P-N Junctions

The p-n junction possesses some interesting properties which have useful applications in modern electronics. P-doped semiconductor is relatively conductive. The same is true of N-doped semiconductor, but the junction between them is a nonconductor. This nonconducting layer, called the depletion zone, occurs because the electrical charge carriers in doped n-type and p-type silicon (electrons and holes, respectively) attract and eliminate each other in a process called recombination. By manipulating this nonconductive layer, p-n junctions are commonly used as diodes: electrical switches that allow a flow of electricity in one direction but not in the other (opposite) direction. This property is explained in terms of the forward-bias and reverse-bias effects, where the term bias refers to an application of electric voltage to the p-n junction. Consider lecture 10, 11.

Doping Profile

A plot of the net doping concentration as a function of position is referred to as the doping profile.

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Unable to render embedded object: File (Doping_profile.PNG) not found.

</center>

Only the doping variation in the immediate vicinity of the metallurgical junction is of prime importance. Two common idealizations are the step junction and the linearly graded junction profiles.

Poisson's Equation

Poisson's equation is a relationship from electricity and magnetism. It can be a starting point in finding quantitative solutions to electrostatic variables.

<center>

<br>

<math>\nabla \cdot E = \frac

Unknown macro: {rho}
Unknown macro: {K_s epsilon_o}

</math>

<br>

</center>

Consider when <math>E = E_x</math>.

<center>

<br>

<math>\frac

Unknown macro: {dE}
Unknown macro: {dx}

= \frac

Unknown macro: {K_s epsilon_o}

</math>

<br>

</center>

Below is the charge density inside a semiconductor assume dopants to be totally ionized.

<center>

<br>

<math>\rho = q(p - n +N_D - N_A)</math>

<br>

</center>

Qualitative Solution

There should be band bending and an internal electric field with a nonuniform doping of a pn junction diode. Assume a one-dimensional step junction. Expect regions far from the metallurgical junction behave identically to an isolated semiconductor. The fermi level is constant under equilibrium.

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Unable to render embedded object: File (Step_junction_I.PNG) not found.

Unable to render embedded object: File (Step_junction_II.PNG) not found.

Unable to render embedded object: File (Step_junction_III.PNG) not found.

</center>

Deduce the functional form of the electrostatic variables. The relationship of V versus x is of the same functional form as the curve of <math>E_c</math>, <math>E_i</math>, or <math>E_v</math> upside-down. Find the <math>E</math> versus <math>x</math> relationship from the derivative of <math>E_c</math> with respect to position. Find <math>\rho</math> versus <math>x</math> from the slope of the plot of <math>E</math> versus <math>x</math>.

<center>

<br>

Unable to render embedded object: File (Step_junction_--_qVbi.PNG) not found.

<br>

Unable to render embedded object: File (Step_junction_--_electrostatic_potential.PNG) not found.

<br>

Unable to render embedded object: File (Step_junction_--_electric_field.PNG) not found.

<br>

Unable to render embedded object: File (Step_junction_--_charge_density.PNG) not found.

<br>

<math>\vec E = \int \frac

Unknown macro: {rho}
Unknown macro: {epsilon}

dx</math>

<br>

<math>V = \int \vec E dx</math>

<br>

</center>

There is a voltage drop across the junction under equilibrium conditions and there is charge near the metallurgical boundary. Consider the source of charge. In a p-material, the positive hole charges balance immobile acceptor-site charges. In an n-material, the electronic charge balances the immobile charge associated with the ionized donors. After the two materials are joined, the holes begin to diffuse from the p-side to the n-side. Electrons diffuse from the n-side to the p-side of the junction. The donors and acceptors are fixed. Unbalanced dopant site charge is left behind. There is a significant non-zero charge in the space charge region or depletion region. The built-up of charge and the associated electric field continues until the diffusion of carriers across the junction is balanced by carrier drift.

The Built-in Potential

Consider a nondegenerately-doped pn junction under equilibrium conditions. Ends of the equilibrium depletion region are at <math>-x_p</math> and <math>x_n</math>.

