Solid State Electr Electronics onics Text Book Ben. G. St Stre reeetm tmaan and and Sa Sanj njay ay Ba Ban ner erje jeee: Soli Solid d St Stat atee Electronic Electr onic Devices Devices,, Prentice-Hall of India Private Limited.
Chapter 3 1
Bonding forces in Solids Bond: The interaction of electrons in neighboring atoms is called bond. Ionic bond: Ionic bonds are a type of chemical bond based on electrostatic forces between two oppositely-charged oppositely-charged ions. In ionic bond formation, a metal donates an electron, due to a low electronegativity to form a positive ion. The bonds between the sodium (Na) and chlorine (Cl) ions are ionic bonds. The electronic structure of Na (Z=11) is [Ne]3 s [Ne]3 s1, and Cl(Z=17) has the structure [Ne]3 s [Ne]3 s23 p5. Na atom gives up its outer 3s electron to a Cl atom, so that the crystal is made up of ions with electronic structures of the inert atoms Ne and Ar. Ar. However, the ions have net electric charges after the electron exchange. The Na+ ion has a net positive charge, having lost an electron, and the Cl- ion has a net negative charge, having gained an electron. Once the electron exchanges have been made between the Na and Cl atoms to form Na+ and Cl- ions, the outer orbits of all atoms are completely filled. Since the ions have the closed-shell configurations of the inert atoms Ne and Ar, there are no loosely bound electrons to participate in current flow; as a result, NaCl is a good insulator. insulator. 2
Metallic bond: Metallic bonding is the bonding within metals. It involves the delocalized sharing of free electrons among a lattice (the periodic arrangement of atoms in a crystal is called the lattice) lattice ) of metal atoms. In a metal atom the outer electron shell is only partially filled, usually by no more than three electrons. In Na has only one electron in the outer orbit. This his electron is loosel sely boun ound and is given up easily in ion formation. In the metal metal the outer electron electron of each alkali alkali atom is contributed contributed to the crystal as a whole, so that the solid is made up of ions with closed shells immersed in a sea of free electrons. The forces holding the lattice together arise from an interaction between the positive ion cores and the surrounding free electrons. This is one type of metallic metallic bonding . The metals have the sea sea of electron rons in common, and these electrons are free to move about the crystal under the influence of 3 an electric field.
Covalent bond: Covalent bonding is an intermolecular form of chemical bonding characterized by the sharing of one or more pairs of electrons between two components, producing a mutual attraction that holds the resultant molecule together. Atoms tend to share electrons in such a way that their outer electron shells are filled. Atom in the Ge, Si, or C diamond lattice is surrounded by four nearest neighbors, each with four electrons in the outer orbit. In these crystals each atom shares its valence electrons with its four neighbors. The bonding forces arise from a quantum mechanical interaction between the shared electrons. This is known as covalent bonding ; each electron pair constitutes a covalent bond. 4
Energy Bands The electron in an isolated atom has discrete energy level but the electron in a solid has a range, or band , of available energies. The discrete energy levels of the isolated atom spread into bands of energies in the solid because in the solid the wave functions of electrons in neighboring atoms overlap, and an electron is not necessarily localized at a particular atom. As isolated atoms are brought together to form a solid, various interactions occur between neighboring atoms. The forces of attraction and repulsion between atoms will find a balance at the proper interatomic spacing for the crystal. In the process, important changes occur in the electron energy level configuration and these changes result in the varied 5 electrical properties of solid.
Fig. (a) shows the curve of wave function which is obtained from the solution of Schrodinger Wave Equation.
(a)
Fig. (b) shows the orbital model of a Si atom, which has a nucleus consisting of 14 protons (with a charge of +14) and neutrons, 10 core electrons in shells n=1 and 2, and 4 valance electrons in the 3 s and 3p subshells. Fig. (c) shows the energy levels of the various electrons in the coulombic potential well of the nucleus . Since unlike charges attract each other, there is an attractive potential between the negatively charged electrons and the positively charged nucleus.
The Coulomb potential varies only with radius in spherical coordinates
6
1
2
Bonding orbital
+
+
Atomic orbitals Anti-bonding
+ _
V (r ) Anti-bonding energy level
In Fig. 3-2, the coulombic potential wells of two atoms close to each other, along with the wave functions of two-electron centered on the two nuclei are schematically shown. By solving the Schrodinger for such an interacting system, it is found that the composite two electron wave functions are linear combinations of the individual atomic orbits (LCAO).
