Vector Operations

A trap is an impurity atom or imperfection in the crystal. Capable of capturing an electron or hole. In simple words, it is an energy level, that can capture either electrons or holes.

There are two types of traps.

(i) One type of traps helps electrons and holes to recombine and thereby assists in restoration of thermal equilibrium. These type of traps are called recombination centers. In the presence of traps, recombination proceeds at much higher rate.

(ii) Second type of traps does not contribute directly in recombination process, but restricts the freedom of motion of holes or electrons. This is shown in Figure 11.19.

FIGURE 11.19 Influence of traps in electron hole recombination

To study the effect of second types of traps on photoconductivity, let us consider a crystal with N electron trap levels per unit volume. Rate of change of electron concentration ‘n’ is given by,

dn /dt – =L -A n (n + N) + Bn t

where recombination coefficient A is same for both electron-hole recombination and electron trap capture. The term Bn1 represent the rate of thermal evaporation of trapped carrier back into conduction band and is neglected.

In steady state dn / dt = 0

Therefore from equation (1), we get

L – A n (n + N) = 0

or                                    L /A = n0 (n0 N)

Limiting Cases

(i) If illumination level L is low, n0 will be small. So n0 << N and hence n0 can be neglected in comparison to N. Therefore equation (2 ) becomes

L / A = n0 N   or  n0 = L /AN

Therefore photocurrent density will be

J = σ E = n0 e µ E

J = L /AN e µ E

which shows that photocurrent is directly proportional to illumination level (L).

(ii) If illumination level L is high, n0 will be high. So n0 >> N. Hence equation (2) becomes

L /A = n02 or n0 =(L/A)1/2

which is the same steady state current carrier concentration in the absence of traps.

When light incident on the crystal is switched off, L = 0. Therefore equation (1), we get

dn  /dt = – A n (n +N)

dn / n (N+n) = – A dt

Integrating both sides, we get

Therefore response time in the absence of traps is t0 =1 / NA in which carrier concentration reduces to 2 of its steady state. e

Equation (7) shows the response time is reduced by presence of traps.

Further  σ = n0 e µ

From equation (3) and (8), we get

σ = L / NA e µ

which shows presence of traps reduced the photoconductivity.

11 .7 .3 Photoconduction in Semiconductors

The conductivity of a semiconductor increases when the radiation is made incident on it. This phenomenon is called photoconductive effect. This may be explained as follows:

The conductivity (cr) of a semiconductor increases with increase of concentration ofcharge carriers as is clear from the relation

σ = (ne µn + nh µh)e

where ne = free electron concentration per m3

nh = hole concentration per m3

µn = free electron mobility

µh = hole mobility

e = elementary charge = 1.6 x l0-19 C.

When the radiant energy falls on a semiconductor, the covalent bonds get broken and electron-hole pair in excess of those generated thermally are created. These increased charge carriers decrease the resistance or increase the conductivity of the material and hence such a device is called a photoresistor or a photoconductor.

Figure 11.20 represents the energy level diagram of semiconductor having both acceptor and donor impurities (i.e., of PN junction). If the photon of light is allowed to fall on the specimen, the following transitions are possible

(i) An electron may jump from valence band Ea  Acceptor to conduction band, thus creating an electron-hole  level pair. This process is called intrinsic excitation.

(ii) A donor electron may jump into conduction band or a valence electron may jump into an acceptor state. These two transitions are called the impurity semiconductor excitations.

As the density of states in the conduction and valence band is much greater than the density of impurity states, therefore the photoconductivity is mainly due to intrinsic excitation.

Spectral Response: The minimum energy of a photon required for intrinsic excitation is the forbidden gap energy (Eg) of the semiconductor material The long wavelength threshold of the material is defined as λc corresponding to the energy gap Eg and is given by

If Eg is expressed in eV and ‘ λc, in  µ we get

λc =6.62 x 10-34 x 3 x108 /Eg x 1.6 x 10-19

= 1.24 /Eg x 10-6 m

= 1.24 /Eg µm

For silicon Eg = 1.1 eV, therefore ‘A, = 1.13 )lm at room temperature. For germanium

Eg = 0.72 eV, therefore \ = 1.73 JD at room temperature.

Figure 11.21 represents the spectral sensitivity curves for Si and Ge. It is noticed that the long wavelength limit is slightly greater than the calculated values. This is due to impurity excitations. As the wavelength is decreased from value ‘A,, the response increases, becomes maximum and again decreases.

