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Every solid has its own characteristic energy band structure. This variation in band structure is responsible for the wide range of electrical characteristics observed in various materials. The diamond band structure for example, can give a good picture of why carbon in the diamond lattice is a good insulator. To reach such a conclusion, we must consider the properties of completely filled and completely empty energy bands in the current conduction process.
Before discussing the mechanisms of current flow in solids further, we can observe here that for electrons to experience acceleration in an applied electric field, they must be able to move into new energy states. This implies there must be empty states (allowed energy states which are not already occupied by electrons) available to the electrons. For example, if relatively few electrons reside in an otherwise empty band, ample unoccupied states are available into which the electrons can move. On the other hand, the diamond structure is such that the valence band is completely filled with electrons at 0 K and the conduction band is empty. There can be no charge transport within the valence band, since no empty states are available into which electrons can move. There are no electrons in the conduction band, so no charge transport can take place there either. Thus carbon in the diamond structure has a high resistivity typical of insulators.
Semiconductor materials at 0 K have basically the same structure as insulatorsa filled valence band separated from an empty conduction band by a band gap containing no allowed energy states. The difference lies in the size of the band gap E_{g} which is much smaller in semiconductors than in insulators. For example, the semiconductor Si has a band gap of about 1.1 eV compared with 5 eV for diamond. The relatively small band gaps of semiconductors allow for excitation of electrons from the lower (valence) band to the upper (conduction) band by reasonable amounts of thermal or optical energy.
For example, at room temperature a semiconductor with a 1 eV band gap will have a significant number of electrons excited thermally across the energy gap into the conduction band whereas an insulator with E_{g} = 10eV will have a negligible number of such excitations. Thus an important difference between semiconductors and insulators is that the number of electrons available for conduction can be increased greatly in semiconductors by thermal or optical energy.
In metals the bands 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. As expected from the metallic band structures, metals have a high electrical conductivity.
When quantitative calculations are made of band structures, 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 a wave vector. The spacedependent wave function for the electron is
where the function U (k_{x}, x) modulates the wave function according to the periodicity of the lattice.
In such a calculation, allowed values of energy can be plotted vs. the propagation constant k . Since the periodicity of most lattices is different in various directions, the ( E ,k) diagram must be plotted for the various crystal directions and the full relationship between E and k is a complex surface which should be visualized in three dimensions.
The band structure of GaAs has a minimum in the conduction band and a maximum in the valence band for the same k value ( k = 0) . On the other hand, Si has its valence band maximum at a different value of k than its conduction band minimum. Thus an electron making a smallestenergy transition from the conduction band to the valence band in GaAs can do so without a change in k value; on the other hand a transition from the minimum point in the Si conduction band to the maximum point of the valence band requires some change in k . Thus there are two classes of semiconductor energy bands direct and indirect. We can show that an indirect transition involving a change in k requires a change of momentum for the electron.
Figure: Direct and indirect electron transitions in semiconductors: (a) direct transition with accompanying photon emission; (b) indirect transition via a defect level.
In 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. On the other hand, an electron in the conduction band minimum of an indirect semiconductor such as Si cannot fall directly to the valence band maximum but must undergo a momentum change as well as changing its energy. For example, 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 an emitted photon. This difference between direct and indirect band structures is very important for deciding which semiconductors can be used in devices requiring light output. For example, semiconductor light emitters and lasers generally must be made of materials capable of direct bandtoband transitions or of indirect materials with vertical transitions between defect states.
As the temperature of a semiconductor is raised from 0 K , some electrons in the valence band receive enough thermal energy to be excited across the band gap to the conduction band. The result is a material with some electrons in an otherwise empty conduction band and some unoccupied states in an otherwise filled valence band. For convenience, an empty state in the valence band is referred to as a hole. If the conduction band electron and the hole are created by the excitation of a valence band electron to the conduction band, they are called an electronhole pair (abbreviated EHP).
After excitation to the conduction band, an electron is surrounded by a large number of unoccupied energy states. For example, the equilibrium number of electronhole pairs in pure Si at room temperature is only about 10^{10} EHP / cm^{3} , compared to the Si atom density of more than10^{22} atoms / cm^{3} . Thus the few electrons in the conduction band are free to move about via the many available empty states.
➤ Effective Mass
The electrons in a crystal are not completely free, but instead interact with the periodic potential of the lattice. As a result, their “waveparticle” motion cannot be expected to be the same as for electrons in free space. Thus, in applying the usual equations of electrodynamics to charge carriers in a solid, we must use altered values of particle mass. In doing so, we account for most of the influences of the lattice, so that the electrons and holes can be treated as “almost free” carriers in most computations. The calculation of effective mass must take into account the shape of the energy bands in threedimensional k space, taking appropriate averages over the various energy bands.