Bonding energy level
The odd or anti-symmetric combination is called the anti- bonding orbital , while the even or symmetric combination is the bonding orbital . It is seen that the bonding orbital Figure 3-2 Linear combination of atomic orbitals (LCAO): The has a higher value of the wave LCAO when 2 atoms are brought together leads to 2 distinct function than the anti-bonding “normal” models-a higher energy anti-bonding orbital, and a state in the region between 7 the
1
2
Atomic orbitals
V (r ) Anti-bonding energy level Bonding energy level
To determine the energy levels of the bonding and anti-bonding states, it is important to recognize that in the region between the two nuclei the coulombic potential energy V (r ) is lowered (solid line in Fig. 3) compared to isolated atoms (dashed line in Fig. 3-2). The potential energy in the region is lowered because an electron here would be attracted by two nuclei; rather than just one.
For the bonding state the electron probability density is higher in this region of lowered potential energy than for the anti-bonding state. As a result, the original isolated atomic energy level would be split into two, a lower bonding energy level and a higher anti-bonding level. 8
We consider N atoms of silicon in a solid. Each isolated silicon atom has an electron structure 1 s22 s22 p63 s23 p2 in the ground state. There will be 2 N , 2 N , 6 N , 2 N , and 6 N states of type 1 s, 2 s, 2 p, 3 s, and 3 p, respectively. As the interatomic spacing decreases, these energy levels split into bands, beginning with the outer (n=3) shell. As the “3 s” and “3 p” bands grow, they merge into a single band composed of mixture of energy levels. This band of “3 s-3 p” levels contains 8 N available states. As the distance between atoms approaches the equilibrium interatomic spacing of silicon, this band splits into two bands separated by an energy gap E g . The upper band (called the conduction band) contains 4 N states, as does 9 the lower (valence) band.
The energy gap is sometimes called a “ forbidden band ,” since in a perfect crystal it contains no electron states. There are 4N electrons in the original isolated n=3 shells (2 N in 3 s states and 2 N in 3 p states).
These 4N electrons must occupy states in the valence band or the conduction band in the crystal. At 0K the electrons will occupy the lowest energy states available to them. In the case of Si crystal, there are exactly 4 N states in the valence band available to the 4 N electrons. Thus at 0K, every state in the valence band will be filled, while the conduction band will be 10 completely empty of the electrons.
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. 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. 11
Semiconductor: A substance whose conductivity lies between insulator and conductor is a semiconductor. A substance for which the width of the forbidden energy region is relatively small (almost 1 eV) is called semiconductor. In a semiconductor 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 at 0 K. As the temperature is increased, some of the valance band electrons acquire thermal 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: A metal is an excellent conductor. 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.
12
Direct and Indirect Semiconductors A single electron is assumed to travel through a perfectly periodic lattice. The wave function of the electron is assumed to be in the form of a plane wave moving. For example, in the x-direction with propagation constant k , also called wave vector . The space-dependent wave function for the electron is k ( x)
U (k x , x)e jk x x (3 1)
where the function U (k x, x) modulates the wave function according to the periodicity of the lattice. Depending on the transition of an electron from conduction band to valance band with respect to the propagation constant, the semiconductor materials are classified as follows: (a) Direct semiconductor and 13 (b) Indirect semiconductor
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. 14
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. 15
Electrons and Holes Hole : An empty state in the valence band is referred to as a
hole. Electron-Hole Pair : If the conduction band electron and
the hole are created by the excitation of the valence band electron to the conduction band, they are called an electron-hole pair (abbreviated EHP).
16
If a filled band, all available energy states are occupied. For every electron moving with given velocity, there is an equal and opposite electron motion elsewhere in the band. If we apply an electric field, the net current is zero because for every electron j moving velocity v j there is a corresponding electron j‟ with the velocity – v j. Fig. 3-8 illustrates this effect in terms of the electron vs. wave vector plot for the valance band. Since k is proportional to electron momentum, it is clear the two electrons have opposite directed velocities. With N electron/cm3 in the band we express the current density using a sum over all of the electron velocities, and including the charge – q on each electron. In a unit volume, N J ( q ) vi 0
(filled band) (3 2a )
17
Fig. 3-8
Now if we create a hole by removing the jth electron, the net current density in the valance band involves the sum over all velocities, minus the contribution of the electron we have removed.
