Chapter 1 are of interest. These are the energy levels where free moving electrons and missing valence electrons are found. in that case. as can be seen in Figure 1. 15, the energy dependence on momentum can be approximated by a square parabolic function. Near the minimum of the conduction band one can thus write E(k)=Emin A(k-kmir (11.34a) Near the maximum of the valence band one can write E(k)=Emax -b(k-kmax (1.1.34b) with A and B being constants. This approximation is called the " parabolic band approximation"and resembles the e(k)relationship found for the free electron model 1. 1.6. Concept of a hole To facilitate the understanding of electrical conduction in a solid one can make a comparison between the flow of electrical charge in the energy bands and the movement of water drops in a pipe. Let us consider (Figure 1. 16.A) two pipes which are sealed at both ends. The bottom pipe is completely filled with water and the top pipe contains no water(it is filled with air). In our analogy between electricity and water, each drop of water corresponds to an electron, and the bottom and top pipes correspond to the valence and the conduction band, respectively. [] Tilting the pipes corresponds to the application of an electric field to the semiconductor. When the filled or empty pipes are tilted, no movement or flow of water is observed. i.e.. there is no electric current flow in the semiconductor. Thus the semiconductor behaves as an insulator(Figure 1.16A) Let us now remove a drop of water from the bottom pipe and place it in the pe,which corresponds to "moving"an electron from the valence to the conduction band pipes are now tilted, a net flow of liquid will be observed, which correspond to an electrical current flow in the semiconductor(Figure 1. 16. B) The water flow he top pipe (conduction band) is due to the movement of the water drop(electron). In addition, there is also water flow in the bottom pipe (valence band) since drops of water can occupy the space left behind as the air bubble moves. It is, however, easier to visualize the motion of the bubble itself instead of the movement of the ater
20 Chapter 1 are of interest. These are the energy levels where free moving electrons and missing valence electrons are found. In that case, as can be seen in Figure 1.15, the energy dependence on momentum can be approximated by a square parabolic function. Near the minimum of the conduction band one can thus write: Near the maximum of the valence band one can write: with A and B being constants. This approximation is called the "parabolic band approximation" and resembles the E(k) relationship found for the free electron model. 1.1.6. Concept of a hole To facilitate the understanding of electrical conduction in a solid one can make a comparison between the flow of electrical charge in the energy bands and the movement of water drops in a pipe. Let us consider (Figure 1.16.A) two pipes which are sealed at both ends. The bottom pipe is completely filled with water and the top pipe contains no water (it is filled with air). In our analogy between electricity and water, each drop of water corresponds to an electron, and the bottom and top pipes correspond to the valence and the conduction band, respectively.[9] Tilting the pipes corresponds to the application of an electric field to the semiconductor. When the filled or empty pipes are tilted, no movement or flow of water is observed, i.e.: there is no electric current flow in the semiconductor. Thus the semiconductor behaves as an insulator (Figure 1.16.A). Let us now remove a drop of water from the bottom pipe and place it in the top pipe, which corresponds to "moving" an electron from the valence to the conduction band. If the pipes are now tilted, a net flow of liquid will be observed, which correspond to an electrical current flow in the semiconductor (Figure 1.16.B). The water flow in the top pipe (conduction band) is due to the movement of the water drop (electron). In addition, there is also water flow in the bottom pipe (valence band) since drops of water can occupy the space left behind as the air bubble moves. It is, however, easier to visualize the motion of the bubble itself instead of the movement of the "valence" water
