CHAPTER 3: CARRIER CONCENTRATION PHENOMENA Part I

SUB-TOPICS IN CHAPTER 3: Carrier Drift Carrier Diffusion Generation & Recombination Process Continuity Equation Thermionic Emission Process Tunneling Process High-Field Effect

Part I Carrier Drift Carrier Diffusion Generation & Recombination Process

CARRIER DRIFT Mobility The electron in s/c have 3 degree of freedom – they can move in a 3-D space. The K.E of electron is given by (1) From the theorem for equipartition of energy, ½ kT unit energy per degree of freedom. mn – effective mass of electron, vth – average thermal velocity (~ 107cm/s at T=300K)

Mobility (cont.) Electron in s/c moving rapidly in all direction, where thermal motion of an individual electron may be visualized as a succession of random scattering from collisions with lattice atoms, impurity atoms, and other scattering centers, as shown in Fig. 3.1(a). Average distance between collisions – mean free path. Average time between collisions – mean free time C. For typical mean free path ~ 10-5cm, C = 10-15/vth~10-12s (or in 1ps).

Mobility (cont.) When small electric field, E, is applied to s/c sample, each electron will experience a force –qE from the field and accelerated along the field (in opposite direction) during the time between collisions – additional thermal velocity component. This additional component called drift velocity. Combination displacement of an electron (due to random thermal motion) & drift component illustrated in Fig. 3.1(b). Note that: net displacement of the electron is in the opposite direction of applied field.

Mobility (cont.) Without electric field hole Figure 3.1. Schematic path of an electron in a semiconductor. (a) Random thermal motion. (b) Combined motion due to random thermal motion and an applied electric field.

Mobility (cont.) The momentum applied to an electron is given by -qEC, and momentum gained is mnvn. Thus, using physics conservation of energy, electron drift velocity: (2) Note that: vn is proportional to E The proportionality factor may be written as (3) The proportionality factor also called electron mobility. A similar expression may be written for holes in valence band may be written as: vp = p E Mobility is very important parameter for carrier transport – it describes how strongly the motion of an electron is influenced by an applied electric field.

Mobility (cont.) From eq. (3), mobility is related directly to mean free time between collisions determined by the various scattering mechanism. Two MOST important mechanisms: lattice scattering and impurity scattering. Lattice scattering – results from thermal vibrations of the lattice atoms at any temperature, T>0K (it becomes dominant at high temp. – mobility decreases with increasing temp.) – theoretically mobility due to lattice scattering L decrease in proportion to T-3/2 Impurity scattering – results when charge carrier travels past am ionized dopant impurity (donor or acceptor). It depend on Coulomb force interaction. Impurity scattering depends on total concentration of ionization impurities (sum of +ve and –ve charge ions). It becomes less significant at higher temperatures.

Mobility (cont.) The probability of a collision taking place in unit time, 1/C, - the sum of the probabilities of collision due to the various scattering mechanism: or (4) (4a) L – lattice scattering mobility I – impurity scattering mobility

Mobility (cont.) Electron mobility as a function of temp. for Si with 5 different donor concentration is given by Fig. 3.2. For lightly doping (i.e 1014cm-3) – lattice scattering dominates and mobility decreases as the temp. increases. For heavily doped (i.e 1019cm-3) – at low temp. impurity scattering is most pronounced. Mobility is increases as temp. increases. For a given temp., mobility decreases with increasing impurity concentration (due to enhanced impurity scattering).

Lightly doped Heavily doped Figure 3.2. Electron mobility in silicon versus temperature for various donor concentrations. Insert shows the theoretical temperature dependence of electron mobility.

Mobility (cont.) Mobility reaches a maximum value at low impurity concentrations corresponds to the lattice scattering limitation. Both electron & hole mobilities decrease with increasing impurity concentration. Mobility of electrons is greater than holes due to the smaller effective mass of electrons. Figure 3.3. Mobilities and diffusivities in Si and GaAs at 300 K as a function of impurity concentration.

EXAMPLE 1 Calculate the mean free time of an electron having a mobility of 2000 cm2/ V-s at room temperature; also calculate the mean free path. Assume mn = 0.22mo in these calculation.