Example: Find the ( E ,k) relationship for a free electron and relate it to the electron mass.
Solution: The electron momentum is p = . Then
Thus the electron energy is parabolic with wave vector k .
The electron mass is inversely related to the curvature (second derivative) of the ( E ,k) relationship, since
Although electrons in solids are not free, most energy bands are close to parabolic at their minima (for conduction bands) or maxima (for valence bands). We can also approximate effective mass near those band extrema from the curvature of the band.
The effective mass of an electron in a band with a given ( E ,k ) relationship is given by
A particularly interesting feature is that the curvature d^{2}E / dk^{2} is positive at the conduction band minima, and is negative at the valence band maxima. Thus, the electrons near the top of the valence band have negative effective mass. Valence band electrons with negative charge and negative mass move in an electric field in the same direction as holes with positive charge and positive mass. We can fully account for charge transport in the valence band by considering hole motion.
In any calculation involving the mass of the charge carriers, we must use effective mass values for the particular material involved. Table given below lists the effective masses for Ge , Si , and GaAs appropriate for one type of calculation. In this table and in all subsequent discussions, the electron effective mass is denoted by m_{n}* and the hole effective mass by m*_{p} . The n subscript indicates the electron as a negative charge carrier, and the p subscript indicates the hole as a positive charge carrier (The free electron rest mass is m_{0}).
A perfect semiconductor crystal with no impurities or lattice defects is called an intrinsic semiconductor. In such material there are no charge carriers at 0 K , since the valence band is filled with electrons and the conduction band is empty. At higher temperatures electronhole 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.
The generation of EHPs can be visualized in a qualitative way by considering the breaking of covalent bonds in the crystal lattice. If one of the Si valence electrons is broken away from its position in the bonding structure such that it becomes free to move about in the lattice, a conduction electron is created and a broken bond (hole) is left behind. The energy required to break the bond is the band gap energy E_{g} . This model helps in visualizing the physical mechanism of EHP creation, but the energy band mode is more productive for purposes of quantitative calculation. One Important difficulty in the “broken bond” model is that the free electron and the hole seem deceptively localized in the lattice. Actually, the positions of the free electron and the hole are spread out over several lattice spacing and should be considered quantum mechanically by probability distributions.
Figure: Electronhole pairs in the covalent bonding model of the Si crystal.
Since the electrons and holes are created in pairs, the conduction band electron concentration n (electrons per cm^{3}) is equal to the concentration of holes in the valence band p (holes per cm^{3}). Each of these intrinsic carrier concentrations is commonly referred to as n_{i}. Thus for intrinsic material
n = p = n_{i}.
At a given temperature there is a certain concentration of electronhole pairs ni . Obviously, if a steady state carrier concentration is maintained, there must be recombination of EHPs at the same rate at which they are generated. Recombination occurs when an electron in the conduction band makes a transition (direct or indirect) to an empty state (hole) in the valence band, thus annihilating the pair. If we denote the generation rate of EHPs as gi, (EHP/cm^{3}) and the recombination rate as r_{i} , equilibrium requires that:
r_{i} =g_{i}
Each of these rates is temperature dependent. For example, g_{i} (T ) increases when the temperature is raised, and a new carrier concentration n_{i} , is established such that the higher recombination rate r_{i} (T) just balances generation. At any temperature, we can predict that 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 p_{0}
The factor α_{r} is a constant of proportionality which depends on the particular mechanism by which recombination takes place.
In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductor purposely by introducing impurities into the crystal. This process, called doping is the most common technique for varying the conductivity of semiconductors. By doping, a crystal can be altered so that it has a predominance of either electrons or holes. Thus there are two types of doped semiconductors, ntype (mostly electrons) and ptype (mostly holes).
When impurities or lattice defects are introduced into an otherwise perfect crystal, additional levels are created in the energy band structure usually within the band gap. For example, 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. This level is filled with electrons at 0 K , and very little thermal energy is required to excite these electrons to the conduction band. Thus at about 50 K − 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. From figure 1.5, we note that the material doped with donor impurities can have a considerable concentration of electrons in the conduction band, even when the temperature is too low for the intrinsic EHP concentration to be appreciable. Thus semiconductors doped with a significant number of donor atoms will have n_{0} >> (n_{i} , p_{0}) at room temperature. This is ntype material.
Atoms from column III (B, Al, Ga, and In) introduce impurity levels in Ge or Si near the valence band. These levels are empty of electrons at 0 K . At low temperatures, enough thermal energy is available to excite electrons from the valence band 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 Ge and Si. Figure indicates, doping with acceptor impurities can create a semiconductor with a hole concentration p_{0} much greater than the conduction band electron concentration n_{0} (this is ptype material).
Figure : Acceptance of valence band electrons by an acceptor level, and the resulting creation of holes.
