Lecture 10 - Design & Application Problems

1. Device Selection & Mechanism:
You must choose the Silicon Schottky barrier diode. A $p$-$n$ junction diode relies on the injection of minority carriers. When switched to reverse bias, these stored minority carriers must be physically swept out or recombined before the junction can block voltage, leading to a significant "storage time" ($t_s$) delay (typically nanoseconds). A Schottky diode is a majority carrier device. Electrons cross from the semiconductor into the metal and are instantly absorbed into the metal's Fermi sea. Since there is no minority carrier storage, the Schottky diode can switch in picoseconds, meeting the $28 \text{ GHz}$ requirement.

2. Diffusion Capacitance ($C_d$):
Diffusion capacitance is mathematically defined by the change in stored minority charge with respect to applied voltage: $C_d = \frac{dQ_{stored}}{dV}$. Because the Schottky diode has negligible minority carrier injection, $Q_{stored} \approx 0$. Therefore, the diffusion capacitance $C_d$ is practically zero, which eliminates the primary high-frequency RC time constant bottleneck present in $p$-$n$ junctions.

3. Trade-off (Reverse Saturation Current):
The primary trade-off is a significantly higher reverse leakage current. The Schottky diode's reverse saturation current is governed by thermionic emission:

Because the thermionic emission barrier ($\Phi_{Bn}$) is generally much smaller than the bandgap ($E_g$) which limits $p$-$n$ junction diffusion current, $J_{sT}$ is typically $10^3$ to $10^5$ times larger than the $J_s$ of a comparable $p$-$n$ junction. The detector will have higher static power dissipation and noise when reverse-biased.

1. C-V Profiling Extraction:
The depletion capacitance per unit area is $C' = \frac{\epsilon_s}{W}$. Substituting the depletion width equation $W = \left[\frac{2\epsilon_s(V_{bi}+V_R)}{eN_d}\right]^{1/2}$ yields:

This is a linear equation $y = mx + c$.
- The slope is $m = \frac{2}{e\epsilon_s N_d}$. By measuring the slope, we can directly calculate the doping concentration $N_d$.
- The x-intercept occurs where $(1/C')^2 = 0$. Solving $0 = V_R + V_{bi}$ shows the $x$-intercept is exactly $-V_{bi}$.

2. Image Force Lowering:
An electron near the metal interface induces a positive image charge inside the metal, creating an attractive Coulomb force. When combined with the strong external electric field $E$ in the depletion region, the peak of the potential energy barrier is physically lowered and pulled closer to the metal. As $V_R$ increases, the electric field $E$ increases, which increases the barrier lowering $\Delta\phi$. Since $J_{sT} \propto \exp(-\frac{e(\Phi_{Bn} - \Delta\phi)}{kT})$, the leakage current exponentially climbs as the barrier shrinks.

3. Scaling of Barrier Lowering:
The equation for image force lowering is:

Because $\Delta\phi$ is proportional to the square root of the electric field ($\Delta\phi \propto \sqrt{E}$), increasing the electric field by a factor of $100$ will increase the barrier lowering by a factor of $\sqrt{100} = 10$.

1. Initial Contact Evaluation:
For an ideal Ohmic contact on a $p$-type semiconductor, we require the metal work function to be greater than the semiconductor work function ($\Phi_m > \Phi_s$) to create a hole accumulation layer. Here, $\Phi_m$ ($4.2 \text{ eV}$) $< \Phi_s$ ($4.8 \text{ eV}$). Therefore, electrons will flow from the metal into the semiconductor to align the Fermi levels. These electrons recombine with the majority holes near the surface, uncovering fixed negative acceptor ions. This creates a depletion region, bending the bands downward and forming a rectifying Schottky barrier, not an Ohmic contact.

2. Processing Solution:
To force Ohmic behavior, you must perform a degenerate $p^+$ implant/diffusion right at the surface before depositing the metal. By heavily doping the semiconductor interface (e.g., $N_a \approx 10^{20} \text{ cm}^{-3}$), you alter the mechanism of conduction.

