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1. Electrical Science-2 (15B11EC211) Unit-4 Introduction to Semiconductor Lecture-4 Dr. Akansha Bansal OLED Technology Solar Cells

2. Conductivity and Mobility The charge carriers in a solid are in constant motion, even at thermal equilibrium. At room temperature, for example, the thermal motion of an individual electron may be visualized as random scattering from lattice vibrations, impurities, other electrons, and defects. Fig. 1(a) Random thermal motion of an electron in a solid [1]

3. Conductivity and Mobility If an electric field Ex is applied in the x-direction, each electron experiences a net force - qE x from the field. This force averaged over all the electrons, however, is a net motion of the group in x-direction. As a result of this electrostatic force , the electrons would be accelerated and the velocity increases indefinitely with time, however due to the inelastic collision with ions electrons loses energy and a steady state condition is reached where a finite value of drift speed v d is attained. This drift velocity is in the direction opposite to that of the electric field.   E x Field Fig. 1b Well directed drift velocity with an applied electric field .

4. Conductivity and Mobility The quantity , called the electron mobility, describes the ease with which electrons drift in the material. Mobility is a very important quantity in characterizing semiconductor materials and in device development. The mobility can be expressed as the average particle drift velocity per unit electric field. We have, The units of mobility are cm 2 /V-s  

5. Effect of Temperature on Mobility The two basic types of scattering mechanisms that influence electron and hole mobility are: Lattice scattering Impurity scattering Fig.2 Temperature dependence of mobility with both lattice and impurity scattering

6. Effect of Temperature on Mobility Lattice scattering: A carrier moving through the crystal is scattered by a vibration of the lattice, resulting from the temperature. The frequency of such scattering events increases as the temperature increases, since the thermal agitation of the lattice becomes greater. Therefore, we should expect the mobility to decrease as the sample is heated. Impurity scattering: S cattering from crystal defects such as ionized impurities becomes the dominant mechanism at low temperatures. Since the atoms of the cooler lattice are less agitated and a slowly moving carrier is likely to be scattered more strongly by an interaction with a charged ion than is a carrier with greater momentum, impurity scattering events cause a decrease in mobility with decreasing temperature. The mobilities due to two or more scattering mechanisms add inversely:  

7. Effect of Doping on Mobility As the concentration of impurities increases, the effects of impurity scattering are felt at higher temperatures. For example, the electron mobility of intrinsic silicon at 300 K is 1350 cm 2 /(V-s).With a donor doping concentration of 10 17 cm -3 , however, is 700 cm 2 /(V-s). Thus, the presence of the 10 17 ionized donors/cm 3 introduces a significant amount of impurity scattering.   Fig.3 Variation of mobility with total doping impurity concentration (N0 + Nd) for Si at 300 K.

8. High Field Effects Upper limit is reached for the carrier drift velocity in a high field and this limit occurs near the mean thermal velocity (10 7 cm/s) and represents the point at which added energy imparted by the field is transferred to the lattice rather than increasing the carrier velocity. The result of this scattering limited velocity is a fairly-constant current at high field. Fig.4 Saturation of electron drift velocity at high electric fields for Si

9. Example Example 1: How long does it take an average electron to drift 1 μ m in pure Si at an electric field of 100V/cm? Repeat for 10 5 V/cm. Assume μ n = 1350 cm 2 /V-s Solution: Low Field: = 1350 x 100=1.35 x 10 5 cm/s t = L/ = 10 -4 /1.35 x 10 5 = 0.74ns High Field: Scattering limited velocity 10 7 cm/s t = L/ = 10 -4 /10 7 = 10ps  

10. Current Density If n electrons are contained in a length L of conductor, and if it takes an electron a time t seconds to travel distance of L meter in the conductor. A = As L/t is the average, or drift speed, v m/s of the electrons. Current Density denoted by symbol J, is the current per unit area of the conducting medium. Assuming a uniform current distribution Where, J is Amperes/Square meter A- Cross-sectional area of the conductor in meters   L

11. Conductivity n=no. of electrons per cubic meter (1) Where, is the conductivity of the metal in (ohm-meter) -1 ] Eq. 1 is recognized as Ohm’s law, namely, the conduction current is proportional to applied voltage.  

12. Example Example 2: A Si bar 0.1 cm long and 100 μ m 2 in cross sectional area is doped with 10 17 cm -3 antimony. Find the current at 300K with 10V applied. Assume μ n = 700cm 2 /V-s Solution: = 1.6 x 10 19 x 700 x 10 17 = 11.2( Ω .cm ) -1 ρ = 1/ =0.0893( Ω .cm ) =  

13. Conductivity of semiconductor Semiconductor contains two types of charge carriers Negative(free electron) with mobility μ n Positive(hole) with mobility μ p These particles move in opposite direction in an electric field E, but since they are of opposite sign, the current of each is in same direction. Hence, Current density J=(n μ n + p μ p ) qE = σ E n = magnitude of free electron concentration p = magnitude of hole concentration σ = Conductivity Hence, σ = (n μ n + p μ p )q

14. Hall Effect If a magnetic field is applied perpendicular to the direction in which holes drift in a p-type bar, the path of the holes tends to be deflected. Fig.1 The Hall effect [1]

15. Hall Effect Using vector notation, the total force on a single hole due to the electric and magnetic fields is F = q(E + v × B) In the y-direction the force is F y = q(E y -v x B z ) The important result of above equation is that unless an electric field E y is established along the width of the bar, each hole will experience a net force (and therefore an acceleration) in the y-direction due to the q v x B z product. Therefore, to maintain a steady state flow of holes down the length of the bar, the electric field E y must just balance the product q v x B z

16. Hall Effect Once the electric field E y becomes as large as q v x B z , no net lateral force is experienced by the holes as they drift along the bar. The establishment of the electric field E y is known as the Hall effect, and the resulting voltage V AB = E y w is called the Hall voltage. is called the Hall Coefficient  

17. Hall Effect A measurement of the Hall voltage for a known current and magnetic field yields a value for the hole concentration p = = = If a measurement of resistance R is made, the sample resistivity ρ can be calculated ρ ( Ω -cm) = = Mobility is simply the ratio of the Hall coefficient and the resistivity = = =  

18. Example Example: Consider a semiconductor bar with w = 0.1mm, t =10 μ m, and L = 5mm. For B=10 -4 Wb/cm 2 and a current of 1mA, we have V AB = -2mV, V CD = 100mV. Find the type, concentration, and mobility of the majority carrier. Solution From the sign of V AB , we can see that the majority carriers are electrons: = =31.25 x 10 17 cm -3 ρ = = = =0.002 Ω .cm = =