Nearly Free Electron Gas model - I

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1 Nearly Free Electron Gas model - I Contents 1 Free electron gas model summary 1 2 Electron effective mass FEG model for sodium Nearly free electron model Primitive lattice Primitive lattice in 3D Reciprocal lattice Brillouin zone BZ in 3D Empty lattice approximation Reduced zone for a 1D lattice Reduced zone for a 2D lattice Free electron gas model summary The formation of energy bands in a metal can be explained by the delocalization of the valence electrons. The simplest model that explains the distribution of valence electrons in this energy band is called the free electron gas model (FEG model). In FEG, the delocalized valence electrons are not influenced by the atoms of the metal and are free to move within the metal. These electrons cannot all occupy the same energy, due to Pauli exclusion principle, and hence there is a distribution of energies, starting from the bottom of the band, up to a certain energy level. This energy level, that separates the filled and empty states, is called the Fermi level (E F ) and represents the highest filled energy level of the electron in the solid. This definition is valid, as long we have a 1

2 Figure 1: Free electron gas model, showing filled and empty states. The Fermi level and vacuum level are marked. There are empty states available for conduction. continuous distribution of energy levels, i.e. in a metal and if we ignore temperature effects i.e. take T = 0 K. The amount of energy required to remove an electron from the Fermi level into vacuum, is called the work function (φ). There are empty available energy levels between E F and φ and these make metals good conductors of electricity, see figure 1. Because of the large number of atoms that come together to form a solid, there are multiple states that have the same energy, i.e. degeneracy. The density of states (DOS) function, represented by g(e), is defined as the number of energy states per unit energy and per unit volume. In the FEG model, the solid is modeled as a three-dimensional particle in a box, and for this system, the DOS is given by g(e) = 8π 2 ( me h 2 ) 3/2 E (1) Thus, g(e), and hence degeneracy, increases monotonically with energy. The band diagram in figure 1 is valid when temperature is not taken into account. At a finite temperature, thermal energy (given by k B T, where k B is Boltzmann constant) can cause electrons to occupy energy levels above the Fermi level. The probability of electron occupying an energy level E at temperature 2

3 T is given by the Fermi function, f(e), f(e) = 1 ( ) (2) E EF 1 + exp k B T When (E E F ) k B T, equation 2 reduces to the simple Boltzmann distribution, p(e), [ p(e) = exp (E E ] F ) k B T The drawbacks of the FEG models is that while it can explain the high conductivity of metals it cannot explain the difference in conductivity between different metals. It does not take into account the crystal structure of the material and hence cannot explain the formation of energy band gaps in a solid. Thus, more sophisticated models have to be developed. 2 Electron effective mass Electrons in the valence band of a solid are delocalized and free to move. This is one of the central approximations of the FEG model. But, the solid atoms have a positive core containing the nucleus and inner shell electrons. These can influence how electrons move in the valence band. Consider a scenario when an external electric field is applied, as shown in figure 2. The electric field goes from positive to negative, so that the electrons travel in the direction opposite to the field. In vacuum, the acceleration of the electrons (a e ) is given by a e (3) = ee x m e (4) where E x is the electric field and m e is the rest mass of the electron (ignoring relativistic effects). This equation is also valid under the FEG model, where the internal potential in the solid is taken to be zero. In the case of an electron in a solid, the interaction with the solid ions should also be taken into account. Let ΣF int represent the sum of the interaction of the electron with the atoms in the solid. Then, the acceleration of the electron, given by using equation 4, is a e = ee x + ΣF int m e (5) 3

4 Figure 2: Electron motion in (a) vacuum and (b) solid. In a solid, the motion is affected by the interaction with the atoms. This is manifest as an effective mass of the electron. Adapted from Principles of Electronic Materials - S.O. Kasap. Equation 5 can be simplified by assigning an effective mass, m e, to the electron so that we can rewrite equation 5 simply as a e = ee x m e (6) The effective mass represents the effect of the internal forces on the motion of the electron. There is no change in the actual mass of the electron. Thus, by using the concept of effective mass it is still possible to use the FEG model, by replacing the mass of the electron (m e ) with effective mass (m e). For metals like Cu, Ag, or Au the valence electrons are nearly free and the effective masses are nearly equal to the rest mass of the electron. In the case of semiconductors, where there is substantial interaction between the electrons in the bonds and the lattice, there is a finite deviation. Some typical values of effective masses are tabulated in table FEG model for sodium Consider the calculation of Fermi energy, E F, for NA, using the FEG model. The experimental E F is 2.8 ev. To calculate E F using the FEG model, we can use the relation between the total number of electrons in a band, to 4

