Solid State Electr Electronics onics Text Book Solid d St Stat atee Ben. G. St Stre reeetm tmaan and and Sa San nja jay y Ba Ban ner erje jee: e: Soli Electronic Electr onic Devices, Prentice-Hall of India Private Limited.
Chapter 3 and 4
Metals, Semiconductors, and Insulators Insulator: A very poor conductor of electricity is called an insulator. In an insulator material the valance band is filled while the conduction band is empty. empty. The conduction band and valance band in the insulator are separated by a large forbidden band or energy gap (almost 10 eV). In an insulator material, the energy which can be supplied to an electron from a applied field is too small to carry the particle from the field valance band into the empty conduction band. Since the electron cannot acquire sufficient applied energy, conduction is impossible in an insulator. insulator.
Semiconductor: A substance whose conductivity lies between insulator and conductor is a semiconductor. semiconductor. A substance substance for which the width of the forbidden energy region is relatively small (almost 1 eV) is called semiconductor. semiconductor. In a semiconductor material, the energy which can be supplied to an electron electron from a applied applied field is too small to carry carry the particle particle from the field valance band into the empty conduction band at 0 K. As the the tem tempera peratu ture re is incr increa ease sed, d, some some of the the vala valanc ncee band band electrons acquire thermal energy energy.. Thus, the semiconductors allow for excitation of electrons from the valance band to conduction band. These are now free electrons in the sense that they can move about under the influence of even a small-applied field.
metal is an excellent excellent conductor conductor.. Metal: A metal In metals the band either overlap or are only partially filled. Thus electrons and empty energy states are intermixed within the bands so that electrons can move freely under the influence of an electric field.
Direct and Indirect Semiconductors Direct Material: The material (such as GaAs) in which a transition of an electron from the minimum point of conduction band to the maximum point of valence band takes place with the same value of K (propagation constant or wave vector) is called direct semiconductor material.
According to Eq. (3-1) the energy ( E ) vs propagation constant (k) curve is shown in the figure. A direct semiconductor such as GaAs, an electron in the conduction band can fall to an empty state in the valence band, giving off the energy difference E g as a photon of light.
Indirect Material: The material (such as Si) in which a transition of an electron from the minimum point of conduction band to the maximum point of valence band takes place with the different values of K (propagation constant or wave vector) is called indirect material.
According to Eq. (3-1) the energy ( E ) vs propagation constant (k) curve is shown in the figure. An electron in the conduction band minimum of an indirect semiconductor cannot fall directly to the valence band maximum but must undergo a momentum change as well as changing its energy. It may go through some defect state ( E t ) within the band gap. In an indirect transition which involves a change in k, the energy is generally given up as heat to the lattice rather than as emitted photon.
Intrinsic Material A perfect semiconductor with no impurities or lattice defect is called an intrinsic material. In intrinsic material, there are no charge carrier at 0K, since the valence band is filled with electrons and the conduction band is empty. At high temperature electron-hole pairs are generated as valence band electrons are excited thermally across the band gap to the conduction band. These EHPs are the only charge carriers in intrinsic material. Since the electrons and holes are crated in pairs, the conduction band electron concentration n (electron/cm3) is equal to the concentration of holes in the valence band p (holes/cm3). Each of these intrinsic carrier concentrations is commonly referred to as ni. Thus for intrinsic material: n= p=ni (3-6) At a temperature there is a carrier concentration of EHPs ni
Recombination is occurs when an electron in the conduction band makes transition to an empty state (hole) in the valence band, thus annihilating the pair. If we denote the generation rate of EHPs as gi (EHP/cm3-s) and the recombination rate ri, equilibrium requires that (3-7a) ri=gi Each of these rates is temperature dependent. gi(T ) increases when the temperature is raised, and a new carrier concentration ni is established such that the higher recombination rate ri(T ) just balances generation. At any temperature, the rate of recombination of electrons and holes ri is proportional to the equilibrium concentration of electrons n0 and the concentration of holes p0: (3-7b) ri= rn0 p0= rni2=gi The factor r is a constant of proportionality which depends on the particular mechanism takes place.
Extrinsic Material When a crystal is doped such that the equilibrium carrier concentrations n0 and p0 are different from carrier concentration ni, the material is said to be extrinsic material. In addition to the intrinsic carriers generated, it is possible to create carriers in semiconductors by purposely introducing impurities into the crystal. This process, called doping, is the most common technique for varying conductivity of semiconductor. There are two types of doped semiconductors, n-type (mostly electrons) and p-type (mostly holes). An impurity from column V of the periodic table (P, As and Sb) introduces an energy level very near the conduction band in Ge or Si.
