6.3 Carrier concentration and mobility

Carrier concentration and mobility are crucial concepts in understanding how materials conduct electricity. These properties determine how many charge carriers are available and how easily they move through a material, directly impacting electrical conductivity.

Intrinsic and extrinsic semiconductors have different carrier concentrations due to doping levels. Temperature and the Fermi level position also influence carrier concentration. Mobility, affected by lattice and impurity scattering, varies with temperature and material properties, impacting device performance.

Carrier concentration

• Carrier concentration refers to the number of charge carriers (electrons or holes) per unit volume in a material
• Intrinsic and extrinsic semiconductors have different carrier concentrations due to their respective doping levels
• The Fermi level position and temperature significantly influence the carrier concentration in semiconductors

Intrinsic vs extrinsic semiconductors

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• Intrinsic semiconductors have equal numbers of electrons and holes, generated by thermal excitation across the bandgap
• Extrinsic semiconductors are doped with impurities to increase the concentration of either electrons (n-type) or holes (p-type)
• Doping levels in extrinsic semiconductors are typically several orders of magnitude higher than intrinsic carrier concentrations (silicon, germanium)

Fermi level and carrier concentration

• The Fermi level represents the energy at which the probability of finding an electron is 0.5
• In intrinsic semiconductors, the Fermi level lies near the middle of the bandgap
• Doping shifts the Fermi level towards the conduction band (n-type) or valence band (p-type), increasing the respective carrier concentration
• The carrier concentration depends exponentially on the difference between the Fermi level and the band edge

Temperature dependence of carrier concentration

• Increasing temperature excites more electrons from the valence band to the conduction band, increasing the intrinsic carrier concentration
• The intrinsic carrier concentration $n_i$ varies with temperature as $n_i \propto \exp(-E_g/2k_BT)$, where $E_g$ is the bandgap and $k_B$ is the Boltzmann constant
• Extrinsic carrier concentration is less sensitive to temperature, as it is primarily determined by the doping level
• At high temperatures, intrinsic carriers may outnumber extrinsic carriers, leading to intrinsic behavior

Carrier concentration in metals vs semiconductors

• Metals have high carrier concentrations ($10^{22}-10^{23} cm^{-3}$) due to the overlap of the valence and conduction bands
• Semiconductors have lower carrier concentrations ($10^{10}-10^{18} cm^{-3}$) due to the presence of a bandgap
• The carrier concentration in semiconductors can be controlled by doping, while metals have a fixed carrier concentration determined by their electronic structure

Carrier mobility

• Carrier mobility quantifies how easily charge carriers move through a material under the influence of an electric field
• Mobility is a key parameter in determining the electrical conductivity and performance of electronic devices

Definition and units of mobility

• Mobility $\mu$ is defined as the ratio of the drift velocity $v_d$ to the applied electric field $E$: $\mu = v_d/E$
• The units of mobility are $cm^2/(V \cdot s)$ or $m^2/(V \cdot s)$
• Higher mobility indicates that carriers can move more easily through the material, resulting in higher conductivity

Factors affecting carrier mobility

• Lattice scattering: Carriers interact with phonons (lattice vibrations), which reduces their mobility
• Impurity scattering: Charged impurities, such as dopants or defects, scatter carriers and lower their mobility
• Carrier effective mass: Lighter carriers (e.g., electrons in GaAs) have higher mobilities than heavier carriers (e.g., holes in Si)
• Temperature: Mobility generally decreases with increasing temperature due to enhanced lattice scattering

Lattice scattering vs impurity scattering

• Lattice scattering dominates at high temperatures, where phonon populations are higher
• Impurity scattering dominates at low temperatures, where lattice vibrations are suppressed
• The relative importance of lattice and impurity scattering depends on the material purity and doping level

Temperature dependence of mobility

• Mobility due to lattice scattering varies as $\mu_L \propto T^{-3/2}$
• Mobility due to impurity scattering varies as $\mu_I \propto T^{3/2}$
• The total mobility is given by Matthiessen's rule: $1/\mu = 1/\mu_L + 1/\mu_I$
• The temperature dependence of mobility can help identify the dominant scattering mechanism

Mobility in metals vs semiconductors

• Metals typically have lower mobilities ($1-100 cm^2/(V \cdot s)$) than semiconductors due to their higher carrier concentrations and stronger electron-phonon interactions
• Semiconductors can have mobilities ranging from $100-10^5 cm^2/(V \cdot s)$, depending on the material and doping level
• High-mobility semiconductors (GaAs, InSb) are used in high-frequency and high-speed electronic devices

Carrier transport

• Carrier transport describes the motion of charge carriers in response to electric and magnetic fields
• Understanding carrier transport is essential for designing and optimizing electronic devices

Drift current and drift velocity

• Drift current is the flow of charge carriers due to an applied electric field
• Drift velocity $v_d$ is the average velocity of carriers in the direction of the electric field
• The drift current density $J$ is given by $J = nqv_d$, where $n$ is the carrier concentration and $q$ is the elementary charge
• The drift velocity is proportional to the electric field: $v_d = \mu E$