5 Table 1: Effective masses of electrons in some selected materials. For metals, effective masses are close to rest mass of electron, while there is a substantial deviation for semiconductors. m e m e Cu Ag Au Na m e Si Ge GaAs ZnO m e density of states and Fermi function. n = g(e) f(e) de (7) band At T = 0 K, this equation reduces to n = EF 0 g(e) de (8) In the FEG model, g(e) is given by equation 1. Substituting and integrating, gives the relation between n and E F as n = 16π 2 3 ( me h 2 ) 3/2 (E F ) 3/2 (9) n is the number of electrons per unit volume. Given that sodium has one electron per atom in the valence shell, from the atomic weight and density of sodium, it is possible to calculate n to be cm 3. This gives, using equation 9, an E F value of 3.15 ev, higher than the experimental value by 15%. But, if m e is replaced by the electron effective mass, m e, in sodium, table 1, then the calculated value is 2.7 ev, much closer to the experimental value. Thus, small deviations from the FEG model can be fixed by using the effective electron mass that includes the lattice effect. But this approach will only work for certain elements, where the lattice effect is weak, i.e. m e and m e are relatively close to each other. 3 Nearly free electron model The nearly free electron model explicitly considers the role of the lattice and consequently, the lattice potential, on the energies of the valence electrons. These electrons are still delocalized, but now a lattice potential is added to the system. The approach works well for crystalline solids, with strict 5

6 periodicity. Defects in the solid can affect the electron energies, but these are taken to be localized and do not affect the overall band picture of the solid. For simplicity, we will consider only ideal crystalline solid, i.e. with no defects. 3.1 Primitive lattice For a crystalline solid, it is possible to define a unit cell, with corresponding unit cell vectors. Consider a three-dimensional periodic array of atoms. Let the unit cell vectors be a, b, and c. Then, any atomic location in the lattice can be written in terms of these unit vectors as l = l1 a + l 2 b + l3 c (10) where l 1, l 2, and l 3 are integers. Thus, the unit cell is the building block of the lattice, since the entire lattice can be built by suitable translation of the unit cell. For a given lattice, it is also possible to define a primitive unit cell, or a minimum volume cell. If a, b, and c are lattice vectors for this primitive unit cell, then the volume is given by a ( b c). There are different ways of constructing the primitive cell. One such construction, is called the Wigner- Seitz cell (WS cell). This is formed by constructing the included area of the perpendicular bisector planes of the translation vectors from a chosen center to the nearest equivalent lattice site. Consider a one-dimensional lattice. This is an array of equally spaced points, with lattice constant a. The translation vectors go to the two nearest neighbors (on either side) and the perpendicular bisectors to these vectors form the WS cell, see figure 3. It is possible to extend this to a two-dimensional lattice, e.g. square lattice. Here, there are 2 lattice vectors and four lines to the nearest neighbors. The perpendicular bisectors to these neighbors also form a square, see figure 4. In both cases, the WS cell encloses one atom (or one basis). 3.2 Primitive lattice in 3D The same construction can be extended to 3D. Consider a simple cubic lattice, with one atom (basis) per unit cell, located at the corners. It is possible to construct a WS cell, from perpendicular bisector planes of vectors going along the sides of the cube. The resulting cell is also simple cubic, but the atom is located at the center of the cell. Consider next, the body centered cubic (bcc) lattice. The conventional 6

7 Figure 3: WS cell for a 1D lattice. If the spacing between lattice points is a, then the perpendicular bisectors are located at a/2 from the central point (marked in red). The WS cell is the region between the dotted lines. Figure 4: WS cell for a 2D lattice. If the spacing between lattice points is a, then the perpendicular bisectors are located at a/2 from the central point, marked in red. The WS cell is defined by the region enclosed by the dotted lines and is a square. 7

