3.4 Temperature dependence of carrier concentration

Intrinsic Carrier Concentration

The intrinsic carrier concentration ($n_i$) describes how many electrons and holes exist in a pure semiconductor at thermal equilibrium. It's one of the most temperature-sensitive parameters in all of semiconductor physics, and nearly every device characteristic you'll study traces back to it.

Temperature Dependence

As temperature rises, more electrons gain enough thermal energy to jump from the valence band to the conduction band. This produces an exponential increase in $n_i$ with temperature.

The governing equation is:

$n_i = \sqrt{N_c N_v} \exp\left(\frac{-E_g}{2k_BT}\right)$

where:

• $N_c$ and $N_v$ are the effective density of states in the conduction and valence bands
• $E_g$ is the bandgap energy
• $k_B$ is the Boltzmann constant
• $T$ is the absolute temperature in Kelvin

The exponential term dominates the temperature behavior. Even modest temperature increases can cause $n_i$ to change by orders of magnitude. For silicon at 300 K, $n_i \approx 1.5 \times 10^{10} \text{ cm}^{-3}$; by 600 K, it's roughly $10^{15} \text{ cm}^{-3}$.

Bandgap Energy

The bandgap energy $E_g$ is the minimum energy an electron needs to cross from the valence band to the conduction band. It sits in the exponent of the $n_i$ equation, so even small differences in $E_g$ produce large differences in carrier concentration at a given temperature.

• Silicon: $E_g = 1.12 \text{ eV}$ at 300 K
• Germanium: $E_g = 0.67 \text{ eV}$ at 300 K
• GaAs: $E_g = 1.42 \text{ eV}$ at 300 K

A wider bandgap means fewer carriers at the same temperature. That's why wide-bandgap materials like SiC (3.26 eV) and GaN (3.4 eV) can operate at much higher temperatures before intrinsic carriers overwhelm the doping.

Note that $E_g$ itself decreases slightly with increasing temperature (the Varshni relation), which further contributes to the rise in $n_i$.

Density of States

The effective density of states $N_c$ and $N_v$ count how many states are available near the band edges for carriers to occupy. Both are proportional to $T^{3/2}$:

$N_c \propto T^{3/2}, \quad N_v \propto T^{3/2}$

This $T^{3/2}$ dependence contributes to the temperature sensitivity of $n_i$, but it's a much weaker effect than the exponential term. The pre-factor $\sqrt{N_c N_v}$ varies slowly compared to $\exp(-E_g / 2k_BT)$.

Different materials have different effective masses for electrons and holes, which changes $N_c$ and $N_v$. III-V semiconductors (GaAs, InP) and II-VI semiconductors (CdTe, ZnSe) can have quite different density-of-states values from silicon, affecting their $n_i$ even at the same bandgap.

Extrinsic Semiconductors

Extrinsic semiconductors are doped with impurity atoms to control the carrier concentration. Donors (n-type) add electrons; acceptors (p-type) add holes. The temperature behavior of an extrinsic semiconductor is more complex than the intrinsic case because you have to account for whether the dopants are actually ionized.

Dopant Ionization

At any finite temperature, dopant atoms can be ionized (releasing or capturing a carrier) or neutral. The ionization energy $E_d$ (for donors) or $E_a$ (for acceptors) is the energy needed to ionize the dopant.

• Shallow dopants (e.g., phosphorus in Si, $E_d \approx 45 \text{ meV}$) ionize easily because their ionization energy is much smaller than $E_g$.
• Deep dopants (e.g., gold in Si) have larger ionization energies and require higher temperatures for full ionization.

At very low temperatures, thermal energy $k_BT$ is too small to ionize many dopants, so the free carrier concentration is well below the doping level.

Freeze-out, Extrinsic, and Intrinsic Regions

The carrier concentration in a doped semiconductor passes through three distinct temperature regions. This is one of the most important concepts in the unit:

1. Freeze-out region (low $T$): Not enough thermal energy to ionize all dopants. The carrier concentration rises steeply with temperature as more dopants ionize. The slope on a log($n$) vs. $1/T$ plot is governed by the dopant ionization energy.

2. Extrinsic (exhaustion/saturation) region (moderate $T$): All dopant atoms are fully ionized. The carrier concentration plateaus at approximately the doping concentration $N_D$ (or $N_A$) and is nearly independent of temperature. This is the normal operating range for most devices.

3. Intrinsic region (high $T$): The intrinsic carrier concentration $n_i$ grows large enough to exceed the doping level. The semiconductor behaves as if it were intrinsic, and the carrier concentration rises exponentially again. Device operation typically fails in this regime.