<center>

<br>

<math>E = - \frac

Unknown macro: {dV}
Unknown macro: {dx}

</math>

<br>

<math>- \int_{-x_p}^

Unknown macro: {x_n}

E dx = \int_

Unknown macro: {V(-x_p)}

^

Unknown macro: {V(x_n)}

dV</math>

<br>

<math>- \int_{-x_p}^

E dx = V(x_n) - V(-x_p)</math>

<br>

<math>- \int_{-x_p}^

Unknown macro: {x_n}

E dx = V_

Unknown macro: {bi}

</math>

<br>

<math>J_N = q \mu_n n E + qD_n \frac

Unknown macro: {dn}

</math>

<br>

<math>0 = q \mu_n n E + qD_n \frac

Unknown macro: {dn}
Unknown macro: {dx}

</math>

<br>

<math>E = - \frac

Unknown macro: {D_N}
Unknown macro: {mu_N}

\frac

Unknown macro: {dn/dx}
Unknown macro: {n}

</math>

<br>

<math>E = - \frac

Unknown macro: {kT}
Unknown macro: {q}

\frac

Unknown macro: {n}

</math>

<br>

<math>V_

Unknown macro: {bi}

= - \int_{-x_p}^

Unknown macro: {x_n}

E dx</math>

<br>

<math>V_

= \frac

Unknown macro: {kT}
Unknown macro: {q}

\int_

Unknown macro: {n(-x_p)}

^

Unknown macro: {n(x_n)}

\frac

Unknown macro: {n}

</math>

<br>

<math>V_

Unknown macro: {bi}

= \frac

Unknown macro: {kT}
Unknown macro: {q}

\ln \left [\frac

Unknown macro: {n(x_n)}
Unknown macro: {n(-x_p)}

\right]</math>

<br>

<math>n(x_n) = N_D</math>

<br>

<math>n(-x_p) = \frac

Unknown macro: {n_i^2}
Unknown macro: {N_A}

</math>

<br>

<math>V_

= \frac

Unknown macro: {kT}
Unknown macro: {q}

\ln \left ( \frac

Unknown macro: {N_A N_D}
Unknown macro: {n_i^2}

\right )</math>

<br>

</center>

Show that <math>qV_

Unknown macro: {bi}

</math> is the barrier to minority carrier injection.

<center>

<br>

<math>p_n = p_p e^{\frac{-qV_{bi}}

Unknown macro: {k_b T}

</math>

<br>

<math>n_p = n_n e^{\frac{-qV_

Unknown macro: {bi}

}

</math>

<br>

</center>

Depletion Approximation

The depletion approximation is a simplifying approximations used in the modeling of devices. It provides a way of obtaining approximate solutions without prior knowledge of the carrier concentrations. Two components of the approximation are below.

  • The carrier concentrations are assumed to be negligible compared to the net doping concentration in a region <math>-x_p \le x \le x_n</math> near the metallurgical junction.
  • The charge density outside the depletion region is taken to be zero

Solution of xn and xp in a step junction

<center>

<br>

<math>x_n = \left [\frac

Unknown macro: {2 K_s epsilon_0}

\frac

Unknown macro: {N_A}
Unknown macro: {N_D (N_A + N_D )}

(V_

Unknown macro: {bi}

- V_A) \right]^

Unknown macro: {1/2}

</math>

<br>

<math>x_p = \frac

Unknown macro: {N_D x_n}

</math>

<br>

<math>W = x_n + x_p</math>

<br>

</center>

Applied Bias

Depletion widths decrease under forward biasing and increase under reverse biasing. A decreased depletion width when there is a forward bias means there is less charge around the junction and a correspondingly smaller electronic field. Reverse biasing creates a large space charge region and a bigger electric field. Potential decreases at all points with a forward bias and increases at all points with a reverse bias. The potential hill shrinks in both size and extent under forward biasing, whereas reverse baising gives rise to a wider and higher potential hill. One may conceive of the diode terminals as providing direct access to the p- and n-ends of the equilibrium Fermi level. Progress from the equilibrium diagram to the forward bias diagram by moving the n-side upward by <math>qV_a</math> while holding the p-side fixed. The reverse bias is obtained from the equilibrium diagram by pulling the n-side Fermi level downward.

Forward-bias

<p>
</p>

Forward-bias occurs when the P-type block is connected to the positive terminal of a battery and the N-type block is connected to the negative terminal, as shown below.

<p>
</p>

With this set-up, the 'holes' in the P-type region and the electrons in the N-type region are pushed towards the junction. This reduces the width of the depletion zone. The positive charge applied to the P-type block repels the holes, while the negative charge applied to the N-type block repels the electrons. As electrons and holes are pushed towards the junction, the distance between them decreases. This lowers the barrier in potential. With increasing bias voltage, eventually the nonconducting depletion zone becomes so thin that the charge carriers can tunnel across the barrier, and the electrical resistance falls to a low value. The electrons which pass the junction barrier enter the P-type region (moving leftwards from one hole to the next, with reference to the above diagram).

