Extended Imperfections, Dislocations & Doped Semiconductors

Unlike 0D point defects (which are thermodynamically stable at $T > 0\text{ K}$), 1D line defects—dislocations—possess high elastic strain energies ($E_{\text{line}} \sim 10^9\text{ eV/cm}$) and are non-equilibrium mechanical imperfections introduced during crystal growth, thermal quenching, or plastic shear.

The Edge Dislocation Concept

An edge dislocation (introduced by Taylor, Orowan, and Polanyi in 1934) can be visualized as inserting an extra half-plane of atoms into an otherwise perfect crystal:

• Dislocation Line ($\mathbf{t}$): The line along the bottom edge of the extra half-plane, forming the boundary between the slipped and unslipped regions of the crystal.
• Stress Distribution: Above the slip plane, atomic planes are squeezed together, creating a localized hydrostatic compression zone. Below the slip plane, atomic bonds are stretched, creating a hydrostatic tension zone.
• Symbol: Represented by $\bot$ (positive edge dislocation, extra half-plane above slip plane) or $\top$ (negative edge dislocation, extra half-plane below).

The Burgers Circuit and Burgers Vector ($\mathbf{b}$)

The fundamental mathematical descriptor of any dislocation is its Burgers vector $\mathbf{b}$ (J. M. Burgers, 1939), determined via the right-hand finish-to-start (RH/FS) convention:

1. In the dislocated crystal, construct a closed atom-to-atom loop (Burgers circuit) encircling the dislocation line $\mathbf{t}$.
2. Traverse the identical sequence of lattice translation vectors in an ideal, defect-free reference lattice.
3. The closure failure vector pointing from the finish point to the start point in the perfect lattice is the Burgers vector $\mathbf{b}$.

• Geometric Orthogonality: For a pure edge dislocation, the Burgers vector is strictly perpendicular to the dislocation line vector:

The slip plane is uniquely defined as the plane containing both $\mathbf{b}$ and $\mathbf{t}$ (normal vector $\mathbf{n} \propto \mathbf{b} \times \mathbf{t}$).

A screw dislocation transforms the parallel atomic planes of a crystal into a continuous helical spiral staircase (Riemann surface) winding around the dislocation axis.

Geometric Characteristics of Screw Dislocations

• Shear Topology: Formed by cutting a crystal halfway through and shifting one side parallel to the cut boundary by one lattice vector.
• Burgers Vector Alignment: For a pure screw dislocation, the Burgers vector is strictly parallel to the dislocation line vector:

• Cross-Slip: Because $\mathbf{b} \parallel \mathbf{t}$, the normal vector $\mathbf{b} \times \mathbf{t} = \mathbf{0}$ does not define a unique slip plane. A screw dislocation can move along any crystallographic plane passing through $\mathbf{t}$ that has high packing density (cross-slip).

Mixed Dislocations

Real dislocations in crystals are generally neither pure edge nor pure screw, but curve continuously through the lattice as mixed dislocations:

• At an arbitrary segment of a curved dislocation with local tangent $\hat{\mathbf{t}}$, the Burgers vector $\mathbf{b}$ can be decomposed into:

where the angle $\theta$ between $\mathbf{b}$ and $\mathbf{t}$ dictates the screw component ($b\cos\theta$) and edge component ($b\sin\theta$).

Dislocation Multiplication: The Frank-Read Source

Plastic deformation requires billions of new dislocations. Charles Frank and W. T. Read (1950) proposed the regenerative dislocation multiplication mechanism:

1. A dislocation segment of length $L$ is pinned at both ends by solute atoms or node intersections.
2. Under an applied resolved shear stress $\tau$, the pinned segment bows outward into a circular arc of radius $R = \frac{G b}{2\tau}$.
3. When $\tau$ exceeds the critical activation stress:

the arc becomes semicircular ($R = L/2$) and becomes unstable, wrapping around the pinning points.

1. Opposite segments meet, annihilate, and detach as an expanding closed dislocation loop, while the original segment regenerates to repeat the cycle indefinitely.

The atomic displacements surrounding a dislocation core induce long-range elastic strain fields governed by continuum linear elasticity theory.

Elastic Stress Fields in Isotropic Media

Using cylindrical coordinates $(r, \theta, z)$ with the dislocation line along the $z$-axis:

1. Pure Screw Dislocation ($\mathbf{b} = (0, 0, b)$):

Exhibits pure shear strain with zero hydrostatic dilatation (no volume change, $\Delta V = 0$):

where $G$ is the shear modulus.

2. Pure Edge Dislocation ($\mathbf{b} = (b, 0, 0)$):

Exhibits both shear and hydrostatic normal stresses:

where $\nu$ is Poisson's ratio. Notice the characteristic $1/r$ divergence as $r \to 0$.