Resistivity Refer to Fig. 3.4. 3.4(a) – n-type s/c & its band diagram at thermal equilibrium. 3.4(b) – when biasing voltage is applied at right-hand-terminal. Assume that contact at both terminals are ohmic (there is negligible voltage drop at each of the contacts). Behavior of ohmic contact – Chapter 7. When E (electric field) is applied to s/c, each electron may experience a force of –qE. Thus, the force is equal to the negative gradient of the potential energy: (5) EC – conduction band energy

Resistivity (cont.) Figure 3.4. Conduction process in an n-type semiconductor (a) at thermal equilibrium and (b) under a biasing condition.

In the gradient of U, any part of the band diagram that is parallel to EC (e.g EF, Ei, and EV) may be used. But it’s convenient to use intrinsic Fermi level Ei (when consider p-n junction in Chapter 4). From (5): (6) Where  - electrostatic potential, and defined as (7) Which represents the relationship between electrostatic potential and potential energy, U.

For homogenous s/c (Fig. 3 For homogenous s/c (Fig. 3.4(b)) – U and Ei decrease linearly with distance, thus electric field constant –ve x-direction. Electrons in cond. band move to the right – electron undergoes a collision, loses some or all of its K.E to the lattice & drops toward its thermal equilibrium position – this process will be repeated many times. Hole behaves in the same manner but in the opposite direction. Transport of carriers under applied electric field – drift current.

From Fig. 3.5, with application of electric field, current density for both electron and hole, Jk may be written as (8) Where for electron, k = n, y= -1; and hole, k=p, y=1 Figure 3.5. Current conduction in a uniformly doped semiconductor bar with length L and cross-sectional area A.

Total current flowing in s/c sample is sum of the electron and hole components, which is (9) From (9), conductivity  = q(nn + pp). Thus, resistivity of semiconductor is given by (10) For extrinsic s/c, generally may be written as (11) For n-type (n>>p), k=n, and p-type (p>>n), k=p

Figure 3.6. Measurement of resistivity using a four-point probe. In practical, to measure resistivity – commonly used the four-point probe method (Fig. 3.6) With thickness, W << d, thus resistivity is govern by (12) Where CF ~ ‘correction factor’ and it is depends on the ratio of d/s, s – probe spacing. Figure 3.6. Measurement of resistivity using a four-point probe.

Room temperature Figure 3.7. Resistivity versus impurity concentration3 for Si and GaAs.

EXAMPLE 2 Find the resistivity of n-type Si doped with 1016 phosphorus atom/cm3 at T = 300K.

THE HALL EFFECT The “Hall effect” was discovered in 1879 by the American physicist, Edwin Hall (1855 – 1938). He discovered the "Hall effect" while working on his doctoral (PhD) thesis in Physics. In 1880, full details of Hall's experimentation with this phenomenon formed his doctoral thesis and was published in the American Journal of Science and in the Philosophical Magazine.

THE HALL EFFECT Hall effect is used to measure the carrier concentration. It is also one of the most convincing methods to show the existence of holes as charge carriers – measurement can give directly the carrier type. Fig. 3.8 show the Hall effect set-up (consider a p-type sample). Using Lorentz force F = qv x B = qvxBz. (B: magnetic field) There is no net current flow along y-direction (in steady-state), thus Ey exactly balances the Lorentz force: Hall field (13) Hall coefficient (14)

Figure 3.8. Basic setup to measure carrier concentration using the Hall effect.

All quantities in RHS can be measured, thus carrier The measurement of the Hall voltage for a known current and magnetic field yields (15) All quantities in RHS can be measured, thus carrier concentration and carrier type can be obtained directly from Hall measurement. RHS: right-hand-side

EXAMPLE 3 A sample of Si is doped with 1016 phosphorus atom/cm3. Find a Hall voltage in a sample with W = 300m, A = 0.0025cm2, I = 1mA, and Bz = 10-4 Wb/cm2.

CARRIER DIFFUSION Diffusion Process Carriers move from a high concentration region to low concentration region ~ called diffusion current. From Fig. 3.9, current density may explain by mathematical formalism below: LHS: RHS: (16) F ~ average electron flow per unit area. l ~ mean free path Dn ~ diffusion coefficient (17)

Figure 3.9. Electron concentration versus distance; l is the mean free path. The directions of electron and current flows are indicated by arrows.

