Aim :

1. Measurement of resistivity of a semiconductor at room temperature.

2. Measurement of variation of resistivity with temperature.

3. Evaluation of band gap of the given semiconductor from the plotting of acquired data.

4. Understanding of the concept of four probe method.Determining the value of specific charge e/m of an electron by Thomson Method

Theory :

Semiconductor

Semiconductor is a very important class of materials because of wide applications in this modern world. The following are the properties which gives a rough description of a semiconductor.

1. The electrical conductivity of a semiconductor is generally intermediate in magnitude between that of a conductor and an insulator. That means conductivity is roughly in the range of 10(3) to 10(−8) siemens per centimeter.

2. The electrical conductivity of a semiconductor varies widely with doping concentration, temperature and carrier injection.

3.Semiconductors have two types of charge carriers, electrons and holes.

4. Unlike metals, the number of charge carriers in semiconductors largely varies with temperature.

5. Generally, in case of semiconductor, increase of temperature enhances conductivity while in case of metals increase of temperature reduces conductivity.

6. The semiconductor can be best understood in the light of energy band model of solid.

Figure 4.1: Band gaps shown for (a) Insulator (b) Alkali Metal (c) Other Metal (d) Gesemiconductor

Energy band structure of solid

Atom has discrete energy levels. When atoms are arranged in a periodic arrangement in a solid the relatively outer shell electrons no longer remain in a specific discrete energy level. Rather they form a continuous energy level, called energy band. In case of semiconductor and insulator, at temperature 0K all the energy levels up to a certain energy band, called valence band, are completely filled with electrons, while next upper band (called conduction band) remains completely empty. The gap between bottom of the conduction band and top of the valence band is called fundamental energy band gap (Eg), which is a forbidden gap for electronic energy states. In case of metals, valence band is either partially occupied by electrons or valence band has an overlap with conduction band, as shown in Fig. 4.1(b and c).

In case of semiconductor, the band gap (~0 − 4eV ) is such that electrons can move from valence band to conduction band by absorbing thermal energy. When electron moves from valence band (VB) to conduction band (CB), it leaves behind a vacant state in valence band, called hole. When electric field is applied, movement of large number of electrons in the valence band can be visualized by the movement of hole as a positive charge particle. The Eg is a very important characteristic property of semiconductor which dictates it’s electrical, optical and optoelectrical properties. There are two main types of

Figure 4.2: Energy band diagram of a semiconductor

semiconductor materials: intrinsic and extrinsic. Intrinsic semiconductor doesn’t contain impurity. Extrinsic semiconductors are doped with impurities. These discrete impurity energy levels lie in the forbidden gap. In p-type semiconductor, acceptor impurity, which can accept an electron, lies close to the valence band and in n-type semiconductor, donor

Figure 4.3: Temperature variation of carrier concentration

impurity, which can donate an electron lies close to conduction band.

Temperature variation of carrier concentration

Fig. 4.3 shows the variation of carrier concentration (concentration of holes) in a ptype semiconductor with respect to 1000/T , where T is the temperature. Initially as temperature increases from 0K (i.e. ionization region), the discrete impurity vacant states starts getting filled up from valence band, which creates holes in valence band. Beyond a certain temperature all the impurity states will be filled up with electrons, which lead to the saturation region.

As temperature increases to further higher values, electrons, in the valence band, get sufficient energy to occupy empty states of conduction band (C.B). This region is called intrinsic region. The temperature above which the semiconductor behaves like intrinsic semiconductor is called “Intrinsic temperature”.

Conductivity of a semiconductor

The conductivity of a semiconductor is given by

(4.1)

Where μn and μp refer to the mobilities of the electrons and holes, and n and p refer to the density of electrons and holes, respectively. The mobility is drift velocity per electric field applied across the material, μ = Vd/E. Mobility of a charge carrier can get affected by different scattering processes.