Dislocation Line Strain Energy

Integrating the elastic strain energy density $w = \frac{1}{2} \boldsymbol{\sigma} : \boldsymbol{\varepsilon}$ from the core radius $r_0 \approx b$ out to the crystal boundary radius $R$:

Because $(1-\nu) \approx 0.7$, an edge dislocation has approximately $40\%$ higher strain energy than a screw dislocation.

The Peach-Koehler Force

The virtual mechanical force per unit length $\mathbf{f}$ exerted on a dislocation by an external stress tensor $\boldsymbol{\sigma}$ is given by the Peach-Koehler equation:

For a pure shear stress $\tau$ resolved in the slip plane in the direction of $\mathbf{b}$, the glide force per unit length is simply $f_{\text{glide}} = \tau b$.

Plastic deformation in crystalline materials proceeds via the conservative motion of dislocations along well-defined slip systems (dislocation slip).

The Peierls-Nabarro Lattice Friction Stress

Even in a pure, defect-free crystal, a dislocation experiences an intrinsic periodic potential barrier as it glides from one lattice valley to the next. The minimum shear stress required to move a dislocation through the crystal without thermal activation is the Peierls-Nabarro stress ($\tau_{\text{PN}}$):

where $w = d/(1-\nu)$ is the dislocation core width, $d$ is the interplanar spacing of the slip planes, and $b$ is the magnitude of the Burgers vector.

• Physical Meaning: $\tau_{\text{PN}}$ decreases exponentially with the ratio $d/b$. Therefore, dislocations slip with the lowest resistance along the most widely spaced crystallographic planes (maximum $d$), which correspond to the densest atomic close-packed planes (e.g., $\{111\}$ in FCC, $\{0001\}$ in HCP).
• In FCC metals (wide core width, $w/b \approx 2-3$), $\tau_{\text{PN}} \approx 10^{-5} G$ (extremely ductile).
• In covalent semiconductors like Si or GaAs (narrow core width due to directional covalent bonds, $w/b \approx 1$), $\tau_{\text{PN}} \approx 10^{-1} G$ (brittle at room temperature).

Schmid's Law and Resolved Shear Stress

Under a uniaxial tensile stress $\sigma$ applied to a single crystal:

• Let $\phi$ be the angle between the tensile loading axis and the slip plane normal $\hat{\mathbf{n}}$.
• Let $\lambda$ be the angle between the tensile axis and the slip direction $\hat{\mathbf{b}}$.

The resolved shear stress $\tau_{\text{RSS}}$ acting on the slip system is:

where $m = \cos\phi \cos\lambda$ is the Schmid factor ($0 \le m \le 0.5$).

• Schmid's Law: Dislocation slip initiates when $\tau_{\text{RSS}}$ reaches a critical threshold characteristic of the material, the Critical Resolved Shear Stress (CRSS) $\tau_{\text{CRSS}}$:

Planar (2D) defects interrupt the continuous periodicity of the crystal lattice across two-dimensional interfaces.

Low-Angle vs High-Angle Grain Boundaries

1. Low-Angle Tilt Boundaries ($ heta < 10^\circ$):

Formed by an array of parallel edge dislocations aligned vertically one above the other with uniform spacing $D$:

Read-Shockley Theory: The interfacial energy per unit area $\gamma(\theta)$ of a low-angle boundary is:

where $E_0 = \frac{G b}{4\pi(1-\nu)}$ and $A$ is an integration constant reflecting the core energy.

2. High-Angle Grain Boundaries ($ heta > 15^\circ$):

Dislocation cores overlap completely ($D \sim b$). The interface becomes an amorphous-like disordered zone characterized by high interfacial energy ($\gamma_{\text{HAGB}} \approx 0.3 - 0.8\text{ J/m}^2$) that serves as sinks for impurities and high-diffusivity pathways.

Stacking Faults and Partial Dislocations

In close-packed FCC structures ($\dots ABCABC \dots$ along $[111]$), a localized error in the close-packing sequence creates a stacking fault:

• Intrinsic Stacking Fault: One layer missing: $\dots ABC | BCABC \dots$ (locally creates four layers of HCP packing $BCBC$).
• Extrinsic Stacking Fault: Extra layer inserted: $\dots ABCA | B | CABC \dots$.
• Shockley Partial Dislocations: A full dislocation $\frac{a}{2}[110]$ dissociates into two Shockley partials bounding a ribbon of stacking fault to minimize total strain energy:

By Frank's energy rule ($E \propto b^2$):

Because $\frac{a^2}{3} < \frac{a^2}{2}$, dissociation is energetically favorable! The equilibrium separation width $d_{\text{eq}}$ between partials is inversely proportional to the Stacking Fault Energy $\gamma_{\text{SFE}}$:

The electronic behavior of extrinsic semiconductors is governed by isolated donor and acceptor states that introduce localized energy levels within the forbidden band gap.

