models

2.2.2 Formulation of the problem

2.2.3 The Boltzman distribution approximation

2.2.4 Other approximations

2.2.5 Analytical formulation

2.2.6 Eciency and losses

2.3 Results of simulations

2.3.1 Validation of the model

2.3.2 Criticality of the transmission range

2.3.3 Optimal extraction energy

2.3.4 Impact of contact conductance

2.4 Conclusion

Bibliography

**3 Experimental study of carrier thermalization **

3.1 Hot carrier spectroscopy

3.1.1 Continuous wave photoluminescence (CWPL)

3.1.2 Ultrafast spectroscopy

3.1.3 Steady state and transient state

3.2 Description of samples

3.2.1 Material selection

3.2.2 Strained samples

3.2.3 Lattice-matched samples

3.3 Experimental setup

3.3.1 Setup A

3.3.2 Setup B

3.3.3 Comparison

3.3.4 Excitation power density

3.4 Experimental results: temperature determination and thermalization rate measurement

3.4.1 Photoluminescence spectra analysis

3.4.2 Qualitative study of carrier thermalization

3.4.3 Carrier thermalization time and thermalization rate

3.4.4 Inuence of the lattice temperature

3.5 Conclusion

Bibliography

**4 Simulation of thermal losses in the absorber **

4.1 Basic mechanisms and models

4.1.1 Electron-phonon and phonon-phonon energy transfer

4.1.2 The phonon bottleneck eect

4.1.3 Carrier energy loss rate reduction

4.2 Introduction of heat losses in the energy balance

4.2.1 Constant characteristic time for thermalization

4.2.2 Thermalization coecient

4.2.3 Charge and energy balance

4.3 Results and discussion

4.3.1 Validation of the model

4.3.2 Inuence of thermalization in the absorber

4.3.3 Thermalization and concentration

4.3.4 Thermalization and selective contacts

4.4 Conclusion

Bibliography

**5 Eciency of realistic cells **

5.1 Absorption

5.2 The hot carrier solar cell in practical conditions: achievable eciency

5.2.1 Numerical resolution

5.2.2 Validation

5.2.3 Achievable eciency

5.3 A roadmap to 50% eciency

5.3.1 Control of thermalization

5.3.2 Carrier extraction

5.3.3 Cell design and synthesis

5.4 Conclusion

Conclusion

A Band gaps of III-V multinary compounds

B Propagation of light in a layered medium

**Bibliography**