ECSE-6230 Semiconductor Devices and Models I Lecture 6

1. ECSE-6230Semiconductor Devices and Models ILecture 6 Prof. Shayla Sawyer Bldg. CII, Rooms 8225 Rensselaer Polytechnic Institute Troy, NY 12180-3590 Tel. (518)276-2164 Fax. (518)276-2990 e-mail: sawyes@rpi.edu March 13, 2014 sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html 1

2. Carrier Motion Define carrier motion of the free electrons Without external fields With low to moderate fields With high fields Net motion and Drift Mobility Conductivity and Resistance In a semiconductor do carriers move without an electric field applied? sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

3. Carrier Transport Basically, carrier transport can be classified into two ways: Drift - Motion under an applied electromagnetic field Diffusion - Motion due to a concentration gradient Most of the transport mechanisms considered here, unless noted, are formulated classically (not quantum mechanically). Often, the drift + diffusion is adequate for many situations and applications. However, in some advanced cases (e.g., submicron MOSFET’s and quantum well devices), more advanced treatment on carrier transport (e.g., Boltzmann Transport Equation) is necessary. sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

4. Thermal Velocity Carriers are almost free in that they are not associated with any particular lattice site (crystal forces in effective mass) From classical thermal physics (Dulong-Petit Law), or  107 cm/s in Si where vth is the thermal velocity, which is the average velocity of carriers due to thermal excitation. sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

5. Net Motion If an electric field, ξx, is applied in the x-direction, each electron experiences a net force -q ξx from the field. This force may be insufficient to alter appreciably the random path of an individual electron…there is a net motion of the group in the x-direction. The force of the field on n electrons/cm3 is: (px, momentum of the group) Is this a continuous acceleration of electrons in the –x direction?

6. Net Motion It is not continuous, there are collisions The net acceleration is balanced in steady-state by the deceleration of the collision process There is net momentum p-x but the net rate of change of momentum when collisions are included must be zero in the case of steady current flow For collisions, consider a group of N0 electrons at time t=0 Define N(t) as the number of electrons that have not undergone a collision by time t The rate of decrease in N(t) at any time t is proportional to the number left unscattered at t,

7. Net Motion The solution to this equation is an exponential function (typical of events dominated by random processes) Thus the rate of change of px due to the decelerating effect of collisions is What is τm?

8. Mobility τm mean free time, time interval between collisions Mobility describes how easily an electron moves in response to an applied field What are the two basic types of scattering mechanisms that hinder mobility?

9. Mobility and Scattering Lattice and impurity scattering Lattice: vibration due to temperature Ionized impurity scattering: slow moving carrier with momentum easily affected by a charged ion Effective Mobility ~ sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

10. Net Motion Sum of acceleration and deceleration effects must be zero for steady state so: The average momentum per electron is Electrons on the average have a constant net velocity in the –x direction mn*conductivity effective mass for electrons

11. Example Calculate the conductivity effective mass of electrons in Si. For Si, ml=0.98 m0 and mt = 0.19 m0 Additional note for GaAs, the conduction band equi-energy surfaces are spherical. So there is only one band curvature effective mass. (The density of states effective mass and the conductivity effective mass are both 0.067 m0)

12. Carrier Transport - Drift Drift Small electric field is applied to the lattice When electrons collide with the lattice there is a loss of energy associated Net carrier velocity in an applied field is the drift velocity vd Electrostatic Force, Mobility describes how easily an electron moves in response to an applied field sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

13. Drift Current density flowing in the direction of the applied field can be found by summing the product of the charge on each electron times its velocity over all electrons per unit volume Analogous argument applies with holes therefore the sum of electron and hole current Term in parenthesis is the conductivityσ = n q un + p q up sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

14. Drift and Resistance If the semiconductor bar in the figure contains both types of carrier then the conductivity is given by the previous equation The resistance of the bar is then Where ρ is the resistivity Electric field Current Hole motion Electron motion Electron motion I sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

15. High Field Transport At low electric field, <vd >   The proportionality constant  that is independent of the electric field f ( ) At sufficiently high fields, <vd > is no longer prop. to  (nonlinearities in mobility) <vd >  vsat Larger fields, impact ionization occurs Why is GaAs decreasing at high fields? sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

16. High Field Transport <vd > vs.  - more complicated for GaAs Velocity overshoot Electrons at the high mobility valley at  (k = 0) have small m*. Electrons are accelerated to gain E and k, reaching beyond the L and X valleys and getting scattered into them. Average drift velocity reaches a maximum, then decreases and saturates to a low value when the L and X minima are populated. Effect is sometimes called? sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

17. Hot Carrier Effects At high electric fields, carriers get more energetic ---- “hot”! Teffective > Tlattice The average energy of the carriers increase as the field increases, acquiring the above effective temperature Te Under steady state condition with an applied field, For Si and Ge, (moderately high fields) Vd starts to deviate from being lienarly dependent on the applied field by a factor of √(T/Te) μ0 is the low field mobility cs the velocity of sound sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

18. Hot Carrier Effects At high fields the drift velocities for Ge and Si become less and less dependent on the applied filed and approach a saturation velocity. Carriers start to interact with optical phonons so previous equation is no longer accurate At high fields Ep is the optical phonon energy sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

19. Example Find the resistivity at 300K for a silicon sample doped with 1.0 x 1014 cm-3 of phosphorus atoms 8.5x 1012 cm-3 of arsenic atoms, and 1.2x1013 cm-3 of boron atoms. Assume that the impurities are completely ionized and the mobilities are μn = 1500 cm2/V-s, μp = 500 cm2/V-s, independent of impurity concentrations. (ni=9.65x109 cm-3) sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html