N J ( q ) vi ( q )v j 0 ( jth electron missing ) (3 2b) i Since the first term of (3-2b) is zero in accordance (3-2a), the net current is J qv j ( Net current due to the missing of jth electron ) The current contribution of the hole is equivalent to that of a positively charged particle with the velocity v j that of the missing electron. The charge transport is actually due to the motion of the new uncompensated electron j’ . Its current contribution (-q)(-v j) is equivalent to that of a positively charged particle with the velocity v j. Thus, it is realized that the current flow in the semiconductor can be account by the motion of electrons and holes of charge carriers. 18
In the valence band, hole energy increases oppositely to electron energy, because the two carriers have opposite charge. Thus hole energy increases downward in Fig. 3-8 and holes, seeking the lowest energy state available, are generally found at the top of the valance band .
In contrast, conduction band electrons are found at the bottom of the conduction band . Holes are found at the top of the valence band because the valence band electrons will rearrange themselves so that they occupy the lowest energy states, leaving only the highest energy level vacant in the valance band. Electrons in the conduction band will similarly rearrange themselves so they occupy the lowest energy states of conduction band.
19
Fig. 3-8
Effective Mass The electrons in a crystal are not completely free, but instead interact with periodic potential of the lattice of a crystal. The electron “wave particle” motion cannot be expected to be the same as for electrons in free space. And, the mass of free electron is not same as for electrons in a solid. The electron momentum can be written as P =mv=ħ k. Then 2 2 2 1 1 P 2 E mv m k 2 2 2m Thus the electron energy is parabolic with wave vector k . 20
The electron mass is inversely related to the curvature (second derivative) of the ( E , k ) relation, since d 2 E 2 d k 2 m
The effective mass of an electron in a band with a given ( E , k ) relationship is found as follows: m*
2
d 2 E / d k 2
Thus, the mass of electron which is obtained from the curvature of the energy band of a solid is called effective mass. 21
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 n i. Thus for intrinsic material: n= p=ni (3-6) At a temperature there is a carrier concentration of EHPs n i 22
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 g i (EHP/cm3-s) and the recombination rate r i, equilibrium requires that r i= g i (3-7a) Each of these rates is temperature dependent. g i(T ) increases when the temperature is raised, and a new carrier concentration ni is established such that the higher recombination rate r i(T ) just balances generation. At any temperature, the rate of recombination of electrons and holes r i is proportional to the equilibrium concentration of electrons n 0 and the concentration of holes p0: r i=a r n0 p0= a r ni2= g i (3-7b) The factor a r is a constant of proportionality which depends 23 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 24 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 n 0>>(ni, p0) at room temperature. This is n-type material.
Fig. 3-12 (a) Donation of electrons from donor level to conduction band. 25
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
26
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 27 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 )
1 e
( E F E F ) / kT
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 ).
28
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.
29
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 E F , the electron 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.
30
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. 31
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 ( E a E F ) / kT 1 e
1
1
0 87 /( 2758 4105 )
1.032
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 v p0 [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 33 (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. 34
Fig. 3.16 (b) Concentration of electrons and holes in n-type material.
35
Fig. 3.16 (a) Concentration of electrons and holes in p-type material.