8 unit cell has 2 atoms, one at the corner and one at the body center i.e. (1/2, 1/2, 1/2). Hence, this is not a primitive unit cell. To construct the WS primitive cell, take the atom located at the body center to be the origin. The nearest neighbors, to this, are the corners and there are 8 corners. If the lattice constant of the bcc unit cell is a, it is possible to define three non-coplanar vectors going from the body center to the corners and these form the vectors of the primitive WS cell. a = a ( ˆx + ŷ + ẑ) 2 a b = (ˆx ŷ + ẑ) 2 c = a (ˆx + ŷ ẑ) 2 (11) where ˆx, ŷ, and ẑ are unit vectors along the three axes. All lattice vectors in the bcc lattice can be generated from the vectors defined above in equation 11 and these form the vectors of the WS cell. The volume of this WS cell is a 3 /2. Similarly, for a face centered cubic (fcc) lattice, the nearest neighbors go from the corner to the face center and there are 12 nearest neighbors. The lattice vector corresponding to the primitive WS cell are a = a (ˆx + ŷ) 2 a b = (ŷ + ẑ) 2 c = a (ẑ + ˆx) 2 (12) The volume of the primitive unit cell here is a 3 /4. The WS cell for bcc and fcc are shown in figure 5. 4 Reciprocal lattice The unit of lattice constant and the primitive vectors associated with the lattice is length, i.e. m or nm. This is called the real lattice. For each real lattice, it is possible to associate a mathematical construct, called the reciprocal lattice, whose dimensions are inverse length, i.e. m 1 or nm 1. The reason for creating the reciprocal lattice is that electron energies in a band are directly proportional to the wavevector, k, which has the dimensions of inverse length. In the FEG model, energy is given by E = 2 k 2 2m e (13) 8

9 Figure 5: WS cell for the (a) bcc lattice and (b) fcc lattice. Both lattices are non-primitive since they have more than one atom per unit cell. Hence, the nearest neighbors are first defined and lattice vectors to these are drawn. The perpendicular bisector planes to these lattice vectors form the WS cell. In the FEG model, only the magnitude of k mattered, since there was no lattice and hence no lattice directions or vectors. In the NFEG model, which includes the periodicity of the lattice, both magnitude and direction of the wavevector matter i.e. k rather than k. It would help to simplify the calculations, if there was a lattice that had the same units as k, i.e. a lattice with dimensions of inverse length. Hence, the concept of reciprocal lattice was introduced. Consider a real lattice, with primitive vectors given by a, b, and c. It is possible to define a reciprocal lattice associated with this, with three vectors a, b, and c, which are related to the real lattice vectors by a = 2π ( b c) a ( b c) b 2π ( c a) = a ( b c) c = 2π ( a b) a ( b c) (14) Here, a ( b c), is the volume of the primitive unit cell and a ( b c ) is the volume of the reciprocal lattice vector. The reciprocal and real lattice 9

10 Figure 6: (a) WS cell for a 1D lattice. (b) Corresponding 1D BZ. The spacing between the reciprocal lattice points is 2π/a and the BZ is constructed from perpendicular bisectors to the nearest neighbors. vectors have the following property a a = b b = c c = 2π a b = a c = b c = b a = c a = c b = 0 (15) Just as any lattice point in the real lattice can be described in terms of the primitive vectors, any reciprocal lattice point can be described in terms of the reciprocal lattice vectors as G = h a + k b + l c (16) where h, k, and l are integers. The primitive unit cell is called Wigner- Seitz cell in the real lattice and the reciprocal lattice counterpart is called a Brillouin zone (BZ). 4.1 Brillouin zone Consider the one dimensional real lattice shown in figure 3. The spacing between the lattice points is a. The reciprocal lattice is also one dimensional, with spacing given by 2π/a. Using the same construction as the WS cell, i.e. perpendicular bisectors to the nearest neighbors, it is possible to construct the BZ, figure 6. It is possible to extend this to 2D. A 2D rectangular real lattice and the corresponding reciprocal lattice are shown in figure 7. Some extra planes are shown for the reciprocal lattice to emphasize the difference in length scales when converting from real to reciprocal lattice. The same concept can be extended to 3D lattices. 10