EINSTEIN RELATION Rewrite Eq. (17) using theorem for equipartition of energy: (18) Using (3), (16), & (18), Einstein relation may be written as (19) (relation of diffusivity & mobility)

DENSITY EQUATIONS Total current density at any point is the sum of the drift & diffusion components: (20) Where k = n, with y=1, and k=p, with y= -1. Total conduction current density is given by Jcond = Jn + Jp

GENERATION & RECOMBINATION Direct Recombination For the direct-bandgap s/c in thermal equilibrium – the continuous thermal vibration of lattice atoms – cause bonds between neighboring atoms to be broken. Bonds broken cause electron-hole pair. Carrier generation – electron to make upward transition to cond. band & leaving a hole in valence band. It represented by the generation rate Gth (number of electron-hole pair generated/cm3/s) – Fig. 3.10(a). Recombination – electron makes transition downward from cond. band. It represented by recombination rate Rth (Fig. 3.10(a)). At thermal equilibrium cond. : Gth = Rth for pn = ni2 to be maintained.

The rate of generation & recombination in n-type is (21) When we shine a light, it produced electron-hole pair at a rate GL, recombination and generation rate (22) (23) nno & pno – electron and hole densities  - proportionality constant Figure 3.10. Direct generation and recombination of electron-hole pairs: (a) at thermal equilibrium and (b) under illumination.

p – lifetime of the excess minority carriers. The net change of hole concentration is given by (24) At steady-state, dpn/dt = 0; (25) And at low level injection, pno << nno, the net recombination is (26) p – lifetime of the excess minority carriers. U is net recombination, defined as U = (nno + pno + ∆p)∆p

From (25) & (26) (in steady-state), generation rate is given by (27) (28) When the light is turn off, t = 0, the boundary cond. pn(0)Eq. (28), and pn()  pno, thus (29) Figure 3.11. Decay of photo excited carriers. a) n-type sample under constant illumination. (b) Decay of minority carriers (holes) with time. (c) Schematic setup to measure minority carrier lifetime.

GENERATION & RECOMBINATION Indirect Recombination The derivation of the recombination rate is more complicated. Et – called the intermediate-level states. There are 4 basic transitions takes place. Example of the indirect-bandgap s/c – Si. After indirect recombination process: (i) Electron capture (ii) Electron emission (iii) Hole capture (iv) Hole emission Figure 3.12. Indirect generation-recombination processes at thermal equilibrium.

The recombination rate is given by (from derivation in Appendix I, Sze, pg. 541): (30) Under low-injection condition in a n-type, so nn >> pn , then (30) can be written as (31) Where, vth – thermal velocity, Nt – concentration of the recombination centre,  - capture cross section (effectiveness of the centre to capture an electron or hole), and

Surface Recombination A large number of localize energy states (generation-recombination centers) may introduced at the surface region. (Fig. 3.13). It may enhance the recombination rate at the surface region by an energy called surface-state. The kinetics of the surface recombination are similar to those in bulk centers. Total number of carrier recombining at the surface per unit area and unit time: (32) And, the low-injection surface recombination velocity is defined as: (33) Where, ps – concentration at surface, Nst – recombination center density per unit area in the surface region.

Figure 3.13. Schematic diagram of bonds at a clean semiconductor surface. The bonds are anisotropic and differ from those in the bulk.

Auger Recombination Occurs by the transfer of the energy & momentum released by the recombination of electron-hole pair to a 3rd particle (either electron or hole). Example shown in Fig. 3.14, the 2nd electron absorb the energy released by direct recombination – becomes an energetic electron. It’s very important – carrier concentration is very high (results from high doping or high injection level). The rate of this recombination can be expressed as (34) B – proportionality constant (strong temperature depending) Figure 3.14. Auger recombination.

Summary of Part 1 In part 1 of carrier transport phenomena, various temperature process include drift, diffusion, generation, and recombination. Carrier drift – under influence of an electric field. At low field, drift velocity is proportional to electric field called Mobility. Carrier diffusion – under influence of carrier concentration gradient. Total current = (drift + diffusion) components. Four types of recombination process: (i) Direct (ii) Indirect (iii) Surface (iv) Auger

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Next Lecture: CHAPTER 3 PART 2: Continuity Equation Thermionic Emission Process Tunneling Process High-Field Effect