Electrons in solids obey FermiDirac statistics. In the development of this type of statistics, one must consider the indistinguishability of the electrons, their wave nature, and the Pauli Exclusion Principle. The rather simple result of these statistical arguments is that the distribution of electrons over a range of allowed energy levels at thermal equilibrium is:
where k is Boltzmann constant. The function f (E), the FermiDirac 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. We notice that, for an energy E equal to the Fermi level energy E_{F}, the occupation probability is
A closer examination of f (E) indicates that at 0 K the distribution takes the simple rectangular form shown in figure. With T = 0 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 0 K every available energy state up to E_{F} is filled with electrons and all states above E_{F} are empty.
At temperatures higher than 0 K , some probability exists for states above E_{F} to be filled. For example, at T =T_{1} 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 Fermi function is symmetrical about E_{F} for all temperatures; that is the probability f (E_{F} +ΔE) that a state ΔE above EF is filled is the same as the probability [1f(E_{f}ΔE) that a state ∆E below E_{F} is 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 concentrations in semiconductors.
For intrinsic material we know that the concentration of holes in the valence band is equal to the concentration of electrons in the conduction band. Therefore, the Fermi level EF must lie at the middle of the band gap in intrinsic material. Since f(E) is symmetrical about E_{F}, the electron probability "tail" of f (E) extending into the conduction band is symmetrical with the hole probability tail [1f(E)] in the valence band. The distribution function has values within the band gap between E_{c} and E_{v} but there are no energy states available, and no electron occupancy results from f (E) in this range.
In ntype material there is a high concentration of electrons in the conduction band compared with the hole concentration in the valence band. Thus in ntype material the distribution function f ( E) must lie above its intrinsic position on the energy scale. Since f (E) retains its shape for a particular temperature, the larger concentration of electrons at Ec in ntype material implies a correspondingly smaller hole concentration at E_{v} . We notice that the value of f (E) for each energy level in the conduction band (and therefore the total electron concentration n_{0} ) increases as E_{F} moves closer to E_{c} . Thus the energy difference ( E_{c} −E_{F}) gives a measure of n .
For ptype material the Fermi level lies near the valence band such that the [1 − f (E)] tail below E_{v} is larger than the f ( E ) tail above E_{c} . The value of ( E_{F} −E_{v}) indicates how strongly ptype the material is.
It is usually inconvenient to draw f (E)vs. E on every energy band diagram to indicate the electron and hole distributions. Therefore, it is common practice merely to indicate the position of E_{F} in band diagrams.
➤ Electron and Hole Concentrations at Equilibrium
The Fermi distribution function can be used to calculate the concentrations of electrons and holes in a semiconductor, if the densities of available states in the valence and conduction bands are known. For example, the concentration of electrons in the conduction band is
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 (n_{0}, p_{0}) indicates equilibrium conditions. The number of electrons per unit volume in the energy range dE is the product of the density of states and the probability of occupancy f ( E ) . Thus the total electron concentration is the integral over the entire conduction band. The function N (E) can be calculated by using quantum mechanics and the Pauli Exclusion Principle.
Since N (E) is proportional to E^{1/ 2} , so the density of states in the conduction band increases with electron energy. On the other hand, the Fermi function becomes extremely small for large energies. The result is that the product f (E)N(E) decreases rapidly above Ec and very few electrons occupy energy states far above the conduction band edge. Similarly, the probability of finding an empty state (hole) in the valence band [1 − f(E)] decreases rapidly below E_{v} and most holes occupy states near the top of the valence band. This effect is demonstrated, which shows the density of available states, the Fermi function, and the resulting number of electrons and holes occupying available energy states in the conduction and valence bands at thermal equilibrium (i.e., with no excitations except thermal energy). For holes, increasing energy points down, since the E scale refers to electron energy.
Figure : Schematic band diagram, density of states, FermiDirac distribution, and the carrier concentrations for (a) intrinsic, (b) ntype, and (c) ptype semiconductors at thermal equilibrium.
The result of the integration of is the same as that obtained if we represent the entire distributed electron states in the conduction band by an effective density of states N_{c} located at the conduction band edge E_{c} . Therefore, the conduction band electron concentration is simply the effective density of states at E_{c }times the probability of occupancy at E_{c}
n_{0} =f (E_{c}) N_{c}
In this expression we assume the Fermi level E_{F} lies at least several kT below the conduction band. Then the exponential term is large compared with unity and the Fermi function f(E_{c}) can be simplified as
Since kT at room temperature is only 0.026 eV , this is generally a good approximation. For this condition the concentration of electrons in the conduction band is
The effective density of states
Thus electron concentration increases as E_{F} moves closer to the conduction band. By similar arguments, the concentration of holes in the valence band is p_{0} = N[1 −f(E_{v})]
where N_{c} is the effective density of states in the valence band.
