**2. p-n junction capacitance**

When voltage across a p-n junction is changed, there is a corresponding change in the depletion region width. This change in width causes a change in the number of free charge carriers on both sides of the junction, resulting in a change in the capacitance. This change has two contributions; a) the contribution due to change in depletion width known as the junction capacitance and b) the contribution due to change in minority carrier concentration called the diffusion capacitance. Junction capacitance is dominant under reverse biased conditions while diffusion capacitance is dominant under forward biased conditions.

Consider a p-n junction with a deep level present having its energy as *ET*. In steady state there is no net flow of charge carriers across the trap. Also the electron and hole densities within the depletion region are negligible. Thus from Shockley and Reed [11] and Hall [12] the relation‐ ship between the total density of deep sates NT and density of filled traps is given by

$$(e\_p n\_T = (e\_n + e\_p) N\_T \tag{1}$$

where ep is the hole emission rate, en is the electron emission rate, nT is the density of filled traps, and NT is the total density of deep sates.

or

$$m\_T = \left(\frac{e\_p}{e\_n + e\_p}\right) \mathbf{N}\_T \tag{2}$$

which gives the density of filled traps nT under steady state conditions.

Now if the system is perturbed, this number changes and will thus cause the total charge in the depletion region to increase or decrease, leading to a corresponding change in the capac‐ itance. This change is only due to deep levels.

For a simple analysis of the response of the diode and interpretation of results, the junction is assumed to be asymmetric. An asymmetric diode is one in which one side of the junction is much more heavily doped than the other, implying that the space charge region is almost exclusively on the low doped side as shown in Figure. 1(a). Here we will consider a p+ n diode with an electron emitting level. The depletion region is thus on the n-side. Figure 1 (b)

information about an impurity level in the depletion region by observation of the capacitance transient originating from the return to thermal equilibrium after a perturbation is applied to the system. A brief description of the capacitance change due to the change in occupancy of

When voltage across a p-n junction is changed, there is a corresponding change in the depletion region width. This change in width causes a change in the number of free charge carriers on both sides of the junction, resulting in a change in the capacitance. This change has two contributions; a) the contribution due to change in depletion width known as the junction capacitance and b) the contribution due to change in minority carrier concentration called the diffusion capacitance. Junction capacitance is dominant under reverse biased conditions while

Consider a p-n junction with a deep level present having its energy as *ET*. In steady state there is no net flow of charge carriers across the trap. Also the electron and hole densities within the depletion region are negligible. Thus from Shockley and Reed [11] and Hall [12] the relation‐

where ep is the hole emission rate, en is the electron emission rate, nT is the density of filled

*p T T n p*

Now if the system is perturbed, this number changes and will thus cause the total charge in the depletion region to increase or decrease, leading to a corresponding change in the capac‐

For a simple analysis of the response of the diode and interpretation of results, the junction is assumed to be asymmetric. An asymmetric diode is one in which one side of the junction is much more heavily doped than the other, implying that the space charge region is almost

with an electron emitting level. The depletion region is thus on the n-side. Figure 1 (b)

exclusively on the low doped side as shown in Figure. 1(a). Here we will consider a p+

*e n N e e* æ ö = ç ÷ ç ÷ <sup>+</sup> è ø

which gives the density of filled traps nT under steady state conditions.

*en e e N pT n p T* = + ( ) (1)

(2)

n diode

**gram of a p<sup>+</sup>**

ship between the total density of deep sates NT and density of filled traps is given by

the deep levels in the depletion region is given below.

diffusion capacitance is dominant under forward biased conditions.

traps, and NT is the total density of deep sates.

itance. This change is only due to deep levels.

or

**2. p-n junction capacitance**

200 Solar Cells - New Approaches and Reviews

Figure 1. Basic concept of thermal emission from a deep level and capacitance transient (a) energy band diagram of a p+ n junction with an electron trap present at energy ET at zero applied bias and at steady reverse bias VR, (b) isothermal capacitance transient for thermal emission of the majority carrier traps. The condition for the trap occupation and free carrier concentration during various phases 1-4 of the transient are also shown. The part not shaded in the insets 1-4 **Figure 1.** Basic concept of thermal emission from a deep level and capacitance transient (a) energy band diagram of a p + n junction with an electron trap present at energy ET at zero applied bias and at steady reverse bias VR, (b) isothermal capacitance transient for thermal emission of the majority carrier traps. The condition for the trap occupation and free carrier concentration during various phases 1-4 of the transient are also shown. The part not shaded in the insets 1-4 shows space charge width (after Lang [1]).

schematically shows four processes of generating a capacitance transient due to majority carrier level.