Saturation of Carrier Concentration

In the extrinsic region, the free carrier concentration equals the dopant concentration ($n \approx N_D$ for n-type). Further temperature increases don't add more carriers because there are no more dopants left to ionize.

Shallow dopants reach saturation at lower temperatures than deep dopants. For phosphorus in silicon, saturation occurs well below room temperature, so at 300 K you can safely assume full ionization.

Fermi Level Position

The Fermi level $E_F$ is the energy at which the probability of electron occupation is exactly 1/2. Its position relative to the band edges determines the electron and hole concentrations through:

$n = N_c \exp\left(\frac{-(E_c - E_F)}{k_BT}\right), \quad p = N_v \exp\left(\frac{-(E_F - E_v)}{k_BT}\right)$

Temperature Dependence

• Intrinsic semiconductors: $E_F$ sits near mid-gap. It shifts slightly with temperature because $N_c$ and $N_v$ (and thus the effective masses) are generally not equal. The intrinsic Fermi level is:

$E_i = \frac{E_c + E_v}{2} + \frac{k_BT}{2}\ln\left(\frac{N_v}{N_c}\right)$

The second term is small, so $E_i$ stays close to mid-gap.

• n-type semiconductors: At low temperatures, $E_F$ sits close to the conduction band (near the donor level). As temperature increases, $E_F$ moves toward mid-gap.
• p-type semiconductors: At low temperatures, $E_F$ sits close to the valence band. It also moves toward mid-gap as temperature rises.

At sufficiently high temperatures, $E_F$ in any extrinsic semiconductor converges to $E_i$ because intrinsic carriers dominate.

Intrinsic vs. Extrinsic Semiconductors

In an intrinsic semiconductor, $n = p = n_i$, and the Fermi level is pinned near mid-gap by this symmetry. In an extrinsic semiconductor, the dopants break that symmetry:

• n-type: $n \gg p$, so $E_F$ is above $E_i$
• p-type: $p \gg n$, so $E_F$ is below $E_i$

The offset from mid-gap can be calculated as $E_F - E_i = k_BT \ln(n/n_i)$ for n-type.

Bandgap Narrowing Effects

In heavily doped semiconductors (typically $> 10^{18} \text{ cm}^{-3}$), the dopant atoms are close enough that their wavefunctions overlap and their discrete energy levels broaden into a band. This effectively reduces the bandgap.

Bandgap narrowing increases $n_i$ (since $E_g$ appears in the exponent) and shifts the Fermi level. This matters most in regions with very high doping, such as the emitter of a bipolar junction transistor, where it can significantly affect current gain calculations.

Carrier Mobility

Carrier mobility $\mu$ quantifies how fast carriers drift per unit electric field (units: $\text{cm}^2/\text{V·s}$). It depends on how frequently carriers scatter off obstacles in the lattice.

Lattice Scattering

Lattice vibrations (phonons) increase in amplitude as temperature rises. Carriers collide with these vibrations more frequently at higher temperatures, reducing mobility. The approximate dependence is:

$\mu_{\text{lattice}} \propto T^{-3/2}$

This is the dominant scattering mechanism at moderate to high temperatures and in lightly doped material.

Impurity Scattering

Ionized dopant atoms create Coulomb potentials that deflect passing carriers. At low temperatures, carriers move slowly and spend more time near each impurity, so they scatter more strongly. As temperature increases, carriers move faster and are deflected less, so mobility from impurity scattering actually increases with temperature:

$\mu_{\text{impurity}} \propto T^{3/2}$

Impurity scattering dominates at low temperatures and in heavily doped semiconductors.

Temperature Dependence of Mobility

The total mobility combines both mechanisms via Matthiessen's rule:

$\frac{1}{\mu} = \frac{1}{\mu_{\text{lattice}}} + \frac{1}{\mu_{\text{impurity}}}$

The result is a mobility curve that:

1. Increases with temperature at low $T$ (impurity scattering weakens)
2. Reaches a peak at some intermediate temperature
3. Decreases with temperature at high $T$ (lattice scattering dominates)

For lightly doped silicon at 300 K, typical values are $\mu_n \approx 1350 \text{ cm}^2/\text{V·s}$ and $\mu_p \approx 480 \text{ cm}^2/\text{V·s}$.

Minority Carrier Concentration

Minority carriers are the less abundant carrier type: electrons in p-type material, holes in n-type. Despite their low concentration, they control the behavior of bipolar devices.

