## Thesis etd-06272016-190534 |

Thesis type

Tesi di laurea magistrale

Author

ERDMAN, PAOLO ANDREA

URN

etd-06272016-190534

Thesis title

Thermoelectric Efficiency of a Multilevel Interacting Quantum Dot

Department

FISICA

Course of study

FISICA

Supervisors

**relatore**Prof. Vicari, Ettore

**relatore**Taddei, Fabio

**relatore**Prof. Fazio, Rosario

Keywords

- quantum dot
- thermoelectric
- mesoscopic transport
- thermoelectric efficiency

Graduation session start date

21/07/2016

Availability

Full

Summary

In this thesis I study the thermoelectric properties of a multilevel interacting quantum dot weakly coupled to two electronic reservoirs kept at different temperatures and chemical potentials. The motivation behind this work was to assess the properties of this system acting as a heat engine, i.e. as a device that converts heat into work. In fact, applying a temperature difference to the quantum dot produces an electric current that can perform work against a potential difference. This problem has a long history and the first studies date back to the 1960s when Ioffe discovered that doped semiconductors have good thermoelectric properties; nonetheless even today the efficiency for heat-to-work conversion of experimental solid state devices remains too low to be competitive with conventional heat engines. In the 1990s two pioneering works from Dresselhaus et al. and Sofo et al. independently proposed to study low dimensional system. The first article showed that decreasing one of the dimensions of a bulk material could lead to a drastic increase of the efficiency; the second article argued that by restricting the energy windows of electrons participating in the transport to a “delta-like” distribution would allow the system to operate at Carnot’s efficiency. This “energy filtering” mechanism can be implemented using low dimensional systems, now experimentally realizable: this motivated our study of a “zero dimensional” system i.e. a quantum dot.

While there is a vast literature on non-interacting systems, much less is known about the impact of electron interactions on the efficiency of nano-heat engines. This will be the main objective of the thesis.

I start by discussing the general properties of heat engines, and by defining the physical quantities used to describe the thermoelectric properties of these system, such as the transport coefficients (electric conductance, thermopower and thermal conductance), the maximum efficiency, the efficiency at maximum power and the maximum power. Then I present the sequential-tunneling formalism which allows me to compute the charge, energy and heat currents as a function of an arbitrary temperature and chemical potential difference.

In the first part of the thesis, I study the system in the linear response regime. I revise and extend the sequential tunneling formalism by deriving the condition under which the expressions of the currents are valid in linear response. Then I study the system in the quantum limit, i.e. when the average thermal energy is much smaller than the typical energy distance between quantum dot energy

levels, and much smaller than the interacting energy. This allows me to obtain simple analytic expressions for the transport coefficients and for the figure of merit ZT which completely characterizes the maximum efficiency and the efficiency at maximum power in the linear regime. I identify analytically optimal system parameters in order to maximize the power and efficiency, and I compare these results with numerical calculations. If the system parameters are such that the typical energy distance between quantum dot energy levels is much larger than the thermal energy, or much smaller than the charging energy, the maximum efficiency approaches Carnot’s efficiency.

Then I study the system numerically beyond the linear response regime. After analyzing the currents, I focus on the maximum power, the maximum efficiency, and on the efficiency at maximum power, showing that applying large temperature gradients allows the system to overcome limits imposed by the linear response regime. In particular, it turns out that the efficiency at maximum power, bounded by half Carnot’s efficiency in the linear response regime, exhibits peaks that approach Carnot’s efficiency.

At last I analyze a real quantum dot system created in a InAs/InP nanowire, whose electronic transport properties were studied. After reproducing the experimental electric conductance, I predict it’s thermoelectric properties: the efficiency and power are maximum in the quantum limit, where large temperature differences allow to overcome the limits imposed by the linear response regime. The spin degeneracy of this system, previously disregarded, creates an asymmetry in the peaks of the maximum power, providing an enhancement of a factor 1.77 with respect to the non degenerate case. I study this enhancement analytically within the quantum limit in the linear response regime, finding that the expression I obtain appears to describe quite accurately also the non linear maximum power.

While there is a vast literature on non-interacting systems, much less is known about the impact of electron interactions on the efficiency of nano-heat engines. This will be the main objective of the thesis.

I start by discussing the general properties of heat engines, and by defining the physical quantities used to describe the thermoelectric properties of these system, such as the transport coefficients (electric conductance, thermopower and thermal conductance), the maximum efficiency, the efficiency at maximum power and the maximum power. Then I present the sequential-tunneling formalism which allows me to compute the charge, energy and heat currents as a function of an arbitrary temperature and chemical potential difference.

In the first part of the thesis, I study the system in the linear response regime. I revise and extend the sequential tunneling formalism by deriving the condition under which the expressions of the currents are valid in linear response. Then I study the system in the quantum limit, i.e. when the average thermal energy is much smaller than the typical energy distance between quantum dot energy

levels, and much smaller than the interacting energy. This allows me to obtain simple analytic expressions for the transport coefficients and for the figure of merit ZT which completely characterizes the maximum efficiency and the efficiency at maximum power in the linear regime. I identify analytically optimal system parameters in order to maximize the power and efficiency, and I compare these results with numerical calculations. If the system parameters are such that the typical energy distance between quantum dot energy levels is much larger than the thermal energy, or much smaller than the charging energy, the maximum efficiency approaches Carnot’s efficiency.

Then I study the system numerically beyond the linear response regime. After analyzing the currents, I focus on the maximum power, the maximum efficiency, and on the efficiency at maximum power, showing that applying large temperature gradients allows the system to overcome limits imposed by the linear response regime. In particular, it turns out that the efficiency at maximum power, bounded by half Carnot’s efficiency in the linear response regime, exhibits peaks that approach Carnot’s efficiency.

At last I analyze a real quantum dot system created in a InAs/InP nanowire, whose electronic transport properties were studied. After reproducing the experimental electric conductance, I predict it’s thermoelectric properties: the efficiency and power are maximum in the quantum limit, where large temperature differences allow to overcome the limits imposed by the linear response regime. The spin degeneracy of this system, previously disregarded, creates an asymmetry in the peaks of the maximum power, providing an enhancement of a factor 1.77 with respect to the non degenerate case. I study this enhancement analytically within the quantum limit in the linear response regime, finding that the expression I obtain appears to describe quite accurately also the non linear maximum power.

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