4.4 Temperature and voltage dependence of electron transport

Electron transport in molecules is heavily influenced by temperature and voltage. These factors affect how electrons move through materials, impacting everything from thermionic emission to resistance changes.

Understanding these dependencies is crucial for designing and analyzing molecular electronic devices. We'll look at how temperature impacts electron emission and energy distributions, and how voltage drives current flow and reveals transport mechanisms.

Thermionic Emission and Thermal Effects

Temperature-Dependent Electron Emission and Transport

• Thermionic emission occurs when electrons gain enough thermal energy to overcome the work function and escape from a material's surface
  • Governed by the Richardson-Dushman equation: $J = A T^2 e^{-\frac{W}{k_B T}}$, where $J$ is the emission current density, $A$ is a material-specific constant, $T$ is the temperature, $W$ is the work function, and $k_B$ is the Boltzmann constant
  • Commonly observed in vacuum tubes and electron guns (cathode ray tubes)
• Thermal broadening of energy levels leads to a distribution of electron energies around the Fermi level
  • Described by the Fermi-Dirac distribution: $f(E) = \frac{1}{e^{(E-E_F)/k_B T}+1}$, where $f(E)$ is the probability of an electron occupying a state with energy $E$, $E_F$ is the Fermi energy, $k_B$ is the Boltzmann constant, and $T$ is the temperature
  • Results in a smearing of the Fermi edge and a non-zero probability of finding electrons above the Fermi level at finite temperatures

Activation Energy and Temperature-Dependent Resistance

• Activation energy is the minimum energy required for a process to occur, such as electron transport or chemical reactions
  • Determines the temperature dependence of the rate of a process according to the Arrhenius equation: $k = A e^{-\frac{E_a}{k_B T}}$, where $k$ is the rate constant, $A$ is a pre-exponential factor, $E_a$ is the activation energy, $k_B$ is the Boltzmann constant, and $T$ is the temperature
  • Can be extracted from the slope of an Arrhenius plot, which is a graph of $\ln(k)$ vs. $1/T$
• Temperature coefficient of resistance (TCR) describes the relative change in resistance with temperature
  • Defined as $\alpha = \frac{1}{R} \frac{dR}{dT}$, where $\alpha$ is the TCR, $R$ is the resistance, and $T$ is the temperature
  • Positive TCR (metals) indicates increasing resistance with temperature due to increased electron scattering
  • Negative TCR (semiconductors) indicates decreasing resistance with temperature due to increased carrier concentration from thermal excitation across the bandgap

Voltage-Dependent Transport

Current-Voltage Characteristics and Differential Conductance

• Bias voltage is the potential difference applied across a device or junction
  • Drives current flow and determines the energy landscape for electron transport
  • Can be used to probe the electronic structure and transport properties of materials and devices
• I-V characteristics describe the relationship between current and voltage in a device
  • Provide information about the transport mechanism, such as ohmic (linear) or non-ohmic (non-linear) behavior
  • Can reveal the presence of energy barriers, such as in a Schottky diode or a tunnel junction
• Differential conductance is the derivative of the current with respect to voltage: $G = \frac{dI}{dV}$
  • Reflects the local density of states and the transmission probability at a given energy
  • Can be measured using lock-in techniques or numerical differentiation of I-V data
  • Peaks in differential conductance correspond to resonant tunneling through discrete energy levels (quantum dots, molecules)

Non-Linear Transport Phenomena

• Non-linear transport occurs when the current-voltage relationship deviates from ohmic behavior
  • Can arise from various mechanisms, such as voltage-dependent tunneling, space-charge limited current, or field emission
  • Examples include negative differential resistance (NDR) in resonant tunneling diodes and current saturation in field-effect transistors
• Voltage-dependent tunneling probability leads to non-linear I-V characteristics in tunnel junctions
  • Described by the Simmons model: $I \propto V e^{-2d\sqrt{\frac{2m\phi}{\hbar^2}}}$, where $I$ is the current, $V$ is the voltage, $d$ is the barrier width, $m$ is the electron mass, $\phi$ is the barrier height, and $\hbar$ is the reduced Planck constant
  • Enables the study of electronic structure and transport mechanisms in molecular junctions and nanoscale devices