Process (1) shows the diode in the quiescent reverse bias condition. Traps in the space charge region are empty because no free carriers are available for capture (t<0). Process (2): reverse **Figure 1. Basic concept of thermal emission from a deep level and capacitance transient (a) energy band dia-**

**VR, (b) isothermal capacitance transient for thermal emission of the majority carrier traps. The condition for the trap occupation and free carrier concentration during various phases 1-4 of the transient are also shown. The** 

Now if the system is perturbed, this number changes and will thus cause the total charge in the depletion region to increase or decrease, leading to a corresponding change in the capacitance. This change is only due to deep levels. For a simple analysis of the response of the diode and interpretation of results, the junction is assumed to be asymmetric. An asymmetric diode is one in which one side of the junction is much more heavily doped than the other,

**part not shaded in the insets 1-4 shows space charge width (after Lang [1]).**

shows space charge width (after Lang [1]).

**n junction with an electron trap present at energy ET at zero applied bias and at steady reverse bias** 

bias is reduced to zero by a majority carrier pulse. The electrons are capture in the deep levels (t=0). The sharp raise in capacitance is due to the collapse of the depletion region. Process (3): when the reverse bias is restored, the capacitance drops to a minimum value because the electrons are trap in the depletion layer (t=0+ ). Process (4): decay of the transient due to thermal emission of the trapped electrons (t>0).

Suppose we have a reverse bias *VR* applied to the sample and it is decreased for a short time to zero. Then electrons will flow into what was previously the depletion region and the levels in this volume will capture electrons (Figure. 1[a]). Neglecting the re-emission of electrons (i.e. temperature is low enough) we get:

$$\frac{d\mathbf{n}\_T}{dt} = \mathbf{c}\_n \left(\mathbf{N}\_T - \mathbf{n}\_T\right) \tag{3}$$

where cn is the capture time constant of electrons

Now if bias pulse is long enough i.e. *tp* > > 1 *cn* , all levels will be filled and *nT* = *NT*. Next the sample is returned to quiescent reverse bias *VR* and thus the depletion region is again depleted of free carriers. The electron emitting traps now start to emit and *nT* will vary with time. This variation is given by with n= p= 0 i.e.

$$\frac{dn\_T}{dt} = e\_p N\_T - \left(e\_n + e\_p\right) n\_T \tag{4}$$

The solution to this is given by

$$\begin{aligned} m\_T(t) &= \left[ \frac{e\_p}{e\_u + e\_p} N\_T + \frac{e\_u}{e\_u + e\_p} N\_T \exp\left(-\left(e\_u + e\_p\right)t\right) \right] \text{for } t > 0\\ \mathbf{N}\_T &\text{ for } t < 0 \end{aligned} \tag{5}$$

Thus nT decreases exponentially with a time constant:

$$\pi = 1/\left(\mathbf{e}\_n + \mathbf{e}\_p\right) \tag{6}$$

Now for an electron emitting centre en » ep. Equ. (5) then reduces to:

$$m\_T(t) = N\_T \exp\left(-e\_n t\right) \tag{7}$$

Thus the amplitude of the transient describing the filled level population gives the measure of trap concentration, while the time constant gives the emission rate of electrons:

Deep Level Transient Spectroscopy: A Powerful Experimental Technique for Understanding the Physics… http://dx.doi.org/10.5772/59419 203

$$
\pi = 1/\mathbf{e}\_n \tag{8}
$$

The variation of occupancy with time gives information about the emission rate but it is not possible to measure the occupancy directly. The simplest indirect method is to measure the capacitance changes of the p-n junction associated with the occupancy changes.

