Energy Band Model in Doped Semiconductors
- The extra electron from a donor atom creates an energy level just inside the band-gap (below \( E_C \)).
- This energy level doesn't extend throughout the lattice; rather it is localized close to the donor atom.
- At low temperature, the extra electron stays in this level; at room temperature, however, it gains enough thermal energy to jump into the conduction band and becomes mobile.
- Conversely, an acceptor atom creates an empty level just above \( E_V \), which accepts an electron from the valence band at room temperature, creating a mobile hole.
Interactive Band & Lattice Model
Increase the temperature to observe carriers gaining thermal energy. Electrons in the donor level (\( E_D \)) jump to the Conduction Band (\( E_C \)), and holes are created in the Valence Band (\( E_V \)) as electrons get trapped in the acceptor level (\( E_A \)).
Drift Current & Effective Mass
"The net motion of electrons in a nearly full band can be described by considering just the empty states, provided there is a positive electronic charge associated with each state."
Electron Effective Mass (\( m^* \))
Current is due to the net flow of charge: \( J = q N v_d \), where \( v_d \) is the drift velocity. When an electric field is applied, the total force on an electron in a lattice is the sum of external and internal forces (\( F_{total} = F_{ext} + F_{int} = m a \)). Because it is incredibly difficult to account for all internal lattice forces, we model the electron as a "free" particle responding only to the external field, but with a modified effective mass (\( m^* \)).
\[ E - E_C = C_1 k^2 \implies \frac{d^2E}{dk^2} = 2C_1 \]
Comparing with a free particle (\( E = \frac{\hbar^2 k^2}{2m} \)): \[ \frac{1}{\hbar^2} \frac{d^2E}{dk^2} = \frac{1}{m^*} \]
E-k Parabolic Approximation
Extending the Concept to Holes
- A positively charged empty state is created when a valence electron is elevated into the conduction band.
- At \( T > 0 \), any valence electron can gain a small amount of thermal energy and hop into an empty state. The movement of an electron is equivalent to the movement of the empty state itself \(\rightarrow\) a hole is created.
The above equation is equivalent to placing a +ve charge in the empty states and assuming all other states are empty or neutrally charged.
Empty States Equivalence
Almost full band is equivalent to positive charges in empty states.
\[ E - E_V = -C_2 k^2 \quad (C_2 > 0) \]
Taking the second derivative and substituting into the effective mass equation: \[ \frac{1}{\hbar^2} \frac{d^2E}{dk^2} = \frac{-2C_2}{\hbar^2} = \frac{1}{m^*} \]
As \( C_2 > 0 \), we find that \( m^* < 0 \).
An electron moving near the top of an allowed energy band behaves as if it has a negative mass (ice cube in water analogy). Because \( \frac{d^2E}{dk^2} > 0 \) near the bottom of the conduction band, the electron effective mass is positive. However, near the top of the valence band, the curvature is negative, giving \( m^* < 0 \). This pseudo-particle accelerates in the same direction as the electric field, behaving precisely like a positively charged particle with positive mass—a hole!
Curvature and Effective Mass Mapping
Analogy Interactive: Marble vs. Ice Cube
Marble in Oil: An electron in the conduction band moving against the E-field.
Ice Cube in Water: A hole in the valence band moving with the E-field (buoyancy equivalent).
Materials & E-k Diagrams
Conditions for Electric Conduction
Before analyzing specific materials, we must identify which band occupancy scenarios lead to an electric current when a small external field is applied. A net current requires carriers to gain momentum, which is only possible if there are nearby empty states they can move into.
Allowed energy band
fully empty
Allowed energy band
fully occupied
Allowed energy band
almost full
Allowed energy band
almost empty
Insulators
Conduction band empty, Valence band completely full. Very large \( E_g \).
Semiconductors
CB almost empty, VB almost full at \( T > 0 \) K. Moderate \( E_g \).
Metals
Partially filled bands or overlapping bands. No bandgap to cross.
Semiconductor Band Diagram
3D Crystals: Direct vs. Indirect Bandgap
The distance between atoms in a 3D crystal changes as the origin direction changes, making E-k diagrams direction-dependent.
- Direct Bandgap (e.g., GaAs): Conduction band minimum and valence band maximum occur at the same k-value. Electrons can transition by absorbing/emitting a photon directly.
- Indirect Bandgap (e.g., Si): CB minimum and VB maximum occur at different k-values. A transition requires a change in crystal momentum (k), necessitating interaction with a phonon (lattice vibration).
Electron Recombination Transition
Left: GaAs (Direct, Photon only). Right: Si (Indirect, Photon + Phonon required).
Interactive 3D E-k Surface Visualization
To fully grasp the difference between direct and indirect bandgaps, we examine the energy bands across the two-dimensional \(k_x, k_y\) reciprocal space plane using Empirical Pseudopotential Method (EPM) reference data. Use the tool below to rotate the 3D Brillouin zone and extract specific 1D profile slices.
GaAs 3D E-k Surface
A topological view of the Energy bands across the \(k_x, k_y\) plane.
Features to Observe:
- Direct Bandgap: Minimum of the CB and maxima of the VBs align perfectly at \((k_x=0, k_y=0)\), the \(\Gamma\) point.
- Secondary Valleys: Moving away from center, the CB bends downwards at the Brillouin zone edges, creating the L-valley along [111] and X-valley along [100].
Extract 2D Slice
Select two directions from \(\Gamma\) to extract a 1D profile.
COMPUTING SURFACES...
Density of States Function \( g(E) \)
To calculate the number of electrons and holes available for conduction (and eventually describe I-V characteristics), we need to know how many allowed quantum states exist at each energy level.
Mathematical Derivation Summary:
- Consider a free electron in a 3D cubical crystal of side \( a \). Using 3D Schrödinger Equation: \( k^2 = k_x^2 + k_y^2 + k_z^2 = (n_x^2 + n_y^2 + n_z^2)\frac{\pi^2}{a^2} \).
- Volume of a single quantum state in k-space is \( (\frac{\pi}{a})^3 \).
- We consider only the positive 1/8th of a sphere in k-space. Differential volume is \( \frac{1}{8}(4\pi k^2 dk) \).
- Total density of states in the differential volume \(\frac{1}{8}(4{\pi}{k^2}dk): g_T(k)dk = \frac{{k^2}{a^3}}{\pi^2} dk \).
- Converting from k-space to Energy space using \( k^2 = \frac{2mE}{\hbar^2} \).
Quantum States in k-Space
(a) 2D array of allowed states (b) Positive 1/8th spherical volume
Density of quantum states = number of quantum states per unit energy per unit volume.
Extended to semiconductors near the band edges (parabolic approximation):
\( g_c(E) = \frac{4\pi (2m_n^*)^{3/2}}{h^3} \sqrt{E - E_c} \)
\( g_v(E) = \frac{4\pi (2m_p^*)^{3/2}}{h^3} \sqrt{E_v - E} \)
Density of States vs. Energy
The density of energy states in the conduction and valence bands as a function of energy.
Practice Question
Determine the number of quantum states in Si between \( E_c \) and \( E_c + kT \) at 300 K.
Answer: \( 2.12 \times 10^{19} \text{ cm}^{-3} \)
Lecture 4 Knowledge Check
Put your understanding of carrier transport, effective mass, density of states, and Fermi-Dirac statistics to the test.