<p>
</p>

This makes an electric current possible. An electron starts flowing around from the negative terminal to the positive terminal of the battery. It starts at the negative terminal, moving towards the N-type block. Having reached the N-type region it enters the block and makes its way towards the p-n junction. The junction barrier can no longer keep the electron in the N-type region due to the forward-bias effect (in other words, the thin depletion zone produces very little electrical resistance against the flow of electrons). The electron will therefore cross the junction and move ahead into the P-type block. Once inside the P-type region, the electron, being thermally free (from bonding)���or mobile���will move through the rest of the crystal, making its way to the positive terminal of the power supply. Please note that the electron does not jump from one hole to the next in the p-region. This actually qualifies as electron-hole recombination which immobilises both hole and electron. The electron can move freely through the crystal without needing to jump into holes which is what happens when electrons do cross the depletion layer. This process will be repeated over and over again, producing a complete circuit path through the junction.

<p>
</p>

The Shockley diode equation models the operation of a p-n junction outside the avalanche region.

<p>
</p>

Reverse Bias

<p>
</p>

Connecting the P-type region to the negative terminal of the battery and the N-type region to the positive terminal, produces the reverse-bias effect. The connections are illustrated in the following diagram:

<p>
</p>

Because the P-type region is now connected to the negative terminal of the power supply, the 'holes' in the P-type region are pulled away from the junction, causing the width of the nonconducting depletion zone to increase. Similarly, because the N-type region is connected to the positive terminal, the electrons will also be pulled away from the junction.

<p>
</p>

This effectively increases the potential barrier and greatly increases the electrical resistance against the flow of charge carriers. For this reason there will be no (or minimal) electric current across the junction.

<p>
</p>

At the middle of the junction of the p-n material, a depletion region is created to stand-off the reverse voltage. The width of the depletion region grows larger with higher voltage. The electric field grows as the reverse voltage increases. When the electric field increases beyond a critical level, the junction breaks down and current begins to flow by avalanche breakdown.

Qualitative Effect of Bias

  • Applying a potential to the ends of a diode does not increase current through drift
  • The applied voltage upsets the steady-state balance between drift and diffusion, which can unleash the flow of diffusion current
  • "Minority carrier device"

<center>

<br>

<math>n_p = n_n e^{\frac{-q(V_

Unknown macro: {bi}
  • V_a)}
    Unknown macro: {k_B T}
    </math>

<br>

<math>p_n = p_p e^{\frac{-q(V_

- V_a)}{k_B T}}</math>

</center>

  • Forward bias, which is a positive potential applied to p and a negative potential applied to n, decreases the depletion region and increases the diffusion current exponentially
  • Reverse bias, which is a negative potential applied to p and a positive potential applied to n, increases the depletion region, and no current flows ideally
  • Solve minority carrier diffusion equations on each side and determine <math>J</math> at the depletion edge.

<center>

<br>

<math>J = q \left (\frac

Unknown macro: {D_e}
Unknown macro: {L_e}

\frac

Unknown macro: {n_i^2}
Unknown macro: {N_a}

+ \frac

Unknown macro: {D_h}
Unknown macro: {L_h}

\frac

Unknown macro: {N_d}

\right ) \left (e^{\frac

Unknown macro: {qV_a}

{k_B T}} - 1 \right )</math>

<br>

<math>J = J_o \left (e^{\frac

{k_B T}} - 1 \right )</math>

<br>

</center>

I-V curve

Consider the equilibrium band diagram for a pn junction. On the quasineutral n-side of the junction there are a large number of electrons a few holes. On the quasineutral p-side of the junction there are a high concentration of holes and a small number of electrons. Most carriers are of insufficient energy to "climb" the potential hill. There are some high-energy electrons that can surmount the hill and travel over to the p-side of the junction. This is the diffusion from the high-electron population n-side of the junction to the low-electron population p-side of the junction.

<p>
</p>

Electrons on the p-side are not restricted. If an electron on the p-side travels into the depletion region, it is rapidly swept over to the other side of the junction. This drift current balances the diffusion current under equilibrium conditions. A significant change resulting from a forward bias is a lowering of the potential hill between the p- and n- sides of the junction. With the potential hill decreased, more n-side electrons and p-side holes can surmount the hill and travel to the opposite side of the junction. Because the potential hill decreases linearly with the applied forward bias and the carrier concentrations vary exponentially as one progresses away from the band edges, the number of carriers of sufficient energy to surmount the potential barrier goes up exponentially with <math>V_A</math>.

<p>
</p>

A major effect of the reverse bias is to increase the potential hill between the p- and n-sides of the junction. Even a small reverse bias, anything greater than a few <math>\frac

Unknown macro: {q}

</math> in magnitude, reduces the majority carrier diffusion across the junction to a negligible level. Reverse biasing gives rise to a current flow directed from the n-side to the p-side of the junction. The reverse current is expected to saturate once the majority carrier diffusion currents are reduced to a negligible level at a small reverse bias. The overall I-V dependence is of the form below.