The Photoelectric Current: If potential difference is applied across the semiconductor bar and light is allowed to fall on it, the charge carriers are generated and those move under the influence of applied potential difference. The· charge carriers, which do not undergo recombination, reach the conducting contacts at the ends of the bar and hence constitute electric current. This current is given by

where P r = rate of charge carriers produced by light

‘t = average life time of newly created carriers

Ti = average transit time for carriers to reach the conducting contacts

e = charge On each Carrier = 1.6 x 10-19 C.

11.7 .4 Photoconductive Cells

The devices with the help of which light energy is converted to electrical energy are called photoelectric cells. Three types of photoelectric cells are: (t) photoemissive cells (ii) photo-voltaic cell and (iii) photoconductive cells.

Principle of Photoconductive Cells: It is based upon the principle that electrical resistance of semiconductors materials like selenium, lead sulphide etc. decreases when they are exposed to light radiations. The amount of this decrease in resistance (or increase in photoconductivity) depend on the light intensity and frequency of the incident light. So, if such a substance is inserted in the circuit and light is allowed to fall on it, its electric resistance (or photoconductivity) will change consequently, there will be a change of current in the circuit in steps accordingly to the variation of incident light intensity. The decrease in resistance (or increase in photoconductivity) of semiconductors on exposing to light may be explained as below:

If a photon striking the surface of such photosensitive material has energy E (= h v) greater than the energy gap between the valence and conduction bands of the material, sufficient energy will imparted to an electron to raise it to the conduction level. Consequently, a hole is left in the valence band. This electron-hole pair is free to serve as current carriers and hence, the conductivity of the material increases or the electrical resistance is reduced.

Construction and Working: The material generally used in photoconductive cells are cadmium sulphide (CdS), Cadmium selenide (CdSe) or lead sulphide (PbS). Commonly used material is CdS. The construction of one form of CdS cell and its circuit symbol is shown in Figure 11.22.

In this cell, a thin film of CdS is deposited on one side of an iron plate and placed below a transparent film of metal. When light radiation of Iron plate sufficient energy falls on transparent metal film, FIGURE 1 1.22 CdS cell and resistance of CdS layel get reduces and hence its its symbol conductance is increased. Consequently, a current starts flowing in the battery circuit connected between the iron plate and the transparent metal film. This external battery is included in the circuit to generate a direction and provide a path for the current flow. Voltages varying from a few volts to several hundred volts and applied depending upon the application of photoconducting cells.

Characteristics and Spectral Response: The following Figure 11.23 shows the illumination characteristics of a CdS cell. It depicts the .relationship between illumination and resistance. It may be seen that when non-illuminated (i.e., in absolute darkness), cells has a resistance in the ranges of 100 kQ (very high) which is known as dark resistance. When illuminated with strong light, the cell resistance falls to only a few hundred ohms. So, the ratio of dark to light resistance of cell is about 1000 : 1.

The spectral response of CdS cell is shown in Figure 11.23 (b). It matches response of human eye. Like human eye, response is sensitive to visible.

(B) Vectorial representation of moments. The moment of a force is a vector which is the product of distance and force. Hence in case of moment* of a force the cross-product of distance and force would be taken. Consider the Fig. 1.16.

Let                   F =Force vector (F)+ F y j+ F z k)

r =Distance (or position) vector with respect to 0

=xi+ y j + z k

M = Moment of force about point 0

M = (y F z – z F) i + (z F x- x F z) j + (x F Y – y F r) k

The moment of the given force about x, y and z-axis are equal to

M x = y f a – z f y , m y = z f x – x f z , m z = x f y – y f x

M x “‘ Moment of F about x-axis

MY= Moment of F about y-axis, and

M = Moment of F about z-axis.

Problem l.10.Aforce F =2i + 4j – 3k is applied at a point P(l, 1, -2). Find the moment of the force F about the point (2, -1, 2).

Sol. Given:

Force F = 2i + 4j – 3k / p r

The position vector r of the point P w.r.t . 0.

= Position vector of point P

– Position vector of point 0 .

= (i + j – 2k) – (2i – j + 2k) 0 (2, -1, 2)

r = (1 – 2)i + [1 + (1)] j + I- 2 – 2]k = – i + 2j – 4k

The moment M is given by