Nearly Free Electron Gas model - II

Transcription

1 Nearly Free Electron Gas model - II Contents 1 Lattice scattering Bloch waves Band gap formation Electron group velocity and effective mass Fermi wavevector and BZ boundary 7 3 NFEG and DOS 12 4 Experimental measurement tools Hall measurements Ultraviolet photoelectron spectroscopy Cyclotron resonance Lattice scattering In part I, we have seen how to construct energy band diagrams for an empty lattice. In an empty lattice, the k values that the electrons can have are continuous, i.e. there is no effect of the atoms. This is because the atom potentials have been neglected in the empty lattice model. Now, we will consider the effect of the lattice. Let us start with the simple 1D lattice, as seen in figure 1. The atoms in the lattice have a potential. Also, since the lattice is periodic the potential is also periodic, given by the lattice constant a. The effect of this periodic potential is to modify the electron wavefunction in the lattice. In the FEG, with the empty lattice approximation, valence electrons are delocalized and form an energy band, with continuous values of energies. Electrons in these bands have an energy or wavevector distribution and are represented by a plane wave equation and these electrons are called free 1

2 Figure 1: 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 atom (marked in red). The WS cell is the region between the dotted lines. electrons. ψ(x) = A exp(ikx) (1) Consider now, electrons in a lattice. For small energies and correspondingly small k values, the electron waves have a large wavelength, since k = 2π/λ. These electrons see the average of several atoms and are only weakly affected by the periodic atomic potentials. For these electrons, the plane wave equation 1 is still valid. 1.1 Bloch waves 1 As the electron energy increases, k increases, the wavelength decreases, and becomes comparable to the spacing between the atoms. Now, the electrons are no longer free but are influenced by the atomic potentials. This leads to a modification of the traveling wavefunction. The potential in a solid is a periodic function i.e. for a 1D solid U(x) = U(x + na), where n is an integer (positive or negative) and a is the spacing between the atoms. For a general 3D solid, this can be written as U( r) = U( r + R), where R is a general lattice vector. Because, of the periodic nature of the potential, the electron wavefunction should also be periodic. In general, the wavefunction is written as a product of the plane wave, equation 1, along with a periodic 1 This subsection can be skipped without loss of continuity 2

3 potential, that reflects the lattice periodicity ψ(x) = exp(ikx) u k (x) in 1D ψ( r) = exp(i k r) u k ( r) in 3D (2) The term, u k ( r), has the periodicity of the lattice i.e. u( r+ R) = u( r), where R is a general lattice vector. This gives ψ( r + R) = exp(i k R) ψ( r). This theorem, which uses the periodicity of the lattice to construct electron wavefunctions, is called Bloch theorem, and these electrons are called Bloch waves. They can be contrasted with plane waves, that are solutions for the free electrons. 1.2 Band gap formation Consider an example of band gap formation, where Bragg reflection at the Brillouin zone (BZ) boundary causes standing waves to be formed in the lattice. We start with a 1D lattice, where for small k values the plane wave solution can be used. As the value of the wavevector increases, the electrons start to feel the effect of the atoms. The simplest wave to depict this is by including a reflected wave, so that the modified wavefunction is ψ(x) = A exp(ikx) ± B exp( ikx) (3) As the value of k increases, the reflected wave component steadily increases and when k = π/a, A = B. This leads to the formation of standing waves at the BZ boundary. There can be two kinds of standing waves ψ + ψ = A exp(ikx) + B exp( ikx) = 2A cos kx = A exp(ikx) B exp( ikx) = 2iA sin kx (4) These two waves, have different energies in the presence of a lattice potential. This is because ψ + concentrates the electrons at the atomic cores and ψ concentrates the electrons between the atomic cores, see figure 2. ψ + has a lower energy than ψ, since the negatively charged electrons are located closer to the positively charged atomic cores. Thus, for the same value of k, there are two standing waves with different energies i.e. a energy gap is created at the Brillouin zone boundary. This process can be extended to higher values of k i.e. π/a, 2π/a, 3π/a and so on. Thus, the continuous Evs.k band diagram for 1D becomes discontinuous at the boundaries. This is shown in figure 3. The magnitude of the gap depends on the atomic potentials. The reasoning that we developed for 1D 3