The energy level very near the conduction band is filled with electrons at 0K, and very little thermal energy is required to excite these electrons to the conduction band ( Fig. 3-12a).
Thus at 50-100K virtually all of the electrons in the impurity level are, “donated” to the conduction band. Such an impurity level is called a donor level and the column V impurities in Ge or Si are called donor impurities. Semiconductors doped with a significant number of donor atoms will have n0>>(ni, p0) at room temperature. This is n-type material.
Fig. 3-12 (a) Donation of electrons from donor level to conduction band.
Similarly, an impurity from column III of the periodic table (B, Al, Ga and In) introduces an energy level very near the valence band in Ge or Si. These levels are empty of electrons at 0K ( Fig. 3-12b). At low temperatures, enough thermal energy is available to excite electrons from the valence into the impurity level, leaving behind holes in the valence band. Since this type of impurity level “accepts” electrons from the valence band, it is called an acceptor level, and the column III impurities are acceptor impurities in the Ge and Si. Doping with acceptor impurities can create a semiconductor with a hole concentration p0 much greater that the conduction band electron concentration n0. This type is p-type material.
Fig. 3.12b
Carrier concentration The calculating semiconductor properties and analyzing device behavior, it is often necessary to know the number of charge carriers per cm3 in the material. To obtain equation for the carrier concentration, Fermi-Dirac distribution function can be used. The distribution of electrons over a range of allowed energy levels at thermal equilibrium is 1
f ( E )
1 e
( E E F ) / kT
where, k is Boltzmann’s constant (k =8.26 10-5 eV/K=1.38 10-23 J/K). The function f ( E ), the Fermi-Dirac distribution function, gives the probability that an available energy state at E will be occupied by an electron at absolute temperature T . The quantity E F is called the Fermi level, and it represents an important quantity in the analysis of semiconductor behavior
For an energy E equal to the Fermi level energy E F , the occupation probability is 1 1
f ( E F )
( E F E F ) / kT 1 e
2
The significant of Fermi Level is that the probability of electron and hole is 50 percent at the Fermi energy level. And, the Fermi function is symmetrical about E F for all temperature; that is, the probability f ( E F + E ) of electron that a state E above E F is filled is the same as probability [1- f ( E F - E )] of hole that a state E below E F is empty. At 0K the distribution takes the simple rectangular form shown in Fig. 3-14. With T =0K in the denominator of the exponent, f ( E ) is 1/(1+0)=1 when the exponent is negative ( E < E F ), and is 1/(1+ )=0 when the exponent is positive ( E > E F ).
This rectangular distribution implies that at 0K every available energy state up to E F is filled with electrons, and all states above E F are empty. At temperature higher than 0K, some probability exists for states above the Fermi level to be filled. At T =T 1 in Fig. 3-14 there is some probability f ( E ) that states above E F are filled, and there is a corresponding probability [1 f ( E )] that states below E F are empty. The symmetry of the distribution of empty and filled states about E F makes the Fermi level a natural reference point in calculations of electron and hole concentration in semiconductors.
For intrinsic material, the concentration of holes in the valence band is equal to the concentration of electrons in the conduction band. Therefore, the Fermi level E F must lies at the middle of the band gap. Since f ( E ) is symmetrical about the electron E F , probability ‘tail’ if f ( E ) extending into the conduction band of Fig. 3-15a is symmetrical with the hole probability tail [1- f ( E ) ] in the valence band.
In n-type material the Fermi level lies near the conduction band (Fig. 3-15b) such that the value of f ( E ) for each energy level in the conduction band increases as E F moves closer to E c.
Fig. 3.15(b) ntype material
Thus the energy difference ( E c- E F ) gives measure of n. Fig. 3.15(c) ptype material
In p-type material the Fermi level lies near the valence band (Fig. 3-15c) such that the [1- f ( E )] tail value E v is larger than the f ( E ) tail above E c. The value of ( E F - E v) indicates how strongly p-type the material is.
Example: The Fermi level in a Si sample at equilibrium is located at 0.2 eV below the conduction band. At T=320K, determine the probability of occupancy of the acceptor states if the acceptor states relocated at 0.03 eV above the valence band.
