Chater Motion and Recombination of Electrons and Holes OBJECTIVES. Understand how the electrons and holes resond to an electric field (drift).. Understand how the electrons and holes resond to a gradient in the carrier concentration (diffusion). 3. Understand the rocess for the recombination and the generation.
Thermal Motion Even without an alied electric field, carriers are not at rest but ossess finite kinetic energies. The kinetic energy of electrons: E - E C Average electron kinetic energy total kinetic energy number of electrons 3 f ( E) D( E)( E E kt C f ( E) D( E) de m n v th ) de Random thermal motion due to scattering from lattice atom, imurities, other electrons, and defects Thermal velocity v th 3kT ( ) m n / 0 7 cm/ sec
Drift Drift is the motion of charge carriers caused by an electric field Electron and Hole Mobilities Assume that the mean free time between collisions is τ m (τ ) and that the carrier loses its entire drift momentum, m v (m n v) after each collision. Consider the case for hole, the drift momentum gained between collision is equal to. qe m E v th due to collision on average + v - An electric field creates a drift velocity that is suerimosed on the thermal velocity
Drude Theory on Mobility Assume that the carrier loses its entire drift momentum, m v, after each collision. for hole v for electron v dv dt 0 v m dv qe m v qe m F m a m qe m dv q Edt 0 m m q m E E m q E ne n q m q m m n Low field mobility
Mechanisms of Carrier Scattering q, m n,, m, amount of scattering Function of temerature and doing concentration What are the imerfections in the crystal that cause carrier collisions or scattering? q h Phonon(or lattice vibration) scattering ( honon, h mean freetime of honon scattering), Crystal vibration distorts the eriodic crystal structure and thus scatters the electron waves. amount of scattering honon densitycarrier thermal velocity T T 3/ h h T / q imurity Ionized imurity scattering ( imurity, imurity mean freetime of imurity scattering), Ionized imurity scattering becomes dominant at low temeratures. At higher temerature, the ionized imurity scattering becomes weaker and hence the mobility becomes higher. imurity imurity Sum of amount 3/ T ionized of scattering imurity T N d 3/ N a An electron can be scattered by an accetor ion (a) and a donor ion (b) in a strikingly similar manner, even though the ions carry oosite tyes of charge. The same is true for a hole.
Probability of being scattered by ionized imurity: Probability of being scattered by honons: Electrical analogy for scattering R h honon R imurity imurity P h Pi imurity honon Total robability of being scattered: P P P... h imurity honon imurity honon imurity P R Rh Rimurity Rh : with low doing and high temerature μ ionized imurity scattering honon scattering R R imurity : with heavily doing and low temerature 3/ T T 3/ honon imurity honon imurity total mobility T
Mobility vs. doant concentration for Si at 300 K Ionized imurity scattering can be neglected. 40 ( cm / V s) 50 7 0.7 [ Na Nd/.60 ] 38 n( cm / V s) 9 7 0.85 [ Na Nd/0 ] Mobility decreases with increasing doing. due to free-carrier screening When the carrier concentration is large, the carriers can distribute themselves to artially screen out the coulomb field of the doant ions.
Mobility vs. doing concentration for Ge and GaAs at 300 K 8000 400 Note that n for both Si and GaAs
Temerature deendence of electron and hole mobility In general, mobility decreases with T (lattice scattering). In case of very high N d, mobility is not dramatically influenced by T
Velocity Saturation and Ballistic Transort Sloe=mobility due to high energy otical honon scattering mvsat Eot 40meV v E m cm s sat 7 ot / 0 / E vd E v sat for holes in Si for electrons in Si EE vsat E 0 Low field region High field region When device size is smaller than the mean free ath, the carriers exerience a nearly scattering-free environment called ballistic transort
Drift Current and Conductivity A P-tye semiconductor bar of unit area is used to demonstrate the concet of current density. q m J, drift qv q E E, v E E m amere coulomb electrons cm 3 cm electron cm s q m where q : hole conductivity m q Jn, drift qnv qnne ne, v ne E nq where nqnn : electron conductivity Jdrift Jn, drift J, drift ( qnn q ) E ( n ) E, Jdrift E, qn q n : conductivity A S( siemens) V cm cm v cm V cm E s / / V s cm
Drift and Resistance L R wt L wt : resistivity A S( siemens) V cm cm cm Conversion between resistivity and doant density of silicon at room temerature.
Hall Effect The Hall effect is related to the force that acts on a charged article that moves in a magnetic field [and electric field]. F q( E v B): Lorentz Force If a magnetic field is alied erendicular to the direction in which holes drift in a P-tye semiconductor bar, the ath of the holes tends to be deflected. z y x Illustration of the Hall effect. (a) The force due to the magnetic field B z deviates the hole trajectory. (b) Accumulated holes create Hall field E H = -E x that counteracts the force from the magnetic field B z.