3. Quantum Mechanical Principle:
The depletion width $W$ is inversely proportional to the square root of the doping concentration ($W \propto 1/\sqrt{N_a}$). By making $N_a$ massive, the depletion width shrinks to incredibly thin dimensions ($\sim 10 \text{ \AA}$). At this nanoscale width, carriers no longer need to acquire thermal energy to leap over the top of the barrier (thermionic emission). Instead, their wavefunctions extend through the barrier, allowing them to quantum tunnel directly through it with near-zero resistance, acting as a highly efficient Ohmic contact.

1. Band Discontinuities & Alignment:
Using the electron affinity rule, the conduction band discontinuity is:

The total bandgap difference is $\Delta E_g = 2.2 \text{ eV} - 1.1 \text{ eV} = 1.1 \text{ eV}$. The valence band discontinuity covers the remainder: Because Material A's conduction band is $0.5 \text{ eV}$ lower than B's, and its valence band is $0.6 \text{ eV}$ higher than B's, the entire $1.1 \text{ eV}$ bandgap of Material A fits strictly inside the $2.2 \text{ eV}$ bandgap of Material B. This defines a Straddling alignment.

2. Charge Transfer at Equilibrium:
Material B is heavily $n$-doped, meaning its Fermi level is near its conduction band edge. Material A is undoped (intrinsic), meaning its Fermi level is near mid-gap. When joined, electrons from the higher-energy states in heavily doped Material B will "spill over" or diffuse across the interface into the lower-energy, empty states of undoped Material A until the global Fermi level is perfectly flat across the entire system. This leaves behind fixed positive donor ions in Material B and creates an accumulation of negative electrons in Material A.

3. Band Edge Shape & Unique Feature:
The transfer of electrons sets up a built-in electric field that bends the energy bands. On the Material B side, the conduction band bends upward (depletion). On the Material A side, the influx of electrons causes the conduction band to bend sharply downward toward the interface. Because the conduction band of Material A dips deeply, it is pulled below the global equilibrium Fermi level. This steep dip, combined with the abrupt vertical wall of the $\Delta E_c$ discontinuity, forms a triangular quantum well. This traps the spilled electrons in a highly confined region, forming a Two-Dimensional Electron Gas (2-DEG).

1. Modulation Doping Architecture:
In standard semiconductors, you dope the material where you want the current to flow. "Modulation doping" breaks this rule. The donor impurities are intentionally placed exclusively in the wide-gap material (N-AlGaAs). The adjacent narrow-gap material (GaAs) is left completely undoped (intrinsic). As the system reaches thermal equilibrium, the electrons donated by the impurities in the AlGaAs spill across the heterojunction interface and fall into the lower-energy conduction band of the undoped GaAs, forming the 2-DEG. Thus, the parent donor ions are in the AlGaAs, but the conducting electrons reside in the GaAs.

2. Solving the Trade-off (Mobility):
In bulk semiconductors, increasing doping (to get more carriers) inherently reduces mobility because the traveling electrons constantly collide with the charged, ionized dopant atoms (Coulomb/impurity scattering). Modulation doping physically spatial-separates the carriers from their parent ions. The dense sheet of electrons in the 2-DEG travels through the undoped GaAs crystal lattice. Because there are no ionized impurities in the GaAs to scatter them, they experience near-pristine lattice conditions. This results in record-breaking mobility and extremely low noise, which is vital for amplifying faint satellite signals without adding static.

3. Quantum Confinement:
The triangular potential well holding the 2-DEG is incredibly narrow in the $z$-direction (perpendicular to the interface)—often on the order of $10 \text{ nm}$ or less, comparable to the de Broglie wavelength of the electron. Because they are physically trapped in such a tight space, quantum mechanics dictates that their allowable energy states in the $z$-direction cannot be continuous; they are quantized into discrete subbands ($E_0$, $E_1$, etc.). However, there are no physical boundaries parallel to the interface (the $x$-$y$ plane), so the electrons are free to move laterally like a classic continuous gas. They are "two-dimensional" because their movement is strictly limited to this plane, with the vast majority occupying the lowest ground state $E_0$.