11 Figure 7: (a) WS cell for a 2D rectangular lattice. (b) Corresponding 2D BZ. In the real lattice, if b > a, then in the reciprocal lattice, 2π/b < 2π/a. 4.2 BZ in 3D Consider the extension of reciprocal lattice and BZ to three dimensions. The reciprocal lattice of a simple cubic real lattice is also simple cubic. This can be easily shown using equation 14. For the simple cubic real lattice, the vectors are aˆx, aŷ, and aẑ, where a is the lattice constant. Using equation 14, the reciprocal lattice vectors are (2π/a) ˆx, (2π/a) ŷ, and (2π/a) ẑ, which is also a simple cubic lattice. Consider next, the fcc lattice. For calculating the BZ, we need the primitive lattice, i.e. the WS cell. We saw earlier that, we can define the primitive lattice based on the nearest neighbors, see equation 12. Now, using equation 14, it is possible to calculate the reciprocal lattice vectors for the fcc real lattice. These will be vectors for the primitive cell in reciprocal space. The calculated vectors are a = 2π a ( ˆx + ŷ + ẑ) b = 2π a c = 2π a (ˆx ŷ + ẑ) (ˆx + ŷ ẑ) (17) Comparing equation 17 and 11, both look similar, except for the constant term in the front. Thus, the fcc real lattice translates to a bcc reciprocal lattice. Similarly, if we start with a bcc real lattice, with primitive vectors given by equation 11, then the reciprocal lattice vectors calculated using 11

12 equation 14 are a = 2π a b = 2π a (ˆx + ŷ) (ŷ + ẑ) (18) c = 2π a (ẑ + ˆx) Again, comparing equations 18 and 12, both look similar except for the constant term. Thus, the bcc real lattice translates to the fcc reciprocal lattice. The shapes of the reciprocal lattices are given in figure 5. For the simple hexagonal lattice, the reciprocal lattice is also hexagonal. Knowing the reciprocal lattice vectors, it is possible to construct the Brillouin zone. This is done similar to the WS cell, i.e. identify the nearest neighbors and draw perpendicular bisector planes to the vectors connecting these nearest neighbors. Consider the bcc reciprocal lattice. The nearest neighbor from the center (origin) is the corner. The general reciprocal lattice vector, G, for this lattice is given by G = 2π a [( h + k + l)ˆx + (h k + l)ŷ + (h + k l)ẑ] (19) If the center of the bcc is taken as the origin, then the (hkl) values of one of the body diagonals is (100). The G for this is 2π/a ( hˆx + ŷ + ẑ) and has magnitude (2π/a) 3. Similarly, G 010, G 001, and G 111, are all body diagonals. Perpendicular bisector planes to these vectors form the BZ. 5 Empty lattice approximation The core of the NFEG model, is that for every real primitive lattice there is a reciprocal lattice, with dimensions of length 1, and with a general reciprocal lattice vector, G hkl. The advantage of using G is that any electron wavevector, K, can be written as the sum of a wavevector within the first Brillouin zone, k, and the general reciprocal lattice vector, G. Since, the wavevector is directly proportional to the energy, the electron energy vs. wavevector plot can be reduced to the first BZ. E = 2 K 2 2m e K = k + G E = 2 k + G 2 2m e (20) 12