4 Figure 2: Schematic of ψ 2 + and ψ 2 along the 1D solid. In both cases the value of k = π/a. The two wavefunctions have different energies, since the spatial distribution of the electron wave is different. This leads to the formation of a band gap. Figure 3: (a) A continuous 1D E vs. k diagram. (b) Formation of band gap whenever k intersects the BZ boundary. The magnitude of the gap depends on the atomic potentials. 4

5 Figure 4: Formation of band gap for a 2D lattice in a reduced zone scheme (a) Continuous k values. (b) Discontinuities are created at the BZ boundaries. These open up forbidden energy gaps in the system. can be extended to 2D lattices, figure 4. For the 2D lattice, gaps are formed when the k in a certain direction intersects the BZ boundary. Since, this occurs at different values of k, we get band gaps at different directions. These band gaps can either overlap or not overlap, see figure 5. When there is no overlap of the gaps, then there is a continuity in E vs. k, the conductivity is high. If the gaps overlap, there is a certain forbidden energy regionin the material, where electrons cannot exist. This is called the energy band gap. The same concept can be extended to 3D lattices. 1.3 Electron group velocity and effective mass Consider a solid where there is a specific relation between E vs. k. For such a system, it is possible to define a quantity called group velocity (v g ), which defines the velocity of the wave packet that represents the electron. Group velocity is given by the formula v g = 1 de dk (5) For an electron obeying the free electron gas (FEG) model, i.e. E = 2 k 2 /2m e, equation 5 given v g = k/m e. Since, k represents the momentum, dividing that by the electron mass, gives the velocity, and it increases linearly with k. For systems that do not obey the FEG model, equation 5 can be used to calculate v g. Starting with equation 5, it is possible to relate a mass term with the energy 5

6 Figure 5: In a 2D lattice band gaps are created for different directions. In this figure, two directions are shown and the gaps can either (a) not overlap (b) overlap. Formation of overlapping energy gaps can lower the conductivity of the material. 6

7 and wavevector. dv g dt dv g dt = 1 ( ) d 2 E dk dt = 1 ( ) d 2 E dk dk 2 dt (6) The left hand term, dv g /dt, is the acceleration, a, (time derivative of the velocity). which can be written as a force term divided by the mass. Rearranging the right had side, a = 1 ( ) ( d 2 E dk ) (7) 2 dk 2 dt The term dk/dt represents the force acting on the electron, hence the mass term in the above equation is given by m e = 2 d 2 E dk 2 (8) This mass term, m e, is called the effective mass of the electron. It represents the effect of the atoms on the motion of the electrons in the lattice. Effective mass depends on the energy of the electron (and hence the wavevector). In the FEG model, m e = m e, i.e. the effective mass is equal to the rest mass of the electron. But, under the NFEG model, effective mass deviates from the rest mass of the electron. For a 1D solid, the relation between E vs. k, group velocity, and effective mass, in the BZ is plotted in figure 6. For low k, the FEG model is valid. In such cases, v g is a linear function of k and m e is a constant and equal to m e. As k approaches the BZ boundary, the effect of the lattice is felt by the electrons. Hence, the group velocity starts to decrease and reaches zero at the boundary (standing waves). Correspondingly, m e also increases and has an inflection point when v g is maximum. The argument for a 1D crystal can be extended to 2D and 3D, but it is important to note that for these solids v g and m e are direction dependent. 2 Fermi wavevector and BZ boundary The nearly free electron gas (NFEG) model is an extension of the FEG model and explains the formation of energy band gaps in a material. It should be 7

8 Figure 6: E vs. k, v g vs. k, and m e vs. k within the BZ for a 1D lattice. For small k, the FEG model holds, so that v g is linear with k and effective mass is constant and equal to m e. Deviations start to appear, when the value of k starts to approach the BZ boundary. 8

9 Figure 7: Depiction of E vs. k diagrams, showing the formation of energy gaps. (a) Extended zone scheme (b) Reduced zone scheme (c) Periodic zone scheme. possible to explain the FEG model, as a particular case of NFEG, where the lattice interactions are neglected. This can be done by comparing the Fermi wavevector, k F, to the wavevector corresponding to the BZ boundary, k BZ. This will also help understand, the filling of the energy band with electrons. Let us go back to a 1D lattice, i.e. a linear chain of atoms. The presence of the atoms creates gaps in the energy band diagram. There are three schemes for depicting this, see figure 7. The extended and reduced zone schemes have been discussed before. The periodic zone scheme is obtained by repeating the reduced zone scheme throughout the lattice. A horizontal line drawn in figure 7(c), will mark constant k or constant E lines. This scheme is easier to visualize electron motion through the lattice. 9