1. Energy Band Theory 21 If, in this water analogy, an electron is represented by a drop of water, a bubble or absence of water in the"valence"pipe represents what is called a hole. Hence, a hole is equivalent to a missing electron in the crystal valence band. a hole is not a particle and it does not exist by itself. It draws its existence from the absence of an electron in the crystal, just like a bubble in a pipe exists only because of a lack of water. Holes can move in the crystal through successive "filling"of the empty space left by a missing electron. The hole carries a positive charge +q, as the electron carries a negative charge-g(g=1.6 10-19 Coulomb) Figure 1. 16: Energy bands and electrical conduction: water analogy 1. 1.7. Effective mass of the electron in a crvstal The mass m of an electron can be defined by the where a is the acceleration the electron undergoes under the influence of an external applied force F. The fact that the electron is in a crystal will influence its response to an applied force. As a result, the apparent, effective"mass of the electron in a crystal will be different from that of an electron in a vacuum In the case of a free electron Relationship 1.1.3 can be used to find the mass of the electron [10] 加2k2 E= 2m d 2e/dk (1.1.35) where m=mo=9.11x10-28 gram is the mass of the electron in a vacuum The mass is a constant since E is a square function of k. Using the rightmost term of 1. 1.35 as the definition of the electron mass and using Equations 1. 1.28 and 1.1.29 which defines the relationship
1. Energy Band Theory 21 If, in this water analogy, an electron is represented by a drop of water, a bubble or absence of water in the "valence" pipe represents what is called a hole. Hence, a hole is equivalent to a missing electron in the crystal valence band. A hole is not a particle and it does not exist by itself. It draws its existence from the absence of an electron in the crystal, just like a bubble in a pipe exists only because of a lack of water. Holes can move in the crystal through successive "filling" of the empty space left by a missing electron. The hole carries a positive charge +q, as the electron carries a negative charge -q Coulomb). 1.1.7. Effective mass of the electron in a crystal The mass m of an electron can be defined by the relationship F=ma where a is the acceleration the electron undergoes under the influence of an external applied force F. The fact that the electron is in a crystal will influence its response to an applied force. As a result, the apparent, "effective" mass of the electron in a crystal will be different from that of an electron in a vacuum. In the case of a free electron Relationship 1.1.3 can be used to find the mass of the electron where gram is the mass of the electron in a vacuum. The mass is a constant since E is a square function of k. Using the rightmost term of 1.1.35 as the definition of the electron mass and using Equations 1.1.28 and 1.1.29 which defines the relationship
Chapter 7 between E and k in a one-dimensional crystal, the mass of an electron within an energy band can be calculated PBE)=ca+ b) and m*s、 (1.1.36) where m* is called effective mass"of the electron in a crystal Unlike the case of a free electron the effective mass of the electron in a crystal is not constant, but it varies as a function of k(Figure 1. 17) 117/ Figure 1. 17: Electron energy and effective a one- dimensional crystal. The first Brillouin zone( shown in gray Additionally, the mass in the crystal will be different for differing energy bands. The following general observations can be made o if the electron is in the upper half of an energy band, its effective mass is negative o if the electron is in the lower half of an energy band, its effective mass is positive
22 Chapter 1 between E and k in a one-dimensional crystal, the mass of an electron within an energy band can be calculated: where m* is called the "effective mass" of the electron in a crystal. Unlike the case of a free electron the effective mass of the electron in a crystal is not constant, but it varies as a function of k (Figure 1.17). Additionally, the mass in the crystal will be different for differing energy bands. The following general observations can be made: if the electron is in the upper half of an energy band, its effective mass is negative if the electron is in the lower half of an energy band, its effective mass is positive
1. Energy Band Theory o if the electron is near the middle of an energy band, its effective mass tends to be infinite The negative mass of electrons located in the top part of an energy band may come as a surprise, but can easily be explained using the concept of a hole. Let us consider the acceleration, a, given to an electron with charge - q and negative mass, -m*, by an electric field, C. It is easy to realize that this acceleration corresponds to a hole with positive mass, +m*, and positive charge, +q, since F-q 96 a with m*>0 (1.1.37) In the case of a three-dimensional crystal the expression of the effective mass is more complicated because the acceleration of an electron can be in a direction different from that of the applied force. In that case the effective mass is expressed by a 3x3 tensor with mxx etc. (11.38) Usually physics of semiconductor devices deals only with electrons situated near the minimum of the conduction band or holes located near the maximum of the valence band in the case of silicon the mass of electrons near the minimum of the conduction band along the [100]kx- direction is equal to mI=0.97 mo, and in the orthogonal directions it is m,=0.19 mo. mi is called the longitudinal mass and m, the transversal mass, while mo is the mass of a free electron in a vacuum. These masses are related to the energy by the following relationship called parabolic energy band approximation E(k)=Ec(k/+.:( x-km x2+.