Conductivity and resistivity

• Conductivity $\sigma$ is a measure of a material's ability to conduct electric current
• Resistivity $\rho$ is the reciprocal of conductivity and quantifies a material's resistance to current flow
• Conductivity is related to carrier concentration and mobility: $\sigma = nq\mu$
• Resistivity is given by $\rho = 1/\sigma = 1/(nq\mu)$

Hall effect and Hall coefficient

• The Hall effect is the generation of a transverse voltage (Hall voltage) in a conductor when a magnetic field is applied perpendicular to the current flow
• The Hall coefficient $R_H$ is defined as the ratio of the induced electric field to the product of the current density and the magnetic field: $R_H = E_y/(J_xB_z)$
• The Hall coefficient is related to the carrier concentration: $R_H = 1/(nq)$
• Measuring the Hall voltage allows the determination of the carrier type (electrons or holes), concentration, and mobility

Magnetoresistance and its applications

• Magnetoresistance is the change in a material's resistance when exposed to a magnetic field
• Ordinary magnetoresistance arises from the deflection of carriers by the Lorentz force, leading to an increased path length and resistance
• Giant magnetoresistance (GMR) occurs in multilayer structures alternating ferromagnetic and non-magnetic layers, used in magnetic sensors and hard drives
• Colossal magnetoresistance (CMR) is observed in certain manganese oxides and has potential applications in magnetic memory devices

Measurement techniques

• Various measurement techniques are used to characterize the electrical properties of materials, including carrier concentration and mobility
• These techniques provide valuable information for material optimization and device design

Hall effect measurements

• Hall effect measurements involve applying a magnetic field perpendicular to a current-carrying sample and measuring the resulting Hall voltage
• The Hall voltage is proportional to the carrier concentration and can be used to determine the carrier type and mobility
• Van der Pauw configuration is often used for Hall effect measurements on thin films and irregular-shaped samples

Four-point probe method

• The four-point probe method is used to measure the resistivity of a material
• Four equally spaced probes are brought into contact with the sample surface, with current passed through the outer probes and voltage measured across the inner probes
• This method eliminates contact resistance and is suitable for both bulk and thin film samples
• The resistivity is calculated from the measured voltage, current, and a geometric factor depending on the sample thickness and probe spacing

Van der Pauw method

• The Van der Pauw method is an extension of the four-point probe technique for measuring the resistivity and Hall coefficient of thin films and irregular-shaped samples
• Four contacts are placed on the sample perimeter, and resistance measurements are taken in different configurations
• The resistivity and Hall coefficient can be calculated from the measured resistances using the Van der Pauw equation
• This method is widely used in the semiconductor industry for characterizing material properties

Capacitance-voltage (C-V) measurements

• Capacitance-voltage (C-V) measurements are used to characterize the carrier concentration profile and interface properties of semiconductor devices
• A voltage is applied to a metal-oxide-semiconductor (MOS) or p-n junction structure, and the capacitance is measured as a function of voltage
• The carrier concentration can be extracted from the slope of the 1/C^2 vs. V plot (Mott-Schottky analysis)
• C-V measurements also provide information on oxide thickness, interface trap density, and Fermi level position

Applications

• Understanding carrier concentration and mobility is crucial for developing and optimizing various electronic and optoelectronic devices
• Tailoring these properties enables the design of high-performance materials and devices for specific applications

High-mobility semiconductors for electronics

• High-mobility semiconductors (GaAs, InGaAs, InSb) are used in high-frequency and high-speed electronic devices, such as radio frequency (RF) transistors and high electron mobility transistors (HEMTs)
• These materials have higher electron mobilities than silicon, allowing for faster switching speeds and lower power consumption
• Applications include 5G wireless communication, radar systems, and satellite communication

Thermoelectric materials and figure of merit

• Thermoelectric materials convert temperature gradients into electrical energy (Seebeck effect) or vice versa (Peltier effect)
• The thermoelectric figure of merit $ZT$ is a measure of a material's thermoelectric efficiency, 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
• High $ZT$ values require a combination of high electrical conductivity, high Seebeck coefficient, and low thermal conductivity
• Strategies for improving $ZT$ include optimizing carrier concentration, enhancing mobility through band engineering, and reducing thermal conductivity through nanostructuring (BiTe, PbTe)

Transparent conducting oxides (TCOs)

• Transparent conducting oxides (TCOs) are materials that combine high electrical conductivity with optical transparency in the visible range
• TCOs are essential for applications such as solar cells, flat-panel displays, and touch screens
• Common TCOs include indium tin oxide (ITO), fluorine-doped tin oxide (FTO), and aluminum-doped zinc oxide (AZO)
• The carrier concentration and mobility of TCOs can be tuned by controlling the doping level and deposition conditions to optimize the balance between conductivity and transparency

Semiconductor devices and carrier control

• Semiconductor devices, such as transistors, diodes, and solar cells, rely on the control of carrier concentration and transport for their operation
• P-n junctions are formed by bringing together p-type and n-type semiconductors, creating a built-in electric field that controls the flow of carriers
• Bipolar junction transistors (BJTs) and field-effect transistors (FETs) use electric fields to modulate the carrier concentration and conductivity in the device channel
• Solar cells convert light into electricity by generating electron-hole pairs and separating them using a p-n junction or other carrier-selective contacts
• Carrier lifetime and diffusion length are important parameters in solar cells, as they determine the collection efficiency of photogenerated carriers