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# 物理代写|半导体物理代写Semiconductor Physics代考|ECE4570 Diffusion

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## 物理代写|半导体物理代写Semiconductor Physics代考|Diffusion

In the preceding section the excess carriers are uniform in space. In this section, we discuss the situations where excess carriers are introduced locally, causing a condition of nonuniform carriers. Examples are local injection of carriers from a junction, and nonuniform illumination. Whenever there exists a gradient of carrier concentration, a process of diffusion occurs by which the carriers migrate from the region of high concentration toward the region of low concentration, to drive the system toward a state of uniformity. This flow or flux of carriers, taking electrons as an example, is governed by the Fick’s law,
$$\left.\frac{d \Delta n}{d t}\right|x=-D_n \frac{d \Delta n}{d x},$$ and is proportional to the concentration gradient. The proportionality constant is called the diffusion coefficient or diffusivity $D_n$. This flux of carriers constitutes a diffusion current, given by $$J_n=q D_n \frac{d \Delta n}{d x},$$ and $$J_p=-q D_p \frac{d \Delta p}{d x} .$$ Physically, diffusion is due to random thermal motion of carriers as well as scattering. Because of this, we have $$D=v{t h} \tau_m .$$

## 物理代写|半导体物理代写Semiconductor Physics代考|Thermionic Emission

Another current conduction mechanism is thermionic emission. It is a majoritycarrier current and is always associated with a potential barrier. Note that the critical parameter is the barrier height, not the shape of the barrier. The most-common device is the Schottky-barrier diode or metal-semiconductor junction (see Chapter 3). Referring to Fig. 26, for the thermionic emission to be the controlling mechanism, the criterion is that collision or the drift-diffusion process within the barrier layer to be negligible. Equivalently, the barrier width has to be narrower than the mean free path, or in the case of a triangular barrier, the slope of the barrier be reasonably steep such that a drop in $k T$ in energy is within the mean free path. In addition, after the carriers are injected over the barrier, the diffusion current in that region must not be the lim- iting factor. Therefore, the region behind the barrier must be another $n$-type semiconductor or a metal layer.

Due to Fermi-Dirac statistics, the density of electrons (for $n$-type substrate) decreases exponentially as a function of their energy above the conduction band edge. At any finite (nonzero) temperature, the carrier density at any finite energy is not zero. Of special interest here is the integrated number of carriers above the barrier height. This portion of the thermally generated carriers are no longer confined by the barrier so they contribute to the thermionic-emission current. The total electron current over the barrier is given by (see Chapter 3 )
$$J=A^* T^2 \exp \left(-\frac{q \phi_B}{k T}\right) .$$
where $\phi_B$ is the barrier height, and
$$A^* \equiv \frac{4 \pi q m^* k^2}{h^3}$$
is called the effective Richardson constant and is a function of the effective mass. The $A^*$ can be further modified by quantum-mechanical tunneling and reflection.

## 物理代写|半导体物理代写Semiconductor Physics代考|Diffusion

$$\left.\frac{d \Delta n}{d t}\right|x=-D_n \frac{d \Delta n}{d x},$$，并与浓度梯度成正比。比例常数称为扩散系数或扩散率$D_n$。载流子的这种通量构成了扩散电流，由$$J_n=q D_n \frac{d \Delta n}{d x},$$和$$J_p=-q D_p \frac{d \Delta p}{d x} .$$给出，在物理上，扩散是由于载流子的随机热运动和散射。因此，我们有$$D=v{t h} \tau_m .$$

## 物理代写|半导体物理代写半导体物理学代考|热离子发射

.

$$J=A^* T^2 \exp \left(-\frac{q \phi_B}{k T}\right) .$$

$$A^* \equiv \frac{4 \pi q m^* k^2}{h^3}$$

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