13 The effect of the atoms of the lattice, is to alter these energies, creating forbidden energy gaps. In the simplest case, the atoms are considered to have zero potential, called empty lattice approximation. For this approximation, there are no forbidden gaps, and the K vectors are continuous. The empty lattice approximation can be used to understand how the E vs. K diagram can be reduced to the first BZ. Later, we will turn on the atomic potentials and show the formation of the forbidden gaps. 5.1 Reduced zone for a 1D lattice Consider a 1D real and reciprocal lattice, as shown in figure 6. There is only one direction and hence the general reciprocal lattice G is h (2π/a), where h is a real integer (positive or negative). Hence, the electron wavevector, K, can be written as K = [k + h ( 2π a )]ˆx with π a k π a (21) Thus, k lies in the first BZ. Thus, any K that lies outside the first BZ, can be brought back to the first BZ, by appropriate choice of k and G. Since, the electron energy depends on wavevector, see equation 20, the energy vs. wavevector plot can be reduced to the first BZ. Both the extended version, i.e. E vs. K and the reduced zone scheme, E vs. k are plotted in figure 8. Consider an electron wavevector, K = 3π/2a. Since, this is a 1D case, magnitude is sufficient. K can be written as 3π/2a = 1.(2π/a) π/2a. The first term represents the G and the second term represent k, which lies in the first BZ. Thus, this vector can be translated back to the first BZ, see figure 8. The same procedure can be adopted for other wavevectors. This type of energy band diagram, is called the reduced zone scheme. The scheme is especially useful for depicting energies in 2D and 3D lattices, since different reciprocal lattice directions can be depicted in the same plot. 5.2 Reduced zone for a 2D lattice Consider a 2D square real lattice, as shown in figure 4. The reciprocal lattice is also a square. Both can be seen in figure 7, where instead of a rectangle we can imagine a square with a = b. Since, this is a 2D lattice, there are two reciprocal lattice vectors and two integers, h, and l, that are used to write the general reciprocal lattice vector, G. G = 2π a (hˆx + lŷ) (22) 13

14 Figure 8: Energy vs. wavevector for a 1D lattice, showing both the extended and the reduced zone scheme. In the reduced zone scheme, all the wavevectors are brought within the first BZ, by appropriate choice of G. Now, multiple directions are possible for K, so that equation 20 can be written as E = K = k + G k = kxˆx + k y ŷ K = (k x + h 2π a )ˆx + (k y + l 2π a )ŷ E = 2 k + G 2 2m e 2 2m e [(k x + h 2π a )2 + (k y + l 2π a )2 ] (23) k x and k y lie in the first BZ, so that they have values between π/a k x, k y π/a. It is now possible to construct different Evs. k diagrams for the different directions of k. Consider the [10] direction in the first BZ. This corresponds to k y = 0 and π/a k x π/a and is represented by Γ X in figure 9. The energy can be written by modifying equation 23 as ( ) 2 E = 2 2π [(k x + h) 2 + l 2 ] (24) 2m e a with the new limits for k x being 1/2 k x 1/2. Thus, it is possible to tabulate the energies for different values of G, see table 2. This data can 14

15 Figure 9: The first BZ in a 2D lattice, showing the k x and k y vectors. The boundaries of the BZ are marked. Γ X is the [10] direction, while Γ C is the [11] direction. Table 2: E vs. k table for different values of G. k is along the [10] direction. G hl E(k) = (k x + h) 2 + l 2 E at Γ E at X units of 2π/a units of 2 /2m (2π/a) 2 (k x = 0) (k x = 1/2) (0,0) kx 2 0 1/4 (1,0) (k x + 1) 2 1 9/4 (-1,0) (k x 1) 2 1 1/4 (0,±1) kx /4 15

16 Figure 10: Energy vs. k for two directions in a 2D reciprocal lattice (a) [10] or Γ X direction (b) [11] or Γ C direction be plotted as energy vs. k x for the [10] direction, see figure 10(a). A similar table and plot can be made for the [11] direction. In this direction, k x = k y and the energy can be written as E = 2 2m e ( ) 2 2π [(k x + h) 2 + (k y + l) 2 ] with k x = k y (25) a Again, the energies can be tabulated for different values of G and plotted, see figure 10(b). The advantage of the reduced zone scheme is that different directions can be plotted together in the same plot, figure 11. Here, both Γ C and Γ X are plotted together. The same can be extended to 3D lattices, G will have three components and the k will also have three components. It is also possible to plot E vs. k for different directions, in the same plot, similar to 2D lattices. The energy band information will be useful when energy gaps are created on application of an atomic potential. When these energy gaps overlap, a forbidden energy gap is created. This is called band gap. On the other hand, when energy gaps are small and do not overlap, then electrons have a continuous pathway for movement. This has implications for the electrical conductivity of the material. 16

17 Figure 11: Energy vs. k for two directions in a 2D reciprocal lattice Γ X direction and Γ C direction, plotted in the same diagram. 17