🔋Thermoelectric Materials and Devices Unit 2 – Transport in Thermoelectric Materials

Fundamentals of Thermoelectric Materials

• Thermoelectric materials convert heat directly into electricity (Seebeck effect) or electricity into heat (Peltier effect)
• Efficiency of thermoelectric materials depends on the figure of merit ($ZT$), which is a function of electrical conductivity ($\sigma$), Seebeck coefficient ($S$), and thermal conductivity ($\kappa$)
  • Higher $ZT$ values indicate better thermoelectric performance
• Ideal thermoelectric materials have high electrical conductivity, high Seebeck coefficient, and low thermal conductivity
  • Achieving this combination of properties is challenging due to the interdependence of these parameters
• Thermoelectric materials are typically semiconductors or heavily doped semiconductors (bismuth telluride, lead telluride)
• Nanostructuring and alloying are common strategies to optimize thermoelectric properties
  • Nanostructuring reduces thermal conductivity by increasing phonon scattering
  • Alloying can enhance the power factor ($S^2\sigma$) by tuning the electronic structure

Electron and Phonon Transport Mechanisms

• Electron transport in thermoelectric materials is governed by the Boltzmann transport equation
  • Describes the distribution of charge carriers in the presence of temperature and electric potential gradients
• Electrical conductivity depends on the carrier concentration ($n$), mobility ($\mu$), and electronic charge ($e$): $\sigma = ne\mu$
• Carrier concentration can be increased by doping, while mobility is influenced by scattering mechanisms
• Phonon transport is responsible for heat conduction in thermoelectric materials
  • Phonons are quantized lattice vibrations that carry heat
• Phonon dispersion relation describes the relationship between phonon frequency and wavevector
  • Acoustic and optical phonon branches have different contributions to thermal conductivity
• Phonon group velocity ($v_g$) determines the speed of heat propagation: $v_g = d\omega/dk$
• Phonon mean free path ($\Lambda$) is the average distance a phonon travels before scattering
  • Shorter mean free paths lead to lower thermal conductivity

Scattering Processes in Thermoelectric Materials

• Scattering processes limit the mobility of charge carriers and reduce the mean free path of phonons
• Electron scattering mechanisms include ionized impurity scattering, acoustic phonon scattering, and polar optical phonon scattering
  • Ionized impurity scattering dominates at low temperatures and high doping levels
  • Acoustic phonon scattering is important at room temperature and above
• Phonon scattering mechanisms include phonon-phonon scattering, phonon-electron scattering, and phonon-boundary scattering
  • Phonon-phonon scattering (Umklapp processes) is the primary mechanism limiting thermal conductivity at high temperatures
  • Phonon-boundary scattering becomes significant in nanostructured materials
• Alloy scattering introduces additional scattering centers due to mass and strain field fluctuations
  • Effective in reducing thermal conductivity without significantly affecting electrical conductivity
• Nanostructuring enhances phonon scattering by introducing interfaces and boundaries
  • Grain boundaries, nanoinclusions, and superlattices are examples of nanostructures that scatter phonons

Thermal Conductivity and Electrical Conductivity

• Thermal conductivity ($\kappa$) is the sum of electronic ($\kappa_e$) and lattice ($\kappa_L$) contributions: $\kappa = \kappa_e + \kappa_L$
  • Electronic thermal conductivity is related to electrical conductivity through the Wiedemann-Franz law: $\kappa_e = L\sigma T$, where $L$ is the Lorenz number
• Lattice thermal conductivity depends on the specific heat ($C_v$), phonon group velocity ($v_g$), and phonon mean free path ($\Lambda$): $\kappa_L = \frac{1}{3}C_vv_g\Lambda$
  • Reducing $\kappa_L$ is a key strategy to improve thermoelectric performance
• Electrical conductivity ($\sigma$) is proportional to the carrier concentration ($n$) and mobility ($\mu$): $\sigma = ne\mu$
  • Increasing carrier concentration through doping enhances electrical conductivity but also increases electronic thermal conductivity
• Mobility depends on the effective mass ($m^*$) and scattering time ($\tau$): $\mu = e\tau/m^*$
  • Higher mobility materials have longer scattering times and lower effective masses
• Optimizing the carrier concentration is crucial to balance electrical conductivity and Seebeck coefficient
  • Typically, the optimal carrier concentration is around $10^{19}$ to $10^{21} cm^{-3}$ for thermoelectric materials