Solution: From above figure, E a- E F ={0.03-(1.1-0.2)} eV= -0.87 eV kT = 8.6210-5 eV/K 320=2758.4 eV
we know that,
f ( E a )
1 1 e
( E a E F ) / kT
1
1
0 87 /( 2758 4105 )
1.0
Electron and Hole Concentrations at Equilibrium The concentration of electron and hole in the conduction band and valance are
n0 E f ( E ) N ( E )dE c E
p0 v [1 f ( E )] N ( E ) dE
(3.12a ) (3.12b)
where N ( E )dE is the density of states (cm -3) in the energy range dE . The subscript 0 used with the electron and hole concentration symbols (n0, p0) indicates equilibrium conditions. The number of electrons (holes) per unit volume in the energy range dE is the product of the density of states and the probability of occupancy f ( E ) [1- f ( E )]. Thus the total electron (hole) concentration is the integral over the entire conduction (valance) band as in Eq. (3.12). The function N ( E ) is proportional to E (1/2), so the density of states in the conduction (valance) band increases (decreases) with electron (hole) energy.
Similarly, the probability of finding an empty state (hole) in the valence band [1- f ( E )] decreases rapidly below E v, and most hole occupy states near the top of the valence band.
This effect is demonstrated for intrinsic, n-type and p-type materials in Fig. 3-16.
Fig. 3.16 (a) Concentration of electrons and holes in intrinsic material.
Fig. 3.16 (b) Concentration of electrons and holes in n-type material.
Fig. 3.16 (a) Concentration of electrons and holes in p-type material.
The electron and hole concentrations predicted by Eqs. (3-15) and (318) are valid whether the material is intrinsic or doped, provided thermal equilibrium is maintained. Thus for intrinsic material, E F lies at some intrinsic level E i near the middle of the band gap, and the intrinsic electron and hole concentrations are
ni N ce
( E c E i ) / kT
, pi N v e
( E i E v ) / kT
(3.21)
From Eqs. (3.15) and (3.18), we obtain
n0 p0 N c e
( E c E F ) / kT
n0 p0 N c N v e
N ve
( E c E v ) / kT
( E F E v ) / kT
N c N ve
E g / kT
(3.22)
From Eq. (21), we obtain
ni pi N c N ve
ni pi N c e
( E c E v ) / kT
( E c E i ) / kT
N ve
N c N v e
E g / kT
( E i E v ) / kT
(3.23)
From Eqs. (3.22) and (3.23), the product of n0 and p0 at equilibrium is a constant for a particular material and temperature, even if the doping is varied.
The intrinsic electron and hole concentrations are equal, ni= pi; thus from Eq. (3.23) the intrinsic concentrations is
ni N c N v e
E g / 2 kT
(3.24)
The constant product of electron and hole concentrations in Eq. (3.24) can be written conveniently from (3.22) and (3.23) as 2 n0 p0 ni (3.25)
At room temperature (300K) is: For Si approximately ni =1.5 1010 cm-3; For Ge approximately ni =2.5 1013 cm-3;
From Eq. (3.21), we can write as
N c ni e
( E c E i ) / kT
N v pi e
( E i E v ) / kT
(3.26)
Substitute the value of N c from (3.26) into (3.15), we obtain ( E c E i ) / kT ( E c E F ) / kT ( E c E i E c E F ) / kT n0 ni e e ni e ( E F E i ) / kT ( E i E F ) / kT (3.27) ni e n0 ni e
Substitute the value of N v from (3.26) into (3.18), we obtain
p0 pi
( E i E v ) / kT ( E F E v ) / kT e e
p0 ni e
( E F E i ) / kT
( E i E v E F E v ) / kT ni e ( E E ) / kT (3.28) ni e i F
It seen from the equation (3.27) that the electron concentrations n0 increases exponentially as the Fermi level moves away from E i toward the conduction band. Similarly, the hole concentrations p0 varies from ni to larger values as
Temperature Dependence of Carrier Concentrations The variation of carrier concentration with temperature is indicated by Eq. (3.21)
ni N c e
( E c E i ) / kT
, pi N ve
( E i E v ) / kT
(3.21)
The intrinsic carrier ni has a strong temperature dependence (Eq. 3.24) and that E F can vary with temperature.
ni N c N v e
E g / 2 kT
(3.24)
The temperature dependence of electron concentration in a doped semiconductor can be visualized as shown in Fig. 3-18.
In this example, Si is doped n-type with donor concentration N d of 1015 cm-3. At very low temperature (large 1/T ) negligible intrinsic EHPs exist, and the donor electrons are bound to the donor atoms.