The electron concentration (in terms of effective density of states N c which is located at the conduction band edge E c) in thermal equilibrium can also be written as follows:
n0 N c f ( E c ) (3.13)
where, N c
3/ 2 * 2 mn kT 2 2
h
It is assumed that the Fermi level E F lies at least several kT below the conduction band. Then e( E c E F ) / kT 1 and the Fermi function f ( E c) can be simplified as 1 ( E c E F ) / kT
f ( E c )
1 e
( E c E F ) / kT
e
(3.14)
For this condition the concentration of electrons in the conduction band is
n0 N ce
( E c E F ) / kT
(3.15)
36
By similar argument, the hole concentration (in terms of effective density of states N v which is located at the valance band edge E v) in thermal equilibrium can also be written as follows:
p0 N v [1 f ( E v )] (3.16)
where, N v
3/ 2 * 2 m p kT 2 2
h
For E F larger than E v by several kT , the probability of finding an empty state at E v is
1 f ( E v ) 1
1 1 e
( E v E F ) / kT
e
( E F E v ) / kT
(3.17)
For this condition the concentration of holes in the valance band is
p0 N ve
( E F E v ) / kT
(3.18)
37
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 ve
( E i E v ) / kT
(3.21)
From Eqs. (3.15) and (3.18), we obtain
n0 p0 N ce
( E c E F ) / kT
n0 p0 N c N ve
N ve
( E c E v ) / kT
( E F E v ) / kT
N c N ve
E g / kT
(3.22) 38
From Eq. (21), we obtain
ni pi N c N ve
ni pi N ce
( E c E v ) / kT
( E c E i ) / kT
N v e
N c N ve
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
n0 p0 ni2 (3.25) At room temperature (300K) is : For Si approximately n i = 1.5 1010 39 cm -3; For Ge approximately n i = 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
n0 ni e
( E c E i ) / kT ( E c E F ) / kT
e
ni e
( E c E i E c E F ) / kT
( E i E F ) / kT ( E F E i ) / kT n0 ni e ni e
(3.27)
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
ni e
( E i E v E F E v ) / kT
ni e( E i E F ) / kT (3.28)
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 40
Example 3-5 A Si sample is doped with 10 17 As atoms/cm3. What is the equilibrium hole concentration p0 at 300K? Where is E F relative to E i? Solution: N d =1017atoms/cm3; kT =0.0259eV; ni(for Si)=1.51010 cm-3;
Since N d >>ni, we can approximate n 0= N d
p0
ni2 n0
(1.5 1010 ) 2
We know that
1017
n0 ni e
2.25 103 cm3
( E F E i ) / kT
n0 E F E i kT ln ni 1017 E F E i 0.0259 ln 10 1.5 10 E F E i 0.407 eV
The resulting band diagram is
41
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 v e
( 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 / 2kT
(3.24)
The temperature dependence of electron concentration in a doped semiconductor can be visualized as shown in Fig. 3-18. 42
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 43
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. 44
Compensation and Space Charge Neutrality If a semiconductor contains both donors and acceptors. For example, Fig. 3.19 illustrates a semiconductor for which both donor and acceptors are present, but N d > N a. The predominance of donors makes the material n-type, and the Fermi level is therefore in the upper part of the band gap. Since E F is well above acceptor level E a, this level is essentially filled with electrons. However, With E F above E i, it cannot be expected that a hole concentration in valence band commensurate with the acceptor concentration. In fact, the filling of the E a states occurs at the expense of the donated conduction band electrons.
Figure 3.19 Compensation in an n-type semiconductor 45 ( N > N )
This mechanism can be visualized as follows: Assume an acceptor state is filled with a valence band electrons with a hole resulting in the valence band. This hole is then filled by recombination with one of the conduction band electrons. Extending this logic to all the acceptor atoms, it is expected that the resultant concentration of electrons in the conduction band to be N d N a instead of the total N d . This process is called compensati on. By this process it is possible to begin with an n-type semiconductor and add acceptors until N a= N d and no donated electrons remain in the conduction band. In such compensated material n0=ni= po, and intrinsic conduction is obtained. With further acceptor doping the semiconductor becomes p-type with a hole concentration of essentially N a- N d . 46
The exact relationship among the electron, hole, donor and acceptor concentrations can be obtained by considering the requirements for space charge neutrality. If the material is to remain electrostatically neutral, the sum of the positive charges (holes and ionized donor atoms) must balance the sum of the negative charges (electrons and ionized acceptor atoms):
p0 N d n0 N a (3.27) Thus in Fig 3.19 the net electron concentration in the conduction band is n p ( N N ) (3.28) 0 0 d a If the material doped n-type (n0>> p0) and all the impurities are ionized, we can approximate Eq. (3-28) by
n0 N d N a
47
Example: A Si is doped with 91015 cm-3 donors and 31015 acceptors. Find the position of the Fermi level at 40C. Solution: N d =91015 cm-3; N a=31015 cm-3; n i=1.51010 cm-3;
Since N d > N a, the Fermi level near to the donor states. We obtain, N rd = N d - N a=61015 cm-3 donor atoms; Thus the Fermi level depends on the donor atoms N rd . Now we consider n 0= N rd because N rd >>ni. T =(273+40)K=313K So we obtain, E F E i
kT lnn0 / ni
5
15
10
E F E i (8.62 10 eV / K )(313 K ) ln (6 10 ) /(1.5 10 ) E F E i 0.348 The resulting band diagram is
48