Seebeck Effect and Peltier Effect

• Seebeck effect is the generation of an electric potential difference ($\Delta V$) due to a temperature gradient ($\Delta T$) in a material
  • Seebeck coefficient ($S$) is defined as $S = -\Delta V/\Delta T$
• Seebeck coefficient depends on the carrier concentration ($n$) and effective mass ($m^*$): $S \propto m^*/n^{2/3}$
  • Higher effective mass and lower carrier concentration lead to higher Seebeck coefficients
• Peltier effect is the reverse of the Seebeck effect, where an electric current induces a temperature gradient
  • Peltier coefficient ($\Pi$) is related to the Seebeck coefficient through the Kelvin relation: $\Pi = ST$
• Peltier cooling and heating occur at the junctions of dissimilar materials when a current is passed
  • Peltier devices (thermoelectric coolers) exploit this effect for solid-state cooling applications
• Thomson effect describes the heating or cooling of a material when an electric current passes through a temperature gradient
  • Thomson coefficient ($\tau$) is related to the Seebeck coefficient: $\tau = T(dS/dT)$

Figure of Merit and Power Factor

• Figure of merit ($ZT$) is a dimensionless quantity that characterizes the efficiency of a thermoelectric material
  • Defined as $ZT = (S^2\sigma/\kappa)T$, where $S$ is the Seebeck coefficient, $\sigma$ is the electrical conductivity, $\kappa$ is the thermal conductivity, and $T$ is the absolute temperature
• Higher $ZT$ values indicate better thermoelectric performance
  • $ZT > 1$ is considered a good thermoelectric material, while $ZT > 3$ is desired for widespread applications
• Power factor ($PF$) is a measure of the electrical power output of a thermoelectric material
  • Defined as $PF = S^2\sigma$, which is proportional to the numerator of the figure of merit
• Optimizing the power factor involves finding the optimal balance between Seebeck coefficient and electrical conductivity
  • Strategies include band engineering, resonant doping, and modulation doping
• Thermoelectric efficiency ($\eta$) is related to the figure of merit through the Carnot efficiency: $\eta = \eta_C(\sqrt{1+ZT}-1)/(\sqrt{1+ZT}+T_C/T_H)$
  • $\eta_C$ is the Carnot efficiency, $T_C$ is the cold-side temperature, and $T_H$ is the hot-side temperature

Advanced Transport Phenomena

• Bipolar effect occurs in narrow-bandgap semiconductors at high temperatures
  • Both electrons and holes contribute to transport, reducing the Seebeck coefficient and increasing thermal conductivity
• Minority carrier effects become significant in heavily doped semiconductors
  • Minority carriers (electrons in p-type, holes in n-type) can diffuse against the temperature gradient, reducing the Seebeck coefficient
• Phonon drag effect enhances the Seebeck coefficient at low temperatures
  • Phonons "drag" charge carriers, leading to an additional contribution to the Seebeck coefficient
• Quantum confinement effects can modify the electronic structure and transport properties in low-dimensional materials (quantum wells, nanowires, quantum dots)
  • Confinement can increase the effective mass and density of states near the Fermi level, enhancing the Seebeck coefficient
• Topological materials (topological insulators, Dirac/Weyl semimetals) exhibit unique transport properties
  • Surface states or linearly dispersing bands can lead to high mobility and large Seebeck coefficients
• Spin Seebeck effect is the generation of a spin voltage due to a temperature gradient in magnetic materials
  • Enables the development of spin caloritronic devices and energy harvesters

Applications and Future Directions

• Thermoelectric generators (TEGs) convert waste heat into electricity
  • Applications in automotive exhaust systems, industrial processes, and space power systems
• Thermoelectric coolers (TECs) provide solid-state cooling for electronic devices, sensors, and medical applications
  • Advantages include compactness, reliability, and precise temperature control
• Wearable thermoelectric devices harvest body heat to power sensors and electronics
  • Flexible and stretchable thermoelectric materials are being developed for wearable applications
• High-temperature thermoelectric materials (skutterudites, half-Heuslers, silicides) are being explored for power generation from high-grade waste heat
  • Stability and performance at elevated temperatures are key challenges
• Nanostructured and low-dimensional materials (superlattices, nanowires, quantum dots) are promising for enhancing thermoelectric performance
  • Reduced thermal conductivity and quantum confinement effects can lead to high $ZT$ values
• Organic and polymer thermoelectric materials offer advantages of low cost, flexibility, and solution processability
  • Potential applications in large-area and printed thermoelectrics
• Computational materials discovery and machine learning are accelerating the search for new thermoelectric materials
  • High-throughput screening and data-driven approaches can identify promising candidates for experimental validation
• Integration of thermoelectric devices with other energy conversion technologies (photovoltaics, fuel cells) can improve overall system efficiency
  • Hybrid systems can exploit the complementary strengths of different energy conversion mechanisms