In the x-direction the force is xfx xe x x y z F q( E v B) yfy q y Ey v x vy 0 zf z ze z 0 0 B z x vb y z F qe qv B x x y z At steady-state, the force along x-axis must be balanced, F qe qv B 0 qv B qe x x y z y z x E y z qv B magnetic force in x direction qe electric force in x direction This establishes a steady-state hole flow along the bar in the x-direction. x If the width of the bar is W, a voltage V H can be measured between the oosite sides and it equals to V H E W x Hall voltage
The Hall voltage is then J y I I IB VH E xw vybzw BzW BzW Bz RH q q t W q t t ( J qv ) Alternatively, R H can be exressed as y y s where R H is known as the Hall coefficient. z R H q given that q A measurement of the Hall voltage from a known current and magnetic field yields a value for the hole concentration.
Diffusion Current Diffusion A rocess whereby articles tend to sread out or redistribute as a result of their random thermal motion. migrating on a macroscoic scale from regions of high article concentration into region of low articles concentration. [Sreading of a ulse of electrons by diffusion] Particles diffuse from high-concentration locations toward low-concentration locations. Within semiconductors, diffusion related carrier (charged article: electron or hole) transort therefore gives rise to article currents J n, diffusion dn dx dn dx qdn J, diffusion d qd dx d dx
n(x) electron diffusion J n,diffusion (x) hole diffusion J,diffusion x steady electron uniform semiconductor injection at surface steady hole injection at surface x Hot oint robe measurement Hot (-) J A Cold Hot (+) J A Cold energetic hole diffuse away -tye negative charge build u n-tye energetic electron diffuse away ositive charge build u
Diffusion-current equation Holes Electrons x x concentration gradient: + hole flux (-x direction): hole diffusion current: The diffusion current density = carrier flux carrier charge J qd diff d x x D x D dx n n n diff concentration gradient: + electron flux (-x direction): electron diffusion current: dn x dx J qd n N
Derivation of Diffusion Current n(x) : mean free ath between collisions small incremental distance n n The electron in segment() have equal chances of moving left or right, and in a mean free time τ one-half of them will move into segment (), vice versa for electrons in (). x x 0 x 0 x 0 The net # of electrons assing x 0 from left to right in one mean free time, n( A) n( A) where A: cross section area ( A): volume The rate of electron flow in the +x direction/unit area electron flux n( x ) { ( n A) ( n A)} ( nn) 0 A (electron flow /unit area sec)
Let the mean free ath n n n( x) n( x x) x n ( x0 ) n( x) n( x x) x be a small differential length,, where x is taken at the center of segment () and x. n( x) n( x x) dn( x) n( x) n( x0 ) lim dx x 0 Similarly for hole, x 0 x dn( x) D n dx d( x) ( x) D dx from the definition of the derivative, where D n is called diffusion coefficient of electron The diffusion current density becomes dn( x) dn( x) Jn, diffusion ( q) Dn qdn dx dx d( x) d( x) J, diffusion ( q) D qd dx dx Jn, diffusion qdnn J qd, diffusion in 3-D
Total electron and hole current density dn( x) Jn Jn, drift Jn, diffusion qnn ( x ) E ( x ) qdn x ˆ : -D dx qnn( x, y, z) E ( x, y, z) qdnn ( x, y, z) : 3-D d( x) J J, drift J, diffusion q ( x ) E ( x ) qd x ˆ : -D dx q ( x, y, z) E ( x, y, z) qd ( x, y, z) : 3-D Total current density flowing in a semiconductor J J n J E (x) + - n(x) (x) J J,, diffusion, drift, J, diffusion, drift, n, diffusion n, drift, J n, diffusion n, drift
Relation between the Energy Diagram and V ( x), E ( x) E C and E V vary in the oosite direction from the voltage. E C and E V are higher where the voltage is lower. E ( ). ( ) C x const qv x E dv ( x) dec( x) dev( x) ( x) dx q dx q dx Energy band diagram of a semiconductor under an alied voltage 0.7 ev is an arbitrary value.
Constancy of the Fermi Level It is always satisfied that no discontinuity on gradient can arrive in the equilibrium Fermi level E F, even if nonuniform doing. Consider two semiconductors attached at x = 0 Material D( E ) f ( E) Material D ( E ) f ( E) x = 0 x There is no current flow at equilibrium. Electron transfer from to must be exactly balanced by the oosite transfer of electron from to. r D ( E) f( E) D ( E)[ f( E)] r D ( E) f( E) D ( E)[ f( E)] D ( E) f( E) D ( E) D ( E) f( E) D ( E) f( E) D ( E) f( E) D ( E) D ( E) f( E) D ( E) f( E) f E) f ( ) ( E ( EEF )/ kt ( EEF [ ] [ )/ kt e e ] def EF EF 0, at equilibrium dx A iece of N-tye semiconductor in which the doant density decreases toward the right.