The probability of finding an empty state at E_{v} is,
for E_{F} larger than E_{v} by several kT.
From these equations, the concentration of holes in the valence band is
The effective density of states in the valence band
Thus hole concentration increases as E_{F} moves closer to the valence band.
The electron and hole concentrations predicted by above equations 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
Note: The intrinsic level E_{i} is the middle of the band gap if the effective densities of states N_{c} and N_{c} are equal. There is usually some difference in effective mass for electrons and holes, however, and N_{c} and N_{c} are slightly different.
The product of n_{0} and p_{0} 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 (since the carriers are created in pairs), n_{i} =p_{i} ; thus the intrinsic concentration is
Law of Mass Action
The constant product of electron and hole concentrations can be written conveniently as
For ntype material the minority concentration (holes)
where N_{D} is donor ion concentration.
For ptype material the minority concentration (electrons)
where N_{A} is acceptor ion concentration.
Another convenient way of writing electron and hole concentration is
This form of the equation indicates directly that the electron concentration is n_{i} when E_{F} is at the intrinsic level E_{i} and that n_{0} increases exponentially as the Fermi level moves away from E_{i} toward the conduction band. Similarly, the hole concentration p_{0} varies from n_{i} to larger values as E_{F} moves from E_{i}, toward the valence band. Since these equations reveal the qualitative features of carrier concentration so directly, they are particularly convenient to remember.
The variation of carrier concentration with temperature is indicated by equations. Initially, the variation of n_{0} and p_{0} with T seems relatively straightforward in these relations. The problem is complicated, however, by the fact that n_{i} has strong temperature dependence and that F_{E} can also vary with temperature. Let us begin by examining the intrinsic carrier concentration.
The exponential temperature dependence dominates n_{i}(T) and a plot of ln (n_{i}) vs 1000/T appears almost linear.
Figure : Intrinsic carrier concentration for Ge, Si, and GaAs as a function of inverse temperature. The room temperature values are marked for reference.
Figure the illustrates a semiconductor for which both donors and acceptors are present, but N_{D} >N_{A}. The predominance of donors makes the material ntype and the Fermi level is therefore in the upper part of the band gap. Since E_{F} is well above the acceptor level E_{a}, this level is essentially filled with electrons. However, with E_{F} above E_{i} we cannot expect a hole concentration in the 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.
The mechanism can be visualized as follows: Assume an acceptor state is filled with a valence band electron, 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, we expect 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 compensation. By this process it is possible to begin with an ntype semiconductor and add acceptors until N_{A} = N_{D} and no donated electrons remain in the conduction band. In such compensated material n_{0} = n_{i }= p_{0} and intrinsic conduction is obtained. With further acceptor doping the semiconductor becomes ptype with a hole concentration of essentially N_{A} − N_{D}.
Figure : Compensation in an ntype semiconductor (N_{D} > N_{A}).
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):
Thus the net electron concentration in the conduction band is
If the material is doped ntype ( n_{0} >>p_{0}) and all the impurities are ionized, we can approximate that n_{0} = N_{D} − N_{A} .
Since the intrinsic semiconductor itself is electrostatically neutral and the doping atoms we add are also neutral, the requirement of equation must be maintained at equilibrium.
Knowledge of carrier concentrations in a solid is necessary for calculating current flow in the presence of electric or magnetic fields. In addition to the values of n and p, we must be able to take into account the collisions of the charge carriers with the lattice and with the impurities. These processes will affect the ease with which electrons and holes can flow through the crystal, that is, their mobility within the solid. As should be expected, these collision and scattering processes depend on temperature, which affects the thermal motion of the lattice atoms and the velocity of the carriers.
Example: The donor concentration in a sample of n type silicon is increased by a factor of 100. Find the shift in the position of the Fermi level at 300 K . (k_{B}T = 25 meV at 300K)
Solution:
Thus shift is ΔE= kT ln (100) = 25 ln (100 ) meV = 115.15 meV
Example: A Si sample is doped with 10^{17} As atoms/cm^{3}. What is the equilibrium hole concentration p_{0} at 300 K? Where is E_{F} relative to E_{i} ? (where n_{i} = 1.5×10^{10}cm^{−3})
Solution: Since N_{D} >> n_{i} we can approximate n_{i} and
Example: A pure Si sample at 300K with intrinsic carrier concentration of 1.5 × 10^{16} / m^{3} is doped with phosphorous. The equilibrium hole concentration and electron mobility is 3
2.25 × 10^{9} / m^{3} and 1350 cm^{2} / Vs respectively. Find the position of Fermilevel relative to the intrinsic level at 300 K .
Solution: Equilibrium electron concentration is
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