As the temperature is raised, these electrons are donated to the conduction band, and at about 100K (1000/T =10) all the donor atoms are ionized. Figure 3-18 Carrier concentration vs. This temperature range is inverse temperature for Si doped with 3. 10 donors/cm 15 called ionization region. Once the donor atoms are ionized, the conduction band electron concentration is n0 N d =1015 cm-3, since one electron is obtained for each
When every available extrinsic electron has been transferred to the conduction band, no is virtually constant with temperature until the concentration of intrinsic carriers ni becomes comparable to the extrinsic concentration N d . Finally, at higher temperature ni is much greater than N d , and the intrinsic carriers dominate. In most devices it is desirable to control the carrier concentration by doping rather than by thermal EHP generation.
Thus one usually dopes the material such that the extrinsic range extends beyond the highest temperature at which the device to be used.
Excess Carrier in Semiconductors The carriers, which are excess of the thermal equilibrium carries values, are created by external excitation is called excess carriers. The excess carriers can be created by optical excitation or electron bombardment.
Optical Absorption Measurement of band gap energy: The band gap energy of a semiconductor can be measured by the absorption of incident photons by the material. In order to measure the band gap energy, the photons of selected wavelengths are directed at the sample, and relative transmission of the various photons is observed. This type of band gap measurement gives an accurate value of band gap energy because photons with energies greater than the band gap energy are absorbed while photons with energies less than band gap
Excess carriers by optical excitation: It is apparent from Fig. 4-1 that a photon with energy hv> Eg can be absorbed in a semiconductor.
Since the valence band contains many electrons and conduction band has many empty states into which the electron may be excited, the probability Figure 4-1 Optical absorption of a photon with of photon absorption is high. hv> E g: (a) an EHP is created during photon Fig. 4-1 indicates, an electron absorption (b) the excited electron gives up excited to the conduction band by optical energy to the lattice by scattering events; (c) absorption may initially have more the electron recombines with a hole in the energy than is common for conduction valence band. band electrons. Thus the excited electron losses energy to the lattice in scattering events until its velocity reaches the thermal equilibrium velocity of other conduction band electrons. The electron and hole created by this absorption process are excess carriers: since they are out of balance with their environment, they must even eventually recombine. While the excess carriers exit in their respective bands, however, they are free
I0
It
If a beam of photons with hv> E g falls on a semiconductor, there will be some predictable amount of absorption, determined by the properties of the material. The ratio of transmitted to incident light intensity depends on the photon wavelength and the thickness of the sample. let us assume that a photon beam of intensity I0 (photons/cm-2-s) is directed at a sample of thickness l as shown in Fig. 4-2. The beam contains only photons of wavelength selected by monochromator. As the beam passes through the sample, its intensity at a distance x from the surface can be calculated by considering the probability of absorption with in any increment dx. The degradation of the intensity – dI ( x)/dx is proportional to the intensity remaining at x: d I( x) I( x) (4.1) dx
The solution to this equation is e x 0
I( x) I
(4.2)
and the intensity of light transmitted through the sample thickness l is I e l t
I
0
(4.3)
The coefficient is called the absorption coefficient and has units of cm-1. This coefficient varies with the photon wavelength and with the material. Fig. 4-3 shows the plot of
Figure 4-3 Dependence of optical absorption coefficient for a semiconductor on the wavelength of incident light.
vs. wavelength.
There is negligible absorption at long wavelength ( hv small) and considerable absorptions with energies larger than E g. The relation between photon energy and wavelength is E =hc/ . If E is given in electron volt and is micrometers, this becomes E =1.24/
Steady State Carrier Generation The thermal generation of EHPs is balanced by the recombination rate that means [Eq. 3.7] (4.10) g (T ) r n2 r n p i 0 0 If a steady state light is shone on the sample, an optical generation rate gop will be added to the thermal generation, and the carrier concentration n and p will increase to new steady sate values. If n and p are the carrier concentrations which are departed from equilibrium: g (T ) g op r np r (n0 n)( p0 p) (4.11) For steady state recombination and no traping, n= p; thus Eq. (4.11) becomes
2 g (T ) g op r n0 p0 r [(n0 p0 ) n n ]
(4.12)
Since g(T )== rn0 p0 and neglecting the n2, we can rewrite Eq. (4.12) as g op r [(n0 p0 ) n] ( n / n )
where, n
1 r ( n0
p0 )
The excess carrier can be written as
(4.13)
is the carrier life time. n
p gop n
(4.14)
