Band bending (Surface defects)
We know every surface has defects because of no periodic lattice anymore. It has different bodings than that of bonds in the bulk of the semiconductor. Therefore, the dispersion relation acts differently on the 2D surface than on the bulk side. We can draw the schematic image of the surface with a band diagram similar to bulk in 2D spatial dimension having many states in the band gap. Figure 1 shows the detail of the surface states and band diagram of an n-type semiconductor.
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| Figure 1: n-type semiconductor and its surface states |
The surface shows many unspecified acceptor states. The Fermi energy of the acceptor and semiconductor will adjust such that the higher Fermi energy will move down-side and defect level Fermi energy will move upwards by delivering electrons to the lower level so that the higher Fermi energy moves to the down-side. Now, if the bulk and surface are in close contact. Three cases will occur in the process.
Case-1: just after the contact, the electrons from the bulk will flow to the empty states of the surface (in this case acceptor level). As a result, the surface will be negatively charged (before there was no charge) as shown in figure 2.
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| Figure 2: Imagine the electron moving from bulk into the surface (like contacting with the surface) |
Case-2: As the surface is negatively charged then the static potential of the surface becomes high and goes (reduced) into bulk as shown in figure 3. Then we get built-in potential (Vb). This shows a path to realize a band bending takes place with the potential energy (-eVb). Further, the negative surface repelled the negative electrons from the bulk into the surface. It is necessary energy to move the electrons to the surface and that is the potential energy.
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| Figure 3: Potential energy distribution of static charges (negative static charge) |
Case-3: The electron transfer continues until the equilibrium state is achieved. This means no difference in the potential energy from the surface side to the bulk where the Fermi energy of the surface equals the Fermi energy of bulk as shown in figure 4.
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| Figure 4: The surface and bulk in equilibrium. |
Space Charge Region
However, this behavior will lead to charge imbalance. The band bending region will not contain any free electrons, but it has static positive charged donor atoms. The fixed or static charges reside in the space charge region (SCR) or static region. The corresponding electric field in the region follows a one-dimensional Poisson relation:
Ex=-dV/dx (1)
This field starts from the positive charge side and ends at the negative charge side. The positive charge side is the ionized donor atoms (inside bulk), and the negative charge is the excess electron on the surface. The SCR makes a formula for the width d of the region. We assume the positive static charge is distributed homogeneously through the SCR volume. Now the capacitance Cs is approximated by a plate capacitor with half of the distance dcap=d/2. Then the capacitance becomes,
CS= 2εsA/d=Q/V (2)
Where capacitor plates area A, Q charge on the plates, V is the potential difference between the plates. The charge is equal to the donor ionized (ND) in the volume (d.A). Then,
Q=eNDdA (3)
Now, we can get the width of the space charge region from equation (2) by using equation (3)
2εsA/d= eNDdA/V
Or, d=(2εsVbi/eND)1/2 (4)
This is correct with the assumption dcap=d/2. The voltage V is nothing but the built-in (Vbi) potential. This is simply the difference of the Fermi energy as mentioned earlier i.e ΔEF/e. If there is an additional potential Vap from outside, then the total potential is:
V=[(ΔEF/e) +(Vap/e)] (5)
The SCR depth:
d=(1/e)[(2εs (ΔEF+eVap)]/ND) 1/2 (6)
We see that the built-in potential is positive. Therefore, an additional positive Vap will enlarge the width of the space charge region. This means the potential difference is increased between bulk and surface. We can have a similar equation from the versatile Poisson equation. We assume the one-dimensional semiconductor where V(x) is varied. Variation of V(x) is due to the charge density ρ(x) distribution into the semiconductor. The ρ(x) is always the net charge density (including electrons, holes, and ionized charges) of the consideration region. In this case, we only have considered static charges (positive and negative both conditions) on the surface.
-εsd2V(x)/dx2=εsdE(x)/dx=ρ(x) (7)
The conduction and valence band can be represented by the electrostatic potential V(x) having potential energy -eV(x). Anyway, the band energy in terms of electrons energy,
EC(x)=-eV(x) (8)
EV(x)=-eV(x) (9)
In case of homogeneously doped semiconductor
However, in a homogeneously doped semiconductor, there are static charges on the surface. There is no current flow and no neutralization of the static charge. The surface side will create an electric field on the bulk side free carriers near the surface resulting in band bending. Here, we expect two different statuses:
Case-1: The polarity of the surface charge is the same as the free majority carrier inside the semiconductor. Because of the same polarity, the surface charges push the inside charge carriers. The pushing force creates a space charge region. This SCR region is called a depletion region figure 5 (b). The band is bent which means the Ec-EF increases towards the surface. The SCR width depends on the dopant density. It is high with low dopant because the dopant atoms cannot reach the interface. We need many dopants to compensate for the surface charge. Also, the accumulation of minority carriers at the valence band results in inversion layer formation. The inversion phenomenon is shown in figure 5(a).
Case-2: If the surface charge is opposite to the majority carrier, then accumulation takes place near the surface region. Then the free charge carriers can compensate for the surface charges. Therefore, the electric field can’t penetrate to a large extent inside the semiconductor. The behavior is similar to accumulation (figure 5 (d)). The majority of carriers and static surface charge have been designated by yellow and blue spheres respectively. Between accumulation and depletion, there is a flat-band case as shown in figure 5 (c).
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| Figure 5: Band bending of n-type semiconductor with respect to its surface static charge carrier |
Now the discussion about the p-n junction can be visualized in terms of their band bending, potential distribution, and electric field distribution as shown in figure 6.
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| Figure 6: Band bending of p-n junction views in a gradual process. |
The equation (6) only takes the form below. Please take care of the sign of Vap for the calculation theoretical calculation.
d=(1/e)[(2εs (1/ND+1/NA)(ΔEF+eVap)]) 1/2 (10)
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