<center>

<br>

<math>I - I_0 \left ( e^{V_A / V_{ref}} - 1 \right )</math>

<br>

</center>

The above equation is identical to the ideal diode equation if V_

Unknown macro: {ref}

is set equal to <math>kT/q</math>.

Notes

Memorize portion of periodic table to learn what elements to use to dope

References

Initial materials selection considerations in interface light emitting devices

  • Bandgap: wavelength of emission
    • Larger atoms, weaker bonds, smaller <math>U</math>, smaller <math>E_g</math>, higher <math>\mu</math>, more costly
  • Nature of gap (direct or indirect): Radiative recombination
  • Lattice matching: Efficiency

Homojunctions and Heterojunctions

Homojunctions

  • Same material on both sides of the junction – different dopant levels
  • After junction is formed, there is constant chemical potential and continuous bands
  • Example: p-n junction in doped <math>Si</math>
  • Lattice matched by definition
  • Impurity scattering

Heterojunctions

  • Different materials on both sides of the junction – different bandgaps and work functions
  • After junction is formed there is constant chemical potential. However bands are discontinuous
  • Example: heterojunction in <math>GaAs/AlGaAs</math>
  • Need to choose lattice matched materials
  • Heterojunctions can be used as a means to create internal potentials
    • Potential barriers of holes and electrons can be created inside a material
    • There are different band gaps and electron affinity/work functions associated with different semiconductor materials
    • Internal fields from doping p-n must be superimposed on the effects.
    • Poisson Solver:

<center>

<br>

<math>\frac

Unknown macro: {dE}
Unknown macro: {dx}

= V</math>

<br>

<math>\frac

Unknown macro: {dx}

= \frac

Unknown macro: {epsilon}

</math>

<br>

</center>

In both cases the chemical potential <math>\mu</math> plays a key role in determining junction characteristics.

Lasers

Below are conditions necessary to achieve lasing action.

  • A system with more than two energy levels
  • Radiative decay path
  • Direct band gap semiconductors
  • Population inversion
  • Forward bias leads to injection of minority carriers
  • Stimulated emission
  • Reflectors for cavity

<math>GaAs</math> p-n junction laser

  • <math>GaAs</math> diode laser is doped with <math>Zn</math> (acceptor) <math>N_A = 10^
    Unknown macro: {20}
    </math> and <math>Te</math> as donor <math>N_D = 10^
    Unknown macro: {18}
    </math>. There are degenerate doping conditions such that the chemical potentials penetrate the bands.
  • <math>t</math> recombination in <math>GaAs</math> <math>10^{-9} s D - 10 cm^2/s</math>
  • <math>l = \sqrt
    Unknown macro: {Dt}
    = 1 \mu m</math>
  • Active region is determined by the large diffusion length and not the small depletion region (~<math>100nm</math>). Light emission is not well confined.
  • Donor tail merges with the conduction band at high impurity concentrations reducing the operational frequency leading to a spectral broadening and inhibiting lasing action
  • High dopant levels increase scattering and decrease carrier mobility
  • High threshold currents lead to cryogenic temperature requirements for operation, which are not practical

The solution is double heterojunction laser structures

  • The active material is <math>GaAs</math> (834 nm) or a quaternary alloy <math>InGaAsP</math>
  • There is close lattice matching of the materials
  • The refractive index of the active layer is <math>3.6</math> vs <math>3.4</math> for <math>Al_
    Unknown macro: {0.3}
    Ga_
    Unknown macro: {0.7}
    As</math>. There is strong confinement of the emitted beam to the active layer.
  • The band gap of <math>GaAs</math> is 1.42 eV while the gap in <math>Al_xGa_
    Unknown macro: {1-x}
    As</math> is about 1.8 (eV) considering <math>x = 0.3</math>. An effect is the confining of carriers to the active layer.
  • Loss is reduced due to the transparency of the <math>AlGaAs</math> layers

Quantum Wells

Approximate a well in a heterojunction structure as containing infinite potential boundaries. Modify electronic transitions through quantum wells.

<center>

<br>

<math>k = \frac

Unknown macro: {n pi}
Unknown macro: {L}

</math>

<br>

<math>E = \frac

Unknown macro: {hbar^2 k^2}
Unknown macro: {2 m^*}

</math>

<br>

<math>E = \frac

Unknown macro: {h^2 n^2}
Unknown macro: {8 m^* L^2}

</math>

<br>

</center>

Solar Cells

When useful

Two-terminal devices:

Three-terminal devices:

Four-terminal devices:

Multi-terminal devices:

Reference

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