Relation to Majority Carriers

The mass action law holds at thermal equilibrium:

$np = n_i^2$

This is true for both intrinsic and extrinsic semiconductors. If you know the majority carrier concentration and $n_i$, you can find the minority carrier concentration directly. For example, in n-type silicon with $N_D = 10^{16} \text{ cm}^{-3}$ at 300 K:

$p = \frac{n_i^2}{N_D} = \frac{(1.5 \times 10^{10})^2}{10^{16}} = 2.25 \times 10^{4} \text{ cm}^{-3}$

Temperature Dependence

Since $n_i^2$ increases exponentially with temperature, so does the minority carrier concentration (assuming the majority carrier concentration stays roughly constant in the extrinsic region). This exponential rise in minority carriers is why bipolar devices become harder to control at elevated temperatures: leakage currents grow rapidly.

At high enough temperatures, $n_i$ exceeds the doping level, and the distinction between majority and minority carriers disappears.

Low Injection vs. High Injection

• Low injection: The injected minority carrier concentration $\Delta n$ (or $\Delta p$) is much smaller than the equilibrium majority carrier concentration. The majority carrier concentration is essentially unchanged, and the mass action law still approximately holds. Most device analysis assumes low injection.
• High injection: The injected carrier concentration is comparable to or exceeds the majority carrier concentration. Both carrier types are present in similar numbers, mobility and recombination dynamics change, and simple equilibrium formulas break down. High injection occurs in devices driven at very high current densities.

Conductivity and Resistivity

The electrical conductivity $\sigma$ of a semiconductor depends on both how many carriers are present and how fast they move:

$\sigma = q(n\mu_n + p\mu_p)$

Resistivity is simply the inverse: $\rho = 1/\sigma$.

Carrier Concentration Effects

In an extrinsic semiconductor under normal conditions, one carrier type dominates. For n-type: $\sigma \approx qN_D\mu_n$. Increasing the doping concentration raises conductivity, provided the dopants are fully ionized.

At very high doping levels, however, the increased impurity scattering reduces $\mu_n$, partially offsetting the gain from higher $N_D$.

Mobility Effects

Since $\sigma$ is proportional to $\mu$, anything that reduces mobility also reduces conductivity. In heavily doped material, impurity scattering can significantly lower mobility, which is why doubling the doping concentration does not simply double the conductivity.

Temperature Dependence

The conductivity vs. temperature curve mirrors the three regions of carrier concentration:

1. Freeze-out region: Conductivity rises with temperature as dopants ionize and carrier concentration increases.
2. Extrinsic region: Carrier concentration is constant ($\approx N_D$), so conductivity is controlled by mobility. Since lattice scattering increases with $T$, conductivity typically decreases with temperature in this range. This gives doped semiconductors a negative temperature coefficient of resistivity in this regime, opposite to metals.
3. Intrinsic region: The exponential rise in $n_i$ overwhelms the mobility decrease, and conductivity increases sharply.

Applications in Devices

The temperature dependencies covered above directly affect how real devices behave. Here's how they show up in three major device types.

Diodes and Rectifiers

The reverse saturation current of a p-n junction diode is proportional to $n_i^2$, so it increases exponentially with temperature. A common rule of thumb: reverse leakage current roughly doubles for every 10°C increase in temperature.

Higher leakage means reduced rectification efficiency and increased power dissipation. Wide-bandgap semiconductors (SiC, GaN) have much smaller $n_i$ at a given temperature, making them better suited for high-temperature rectifier applications.

Bipolar Junction Transistors (BJTs)

BJT operation depends heavily on minority carrier injection. Temperature affects BJTs in several ways:

• The base current increases with temperature due to rising $n_i$, which reduces the current gain $\beta$.
• Mobility changes alter the collector current magnitude and frequency response.
• The base-emitter voltage $V_{BE}$ decreases by roughly $-2 \text{ mV/°C}$ at constant collector current.

Thermal runaway is a real concern: increased temperature raises collector current, which raises power dissipation, which raises temperature further. Proper biasing with emitter resistors or current mirrors helps stabilize BJT circuits against temperature variations.

Metal-Oxide-Semiconductor Field-Effect Transistors (MOSFETs)

MOSFETs are majority-carrier devices, so they're generally less temperature-sensitive than BJTs, but temperature still matters:

• Threshold voltage decreases with temperature because $n_i$ increases and the Fermi potential changes.
• Off-state leakage increases exponentially with temperature due to the rising $n_i$.
• On-state current decreases at high temperatures because mobility drops (lattice scattering).

The mobility reduction and threshold shift have competing effects on drain current. At a specific bias point called the zero temperature coefficient (ZTC) point, these effects cancel and the drain current is temperature-independent. This property is useful for designing temperature-stable bias circuits.