The Hydrogenic Effective-Mass Model

Consider a pentavalent phosphorus atom ($\text{P}$) substituting on a tetravalent silicon site. Four valence electrons participate in tetrahedral covalent $sp^3$ bonding, while the fifth electron experiences the Coulomb attraction of the net $+1e$ core of the $\text{P}^+$ nucleus. Because this electron moves through the host dielectric medium of relative permittivity $\varepsilon_r$, its potential is:

Adapting the Bohr model of the hydrogen atom by replacing electron rest mass $m_0$ with conduction-band effective mass $m_e^*$ and free-space permittivity with $\varepsilon_0 \varepsilon_r$:

1. Donor Binding Energy ($E_d$):

For silicon ($arepsilon_r = 11.7$, $m_e^* \approx 0.26 m_0$):

Because $E_d \sim k_B T_{\text{room}} = 25.9\text{ meV}$, virtually $100\%$ of donors are ionized at room temperature.

2. Donor Bohr Radius ($a_d$):

The electron wavepacket spans dozens of silicon unit cells, justifying the macroscopic dielectric continuum approximation.

Impurity Compensation

When both donors ($N_d$) and acceptors ($N_a$) are present simultaneously in the same semiconductor crystal:

• If $N_d > N_a$, electrons from the donors fill all the acceptor states ($N_a^-$), leaving $N_d - N_a$ active net donors (compensated $n$-type):

• The Fermi level is pinned closer to midgap, and ionized impurity scattering increases, reducing carrier mobility.

The macroscopic movement of charge carriers in semiconductors is governed by two complementary transport mechanisms: drift (driven by electric fields) and diffusion (driven by concentration gradients).

Drift Transport and Carrier Mobility

Under an applied electric field $\mathbf{E}$, electrons and holes experience electrostatic forces and drift with average velocities:

where $\mu_e$ and $\mu_h$ are the electron and hole drift mobilities, related to the momentum relaxation time $\tau$ by:

Total drift current density is:

Mobility is limited by two primary microscopic scattering mechanisms (Matthiessen's rule, $1/\mu = 1/\mu_{\text{phonon}} + 1/\mu_{\text{impurity}}$):

1. Acoustic Phonon Scattering: Increases with temperature ($T$) as lattice vibrations increase: $\mu_{\text{ph}} \propto T^{-3/2}$.

2. Ionized Impurity Scattering (Brooks-Herring): High-velocity carriers at high $T$ spend less time near charged ions: $\mu_{\text{imp}} \propto T^{+3/2} / N_I$.

Diffusion Transport and The Einstein Relation

When a spatial concentration gradient exists, random Brownian thermal motion produces a net diffusion current density (Fick's first law):

where $D_n$ and $D_p$ are diffusion coefficients. In thermodynamic equilibrium under an internal built-in electric field (such as inside a $p-n$ junction), the drift and diffusion current densities must cancel exactly:

Using the Maxwell-Boltzmann distribution $n(x) = n_0 e^{-e V(x)/k_B T}$ where $E = -dV/dx$:

Substituting into the equilibrium current condition yields the Einstein relation:

At room temperature ($300\text{ K}$), the thermal voltage is $k_B T / e = 0.02585\text{ V} \approx 26\text{ mV}$.

Edwin Hall (1879) discovered that when a magnetic field is applied perpendicular to a current-carrying conductor, a transverse electric field develops across the material. This provides the standard method to determine carrier sign, density, and mobility.

Derivation of the Hall Voltage

Consider a rectangular semiconductor bar of thickness $t$, width $w$, and length $L$.

• A direct current $I_x$ flows along the $+x$-direction (current density $J_x = I_x / (w t)$).
• A uniform magnetic field $B_z$ is applied along the $+z$-direction.
• Moving charge carriers with drift velocity $v_x$ experience the Lorentz force:

• Along the transverse $y$-direction, the magnetic force $F_y = -q v_x B_z$ deflects carriers toward one edge until an opposing transverse electrostatic field (Hall field $E_H$) builds up to balance it:

Since $J_x = n q v_x \implies v_x = \frac{J_x}{n q}$:

The Hall Coefficient ($R_H$)

The Hall coefficient $R_H$ is defined as:

• Sign of Carriers: The polarity of the measured transverse Hall voltage $V_H = E_H w$ directly establishes whether conduction is dominated by electrons ($V_H < 0$) or holes ($V_H > 0$).
• Expressing in terms of measurable experimental quantities ($J_x = I_x / (wt)$):

Hall Mobility ($\mu_H$)

Combining the Hall coefficient with the longitudinal electrical conductivity $\sigma = n e \mu$:

This allows direct decoupled determination of both carrier concentration ($n = 1/(e |R_H|)$) and carrier mobility without requiring assumptions about effective mass or relaxation times.