The equation governing the capacitance of a p-n junction is the same as that of a parallel plate capacitor i.e.

$$\mathbf{C} = \frac{\varepsilon A}{W} \tag{9}$$

where

bias is reduced to zero by a majority carrier pulse. The electrons are capture in the deep levels (t=0). The sharp raise in capacitance is due to the collapse of the depletion region. Process (3): when the reverse bias is restored, the capacitance drops to a minimum value because the

Suppose we have a reverse bias *VR* applied to the sample and it is decreased for a short time to zero. Then electrons will flow into what was previously the depletion region and the levels in this volume will capture electrons (Figure. 1[a]). Neglecting the re-emission of electrons (i.e.

> ( ) *<sup>T</sup> nT T dn cN n*

> > > > 1 *cn*

( ) *<sup>T</sup> pT n p T dn eN e e n*

é ù = ê ú + - + + + ë û

Thus the amplitude of the transient describing the filled level population gives the measure

of trap concentration, while the time constant gives the emission rate of electrons:

*dt*

( exp

Now for an electron emitting centre en » ep. Equ. (5) then reduces to:

*ee ee*

*T T T np np np*

*<sup>e</sup> <sup>e</sup> nt N N e e t*

) *<sup>p</sup> <sup>n</sup>*

sample is returned to quiescent reverse bias *VR* and thus the depletion region is again depleted of free carriers. The electron emitting traps now start to emit and *nT* will vary with time. This

). Process (4): decay of the transient due to thermal

*dt* = - (3)

= -+ (4)

*t*

τ = 1/ e + e ( n p ) (6)

*n t N et TT n* ( ) exp = -( ) (7)

>

(5)

( ( ) ) for 0

, all levels will be filled and *nT* = *NT*. Next the

electrons are trap in the depletion layer (t=0+

where cn is the capture time constant of electrons

Now if bias pulse is long enough i.e. *tp*

variation is given by with n= p= 0 i.e.

The solution to this is given by

=

for

*tN*

*T*

0

<

Thus nT decreases exponentially with a time constant:

emission of the trapped electrons (t>0).

202 Solar Cells - New Approaches and Reviews

temperature is low enough) we get:

$$\mathcal{W}^2 = \frac{2\varepsilon (V\_b - V)(N\_D + N\_A)}{qN\_D N\_A} \tag{10}$$

ε is the relative permittivity of the material, Vb is the built-in voltage, V is the applied voltage, q is the electronic charge, ND and NA are the donor and acceptor concentrations, W is the depletion region width and A is the junction area.

For a p<sup>+</sup> n junction, including the contribution of the filled traps in the depletion region, this becomes:

$$\mathcal{W}^2 = \frac{2\varepsilon \left(V\_b - V\right)}{qN\_D^\*}\tag{11}$$

where *ND* \* <sup>=</sup> *ND* <sup>−</sup>*nT*

Now for nT « ND one can expand and get the following result:

$$\mathbf{C} = \mathbf{C}\_0 \left( \mathbf{1} - \frac{n\_T}{2N\_D} \right) \tag{12}$$

where C0 is the capacitance at reverse bias (VR).

By taking into consideration the time variation of nT, we get:

$$\mathbf{C}(t) = \mathbf{C}\_0 \left[ 1 - \frac{N\_T}{2N\_D} \exp\left(-\frac{t}{\tau}\right) \right] \tag{13}$$

Thus the emission rates and trap concentrations can be determined from the changes in the capacitance of a p-n junction due to bias pulses. These changes are in the form of capacitance transients.

d. Cature cross-sections

**obtained from the peak positions ( [31].** 

**Figure 2. Diagram illustrating the basic principles of DLTS (a) the rate window concept (after Lange [1]), (b) application of the rate window concept using a time filter such as dual-gate box car shown here (after Lange [1]), (c) showing the shift Figure 2.** Diagram illustrating the basic principles of DLTS (a) the rate window concept (after Lange [1]), (b) applica‐ tion of the rate window concept using a time filter such as dual-gate box car shown here (after Lange [1]), (c) showing the shift of the peak positions in temperature with the rate window and the Arrhenius plot obtained from the peak positions ([31].

**of the peak positions in temperature with the rate window and the Arrhenius plot** 