Effects of temperature on conductivity of a semiconductor

In the semiconductor, both mobility and carrier concentration are temperature dependent. So conductivity as a function of temperature can be expressed by

(4.2)

One interesting special case is when temperature is above intrinsic temperature where mobility is dominated by only lattice scattering (/ T−3/2). That means in this temperature region mobility decreases with increase of temperature as T−3/2 due to increase of thermal vibration of atoms in a solid.

In the intrinsic region, n ~ p ~ ni, where ni is the intrinsic carrier concentration. The intrinsic carrier concentration is given by

(4.3)

where, m(n)* and m(p)* are effective mass of electron and hole. Here the exponential temperature dependence dominates ni(T). The conductivity can easily be shown to vary with temperature as

(4.4)

In this case, conductivity depends only on the semiconductor band gap and the temperature. In this temperature range, plot of ln σ vs 1000/T is a straight line. From the slope of the straight line, the band gap (Eg) can be determined. The procedure of measurement of conductivity is given below.

Four probe technique

Four probe technique is generally used for the measurement of resistivity of semiconductor sample. Before we introduce four probe technique, it is important to know two probe techniques by which you measured resistivity of a nicrome wire. In two probe technique, two probes (wires) are connected to a sample to supply constant current and measure voltage. In the case of nicrome wire (1st experiment), connections are made by pressing the multimeter probes on nichrome wire. The contact between metal to metal probe of multimeter does not create appreciable contact resistance. But in the case of semiconductor the metal – semiconductor contact gives rise to high contact resistance. If two probe configuration is followed for semiconductor sample, voltmeter measures the potential drop across the resistance of the sample as well as the contact resistance. This is shown in the Fig. 4.4(a).

The potential drop across high contact resistance can be avoided by using four probe technique. In the four probe configuration, two outer probes are used to supply current and two inner probes are used to measure potential difference. When a digital voltmeter with very high impedance is connected to the inner two probes, almost no current goes through the voltmeter. So, the potential drop it measures, is only the potential drop across the sample resistance. This is shown in the equivalent circuit diagram given in Fig. 4.4(b). From the measurement of current supplied and voltage drop across the sample, the resistance can be found out. Resistivity of a sample is given by  = cV/I, where c is a constant

For the specific arrangement, where the probes are equispaced with the distance between two successive probes as a, and the thickness of the sample is h, the resistivity can be calculated by the following formulas.

CASE-1: h >> a. In this case it is assumed that the four probes are far from the edge of the sample and the sample is placed on an insulating material to avoid leakage current. The resistivity in this case is given by

(4.5)

This is the setup used for our experiment.

CASE-2: h << a. In this case the resistivity is given by

(4.6)

Derivation for this is given at the end.

Figure 4.4: (a) Equivalent circuit for two probe measurement. R1,R2 are the contact resistances (b) Equivalent circuit for four probe measurement. R1,R2 and R3,R4 are the contact resistances of current and voltage probes.

Once resistivity (ρ) is determined, conductivity (σ) can be calculated by taking reciprocal of it (σ = 1/ρ).

Advantages of using four probe method

• The key advantage of four-terminal sensing is that the separation of current and voltage electrodes eliminates the impedance contribution of the wiring and contact resistances.
• If the probes are separated by equal distance a, and a << h then resistivity can be found out without knowing the exact shape and size of the sample.

Figure 4.5: Pictorial representations of field lines generated by the applied potential.

Description of the experimental set-up

Probes arrangement It has four individually spring loaded probes. The probes are collinear and equally spaced. The probes are mounted in a teflon bush, which ensure a good electrical insulation between the probes. A teflon spacer near the tips is also provided to keep the probes at equal distance. The whole arrangement is mounted on a suitable stand and leads are provided for the voltage measurement.

Sample Germanium crystal in the form of a chip.

Oven It is a small oven for the variation of temperature of the crystal from the room temperature to about 200oC (max).

Figure 4.6: Four probe experimental setup.