+ny (11.39) 2 2m where Ec(km) is the lowest energy state in the conduction band along the [100] or [-100] kx-directions(Figure 1. 18). In most practical cases, for the sake of simplicity, the effective mass is considered to be constant. In that case m is approximated by a scalar value
1. Energy Band Theory 23 if the electron is near the middle of an energy band, its effective mass tends to be infinite The negative mass of electrons located in the top part of an energy band may come as a surprise, but can easily be explained using the concept of a hole. Let us consider the acceleration, a, given to an electron with charge -q and negative mass, -m *, by an electric field, It is easy to realize that this acceleration corresponds to a hole with positive mass, +m*, and positive charge ,+q, since: In the case of a three-dimensional crystal the expression of the effective mass is more complicated because the acceleration of an electron can be in a direction different from that of the applied force. In that case the effective mass is expressed by a 3×3 tensor: Usually physics of semiconductor devices deals only with electrons situated near the minimum of the conduction band or holes located near the maximum of the valence band. In the case of silicon the mass of electrons near the minimum of the conduction band along the direction is equal to and in the orthogonal directions it is is called the longitudinal mass and the transversal mass, while is the mass of a free electron in a vacuum. These masses are related to the energy by the following relationship called "parabolic energy band approximation": where is the lowest energy state in the conduction band along the [100] or [-100] (Figure 1.18). In most practical cases, for the sake of simplicity, the effective mass is considered to be constant. In that case m * is approximated by a scalar value
0 Figure 1. 18: Energy bands E(k) along the [100] and [-100 crystallographic directions in silicon. Eg1.12 eV In a one-dimensional case the square-law dependence of the energy on k E(k)=Ec(km)+-*(hx-km, x 2 is illustrated by Figure 1. 19.A There are 2 two vectors km t dk and km -dk which correspond to a same energy value Ec(km dk). In a two-dimensional crystal(Figure 1. 19. B )the locus of (kx, ky) values corresponding to the energy level Ec(km dk )is an ellipse in the(kx, ky) pl: The three-dimensional case cannot be drawn on a sheet of paper, but extrapolating from the ID and 2d cases it is easy to conceive that the k values corresponding to the energy level Ec(km dk form ellipsoids in the(kx, ky, kz)space(Figure 1.19.C). In a three-dimensional crystal such as silicon there are 6 equivalent crystal directions ([100],[-1001,[010], [0-10], [001] and [00-lD which present an energy minimum(conduction are the six k-values corresponding to the conduction band energy minima. For simplification the ellipsoids can be approximated by spheres(Figure 1.19. D), which is equivalent to equating the transverse and th longitudinal mass (mI=m,). The energy in the vicinity of the maximum of the valence band is given by E(k)=Ev0)-,*(kx+k+ (1.1.40)
24 Chapter 1 In a one-dimensional case the square-law dependence of the energy on k , is illustrated by Figure 1.19.A There are two vectors and which correspond to a same energy value In a two-dimensional crystal (Figure 1.19.B) the locus of values corresponding to the energy level is an ellipse in the plane. The three-dimensional case cannot be drawn on a sheet of paper, but extrapolating from the 1D and 2D cases it is easy to conceive that the k values corresponding to the energy level form ellipsoids in the space (Figure 1.19.C). In a three-dimensional crystal such as silicon there are 6 equivalent crystal directions ([100], [-100], [010], [0-10], [001] and [00-1]) which present an energy minimum (conduction band minimum). The locus of k-values corresponding to a particular energy value is 6 ellipsoids (Figure 1.19.C). The center of these ellipsoids are the six k-values corresponding to the conduction band energy minima. For simplification the ellipsoids can be approximated by spheres (Figure 1.19.D), which is equivalent to equating the transverse and the longitudinal mass The energy in the vicinity of the maximum of the valence band is given by: