— NEW/EDU — p-n junction in the dark

Attention

This tutorial is under construction

Introduction

In this tutorial, you can learn fundamentals of p-n junction. We refer to §6 in [NelsonPSC2003] and §2 in [Sze_Kwok_2007] to make this tutorial. We look into the physical properties of the GaAs p-n junction at equilibrium first. Then, we apply forward bias and investigate the current-voltage characteristics. We apply the p-n junction to a solar cell and explain the basic principles of the solar cell in — EDU — p-n junction under illumination. If you are interested in simulation of solar cells, we recommend that you read it too.

At equilibrium

Figure 2.4.63 shows the schematic illustration of the p-n junction.

../../../../_images/1D_gaas_edu_pn_junction_schematic_ill.jpg

Figure 2.4.63 The schematic illustration of the p-n junction.

At equilibrium, the built-in-potential Vbi is formed across the space charge region. The process of forming the built-in-potential is explained below. First, the carrier density gradients arise across the junction when p-doped GaAs and n-doped GaAs are joined. Then, the free electrons in n-doped GaAs diffuse and combine with holes in p-doped GaAs. Similarly, the free holes in p-doped GaAs diffuse and combine with electrons in n-doped GaAs. On the other hand, the ionized dopants, such as negatively charged acceptor and positively charged donor, cannot move and are fixed at their initial positions. Therefore, the ionized dopants in the region where the carriers are depleted form the electric field and the built-in-potential Vbi that impede the diffusion of majority carriers. The space charge region (the width: wd) denotes the region that is charged and loses the mobile carriers.

Figure 2.4.64 shows the basic characteristics of the diode at equilibrium.

../../../../_images/1D_gaas_edu_pn_junction_basics.svg

Figure 2.4.64 Some characteristics are shown across a p-n junction. (a) shows the dopant profile. (b) and (c) are the electric field and the potential across the space charge region, respectively.

Note that we assume the all dopants are ionized in the result to be consistent with Fig. 6.3. in [NelsonPSC2003]. You can see that the electric field is formed within the space change region and the voltage is equivalent to Vbi from Figure 2.4.64 (b) and (c).

Figure 2.4.65 shows (a) the band profiles and (b) the carrier densities at equilibrium. bandedges.dat, density_electon.dat, and density_hole.dat are used to produce this figure.

../../../../_images/1D_gaas_edu_pn_junction_0.0V.svg

Figure 2.4.65 The band profiles are plotted in (a). The carrier densities are plotted in (b). The hole density is shown in violet, whereas the electron density is in green.

In (a), CB and VB represent conduction and valence band, respectively. EFn and EFp are the electron quasi Fermi level and the hole quasi Fermi level. The results are in a good agreement with Fig. 6.5. in [NelsonPSC2003].

Vbi can be calculated at potential.dat and it is 2.78481.5779=1.207V in this case. The width of the space charge region wscr can be acquired by the following procedures.

Since

(2.4.2)wp=1Na2ε0εVbiq(1Na+1Nd)

and

(2.4.3)wn=1Nd2ε0εVbiq(1Na+1Nd),

Thus,

(2.4.4)wscr=wp+wn=2ε0εq(1Na+1Nd)Vbi

Each parameter corresponds to a value in the table below.

Parameter

Value

q (Elementary charge)

1.6022×1019C

ε0 (Vacuum permittivity)

8.854×1012C/(Vm)

ε (Relative permittivity of GaAs)

12.93

Na

1.0×1017cm3

Nd

1.0×1016cm3

Thus, wscr is:

wscr=28.854×1014C/(Vcm)12.931.207V1.6022×1019C1.0×1017cm3+1.0×1016cm31.0×1017cm31.0×1016cm3=4.356×105cm=435.6nm

From the equation (2.4.4), you can see that the higher the dopant concentration is, the thinner wscr becomes.

The derivation of the equations is explained in §6 in [NelsonPSC2003].

Under applied bias

We look into the case of the diode under forward bias. Figure 2.4.66 shows animation of (a) the band profiles, (b) the electric field, and (c) the space charge, respect to the applied bias.

../../../../_images/1D_gaas_edu_pn_junction_characters.gif

Figure 2.4.66 Some characters, (a) the band profiles, (b) the electric field, and (c) the space charge, respect to the applied bias.

As you see in Figure 2.4.66, the width wscr decreases as the forward bias is applied. The width wscr under the forward bias can be represented as follows.

(2.4.5)wscr=wp+wn=2ε0εq(1Na+1Nd)(VbiV)

As the width wscr decreases, the electric field that prevents the diffusion of majority carriers also decreases.

Whereas the current density across the diode is 0 at equilirium, applied bias enables majority carriers to diffuse across the junction. This means that a net current of electrons flow from n to p, and a net current of holes from p to n.

To see the effects of applied bias more clearly, let us look at the band profiles and carrier densities at 0.5V.

../../../../_images/1D_gaas_edu_pn_junction_0.5V.svg

Figure 2.4.67 The band profiles are plotted in (a). The carrier densities are plotted in (b). The hole density is shown in violet, whereas the electron density is in green.

The results are consistent with Fig. 6.6. in [NelsonPSC2003] with high accuracy. The built-in-potential is reduced to VbiV=1.2070.5=0.707V. Here, the difference between the quasi Fermi levels within the space charge region is equivalent to qV.

Thus,

(2.4.6)qV=EFnEFp

This relation can be seen from Figure 2.4.67 (a).

J-V curve

In this section, we sweep forward bias to acquire J-V curve. You can refer to — DEV — I–V characteristic of GaAs p–n junction | 1D/2D/3D to understand how to apply bias in nextnano++.

Figure 2.4.68 shows the J-V curve of the diode. IV_characteristics.dat is used to produce this figure.

../../../../_images/1D_gaas_edu_pn_junction_IV_curve.svg

Figure 2.4.68 J-V curve of the diode. (i) space charge recombination current region, (ii) diffusion current region, (iii) high-injection region, (iv) series-resistence effect region.

The light-blue curve shows the numerical result in nextnano++. The violet and orange dashed-dotted curves are acquired analytically. They correspond to Jscr and Jdiff in Fig. 6.7. in [NelsonPSC2003], respectively.

Jscr is called the recombination current density and expressed in the following equation:

(2.4.7)Jscr(V)=Jscr,0(exp(qV/2kBT)1),

where

(2.4.8)Jscr,0=qni(wp+wn)τnτp

Jdiff is called the diffusion current density and expressed in the following equation:

(2.4.9)Jdiff(V)=Jdiff,0(exp(qV/kBT)1),

where

(2.4.10)Jdiff,0=qni2(DnNaLn+DpNdLp)

The parameters used in the expressions above are in the table.

Parameters

Description (unit)

Value used for the analytical J-V curve

kB

Boltzmann constant (J/K)

1.3806E-23

T

The temperature (K)

300

ni

The intrinsic carrier density (cm-3)

2.318E+6

τn/p

The lifetimes of electrons/holes (s)

3.333 × 10-9 for Jscr and 1.0 × 10-10 for Jdiff (*)

Dn/p

The diffusion coefficients of electrons/holes (cm2/Vs)

219.73 / 20.681

Ln/p

The diffusion lengths of electrons/holes (cm)

1.4823 × 10-4 / 4.5476 × 10-5

Attention

(*) There seems to be some errors related to the units in Fig. 6.7. in [NelsonPSC2003]

Therefore we used the lifetimes as fittig parameters.

The derivation of those equations above are described in §6 in [NelsonPSC2003]. Ja is the sum of Jscr and Jdiff (Ja=Jscr+Jdiff). Our result (the light-blue curve) is in a good agreement with Ja until V1.2(V).

Our result shows the four distinct regions as marked Figure 2.4.68 (region (i), (ii), (iii), (iv)). In the next section, we identify the origins of the appearance of the regions.

Recombination current region

The region (i) is attributed to the recombination current region, where the contribution of Jscr is dominant. In this region, electrons and holes recombine within the space charge region since the region still exists. Therefore, the recombination current flows to compensate externally for the disappearance of the recombined carriers. As you can see from (2.4.7), in the semi-log plot log(J)vsV, the slope in the region (i) is qV/2kBT.

Diffusion current region

The region (ii) is the diffusion current region. The contribution of Jdiff is large in this region. Since the space change region almost disappears, a large amount of carriers starts to diffuse. This means that electrons are injected into p-doped GaAs and holes are injected into n-doped GaAs (minority carriers injection). As you can see from (2.4.9), in the semi-log plot log(J)vsV, the slope in the region (ii) qV/kBT.

High-injection region

With increasing the forward bias towards Vbi, the injected hole density becomes comparable to the electron density at the n-side of the junction. You can see it in Figure 2.4.69 (b), where 1.2V is applied to the diode.

../../../../_images/1D_gaas_edu_pn_junction_1.2V.svg

Figure 2.4.69 The band profiles are plotted in (a). The carrier densities are plotted in (b). The hole density is shown in violet, whereas the electron density is in green.

Then, the electron density must increase to maintain the neutrality. As a result, np holds.

Because of the law of the junction,

(2.4.11)np=ni2exp(qV/kBT),

we acquire the equation as follows.

(2.4.12)n=p=niexp(qV/2kBT)

Therefore, the current density becomes roughly proportional to exp(qV/2kBT).

Series-resistance effect

At large currents, the voltage drop outside the space charge region becomes too large to ignore. This is equivalent to considering a single resistance (R) added in series to the ideal diode and corresponds to the region (iv). In this region, the diffusion current density becomes proportional to the applied voltage to the diode (V).

(2.4.13)Jdiff,region(iv)Jdiff,0qVkBT,

where

(2.4.14)V=VIR

Numerical control

Since we solve the current equation and the poisson equation (explanation: Optoelectronic characterization) self-consistently, we need some techniques to make the calculations more stable.

In this section, we introduce the effects of minimum_density_electrons, minimum_density_holes, maximum_density_electrons, and maximum_density_holes.

You should also check Convergence.

In Figure 2.4.68, we divide the simulation scheme into 3, depending on the magnitudes of minimum and maximum carrier densities (scheme (A), (B), and (C)). Scheme (A): 00.4V Scheme (B): 0.40.7V Scheme (C): 0.71.5V

First, the code below defines the magnitudes of the minimum and maximum carrier densities. Note that we use the variables $min_density and $max_density for convenience.

178currents{
179    minimum_density_electrons  = $min_density
180    minimum_density_holes      = $min_density
181    maximum_density_electrons  = $max_density
182    maximum_density_holes      = $max_density
183
184}

Usually, you can set the values of $min_density_* and $max_density_* by referring to bias_XXXXX\density_electon.dat and bias_XXXXX\density_hole.dat. In the scheme (C), the maximum electron and hole densities are about 1.0×1018(cm3). Therefore, it is set to 1.0E+20. Similarly, you can set $minimum_density_*. Since the minimum electron and hole densities are about 1.0×100(cm3), $minimum_density_electrons = 1.0E-2 and $minimum_density_holes = 1.0E-2 are enough low to evaluate the current density accurately. Note that x=0(nm) and x=3000(nm) correspond to the positions of the interfaces of diode/contact. Therefore, we do not include the carrier densities at the positions into the procedures.

In the scheme (B), $minimum_density_electrons = 1.0E-2 and $minimum_density_holes = 1.0E-2 is enough low as well. However, you have to take care of the magnitude of $minimum_density_*.

Figure 2.4.70 (a) shows the effect of the magnitude of $minimum_density_* on the current density under 0.5V in the scheme (B).

../../../../_images/1D_gaas_edu_pn_junction_numerical_control.svg

Figure 2.4.70 The effect of the magnitude of $minimum_density_* on the current density. (a) is under 0.5V in the scheme (B). (b) is under 0.1V in the scheme (A).

Although the current density has to be constant through the diode, it becomes unstable at minimum_density_* set to 1.0E+16 and minimum_density*_ set to 1.0E+20. Thus, you should set $minimum_density_* to 1.0E+15, which shows the constant current density.

In the scheme (A), the same techniques should be applied. Figure 2.4.70 (b) shows the effect of the magnitude of $maximum_density_* on the current density under 0.1V in the scheme (A). As you can see, maximum_density_* should be set to 1.0E+12 to keep the current density constant through the diode.

Exercises

under construction


Last update: 16/07/2024