Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 1 Chap 5. Carrier Motion  Carrier Drift  Carrier Diffusion  Graded Impurity Distribution  Hall Effect  Homework

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 2 Carrier Drift  When an E-field (force) applied to a semiconductor, electrons and holes will experience a net acceleration and net movement, if there are available energy states in the conduction band and valence band. The net movement of charge due to an electric field (force) is called “drift”.  Mobility: the acceleration of a hole due to an E-field is related by If we assume the effective mass and E-field are constants, the we can obtain the drift velocity of the hole by where v i is the initial velocity (e.g. thermal velocity) of the hole and t is the acceleration time.

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 3 Mobility E = 0  In semiconductors, holes/electrons are involved in collisions with ionized impurity atoms and with thermally vibration lattice atoms. As the hole accelerates in a crystal due to the E-field, the velocity/kinetic energy increases. When it collides with an atom in the crystal, it lose s most of its energy. The hole will again accelerate/gain energy until is again involved in a scattering process.

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 4 Mobility  If the mean time between collisions is denoted by  cp, then the average drift velocity between collisions is where  p (cm 2 /V-sec) is called the hole mobility which is an important parameter of the semiconductor since it describes how well a particle will move due to an E-field.  Two collision mechanisms dominate in a semiconductor: –Phonon or lattice scattering: related to the thermal motion of atoms;  L  T -3/2 –Ionized impurity scattering: coulomb interaction between the electron/hole and the ionized impurities;  I  T 3/2 /N I., : total ionized impurity conc. ,  I  If T , the thermal velocity of hole/electron  carrier spends less time in the vicinity of the impurity.  less scattering effect   I 

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 5 Mobility Electron mobilityHole mobility

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 6 Drift Current Density  If the volume charge density of holes, qp, moves at an average drift velocity v dp, the drift current density is given by J drfp = (ep) v dp = e  p pE. Similarly, the drift current density due to electrons is given by J drfn = (-en) v dp = (-en)(-  n E)=e  n nE  The total drift current density is given by J drf = e(  n n+  p p) E

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 7 Conductivity  The conductivity  of a semiconductor material is defined by J drf   E, so  = e(  n n+  p p) in units of (ohm-cm) -1  The resistivity  of a semiconductor is defined by   1/ 

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 8 Resistivity Measurement  Four-point probe measurement

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 9 Velocity Saturation  So far we assumed that mobility is indep. of E-field, that is the drift velocity is in proportion with the E-field. This holds for low E-filed. In reality, the drift velocity saturates at ~10 7 cm/sec at an E-field ~30 kV/cm. So the drift current density will also saturate and becomes indep. of the applied E-field.

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 10 Velocity Saturation of GaAs  For GaAs, the electron drift velocity reaches a peak and then decreases as the E-field increases.  negative differential mobility/resistivity, which could be used in the design of oscillators.  This could be understood by considering the E-k diagram of GaAs.

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 11 Velocity Saturation of GaAs  In the lower valley, the density of state effective mass of the electron m n * = 0.067m o. The small effective mass leads to a large mobility. As the E-field increases, the energy of the electron increases and can be scattered into the upper valley, where the density of states effective mass is 0.55m o. The large effective mass yields a smaller mobility.  The intervalley transfer mechanism results in a decreasing average drift velocity of electrons with E-field, or the negative differential mobility characteristic.

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 12 Carrier Diffusion  Diffusion is the process whereby particles flow from a region of high concentration toward a region of low concentration. The net flow of charge would result in a diffusion current.

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 13 Diffusion Current Density  The electron diffusion current density is given by J ndif = eD n dn/dx, where D n is called the electron diffusion coefficient, has units of cm 2 /s.  The hole diffusion current density is given by J pdif = -eD p dp/dx, where D p is called the hole diffusion coefficient, has units of cm 2 /s.  The total current density composed of the drift and the diffusion current density. 1-D or 3-D

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 14 Graded Impurity Distribution  In some cases, a semiconductors is not doped uniformly. If the semiconductor reaches thermal equilibrium, the Fermi level is constant through the crystal so the energy-band diagram may qualitatively look like:   Since the doping concentration decreases as x increases, there will be a diffusion of majority carrier electrons in the +x direction.  The flow of electrons leave behind positive donor ions. The separation of positive ions and negative electrons induces an E-field in +x direction to oppose the diffusion process.

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 15 Induced E-Field  The induced E-field is defined as that is, if the intrinsic Fermi level changes as a function of distance through a semiconductor in thermal equilibrium, an E-field exists.  If we assume a quasi-neutrality condition in which the electron concentration is almost equal to the donor impurity concentration, then  So an E-field is induced due to the nonuniform doping.

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 16 Einstein Relation  Assuming there are no electrical connections between the nonuniformly doped semiconducotr, so that the semiconductor is in thermal equilibrium, then the individual electron and hole currents must be zero.  Assuming quasi-neutrality so that n  N d (x) and  Similarly, the hole current J p = 0

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 17 Einstein Relation  Einstein relation says that the diffusion coefficient and mobility are not independent parameters. Typical mobility and diffusion coefficient values at T=300K (  = cm 2 /V-sec and D = cm 2 /sec)  n D n  p D p Silicon GaAs Germaium

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 18 Hall Effect  The hall effect is a consequence of the forces that are exerted on moving charges by electric and magnetic fields.  We can use Hall measurement to –Distinguish whether a semiconductor is n or p type –To measure the majority carrier concentration –To measure the majority carrier mobility

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 19 Hall Effect  A semiconductor is electrically connected to V x and in turn a current I x flows through. If a magnetic field B z is applied, the electrons/holes flowing in the semiconductor will experience a force F = q v x x B z in the (-y) direction.  If this semiconductor is p-type/n-type, there will be a buildup of positive/negative charge on the y = 0 surface. The net charge will induce an E-field E H in the +y-direction for p-type and -y-direction for n-type. E H is called the Hall field.  In steady state, the magnetic force will be exactly balanced by the induced E-field force. F = q[E + v x B] = 0  E H = v x B z and the Hall voltage across the semiconductor is V H = E H W  V H >0  p-type, V H < 0  n-type

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 20 Hall Effect  V H = v x W B z, for a p-type semiconductor, the drift velocity of hole is  for a n-type,  Once the majority carrier concentration has been determined, we can calculate the low-field majority carrier mobility.  For a p-semiconductor, J x = ep  p E x.  For a n-semiconductor,

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 21 Hall Effect Hall-bar with “ear” van deer Parw configuration

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 22 Homework  5.14  5.20

Solid-State Electronics Chap. 5 Instructor: Pei-Wen Li Dept. of E. E. NCU 23