10 Figure 8: E vs. k for a 1D solid, with electron filling different energy levels, depending on the valency. Odd number fillings are metals, while even nos are non-metal. Let there be N atoms in this 1D lattice. Then, there are N orbitals and each orbital can take 2 electrons (of opposite spin). Let the valency of the atom be 1, i.e.each atom contributes one electron. Then, the first band is half full and there are empty states available in that band, for the electron to occupy. This makes a 1D chain of atoms, with valency 1, a metal. If the valency is 2, then the first band is completely full, since there are 2N electrons, and there is an energy band gap separating the full and empty states, see figure 7, and the 1D solid with valency 2 is an insulator/semiconductor. The magnitude of the gap depends on the strength of the atomic potential. If there are 3 electrons, the first band is completely full and the second band is half empty, once again a metal, and so on. Therefore, for a simple 1D solid, odd valency atoms are metals, while even valency atoms are non-metals. The electron filling is shown in figure 8. The same concept can be extended to 2D and 3D lattices. For these lattices, there are different E vs. k diagrams for the different directions. If the k values in the different directions do not overlap, there is a forbidden energy gap that extends in all directions and we have an insulator/semiconductor. If there is considerable overlap, such that we have continuous values of k, by choosing different directions, then we get a metal and if there is a small overlap, we get a semi-metal. Whether a given metal behaves as a free electron gas, depends 10

11 on the relation between the magnitude of the Fermi wavevector, k F, and its relation to the BZ boundary, k BZ. Consider two solids that have bcc structure, Na and Ba, with valency 1 and 2 respectively. Let n be electron density per unit volume. If a is the lattice constant, then n is related to a by n = 2Z a 3 (9) where Z is the valency. The factor, 2, comes from the fact that a bcc lattice has 2 atoms in the unit cell, or the volume of the primitive unit cell is a 3 /2. According to the FEG model, n can be related to k F, the Fermi wavevector by k F = (3π 2 n) 1/3 = (6π2 Z) 1/3 (10) a Given that the real lattice is bcc, then the reciprocal lattice is fcc, with a unit cell vector a = 2π/a (ˆx + ŷ). Hence, the distance from the center of the reciprocal lattice to the BZ boundary, k BZ is half the magnitude of a, equal to (π/a) 2. Compare the two quantities, k F and k BZ, for the two elements. For Na, k F < k BZ. Hence, the Fermi sphere lies within the BZ and is not influenced by the formation of energy gaps at the BZ boundary. So, the FEG model works well for Na, with the lattice effect incorporated into the electron effective mass. For Ba, k F > k BZ and the Fermi level penetrates the BZ boundary and lies in the 2nd BZ (formed using the second nearest neighbors). Hence, the Fermi surface is distorted by the energy gap at the BZ boundary. Hence, Ba, with 2 electrons per atom, has a lower conductivity than Na. The difference is nearly an order of magnitude, since only those electrons in the second BZ of Ba can conduct. The BZ is constructed by taking the vectors from the center to the nearest neighbors and drawing perpendicular bisectors to them. This is called the first Brillouin zone. Similarly, vectors can be drawn to the next nearest neighbors and perpendicular bisectors drawn to those. This would be called the second BZ and so on. It is easier to see these BZs for a 2D square lattice. The first and second BZ for a 2D square lattice is shown in figure 9(a). k F represents the Fermi wavevector for a given solid. In the FEG model, a constant k F surface would be the surface of a sphere for a 3D lattice and and the circumference of a circle for a 2D lattice. When the Fermi surface lies well within the BZ boundary, it is not affected by the lattice and the Fermi surface remains spherical. This is the case when the FEG model can be used for that particular material. As k F starts approaching the BZ boundary, it starts to get distorted due to the lattice 11

12 Figure 9: (a) The first and second BZ for a 2D square lattice. The second BZ is marked. (b) Different values of k F and its relation to k BZ. As k F approaches closer to k BZ, the Fermi surface starts to get distorted. potential. This is direction specific since the shortest vectors in 2D are k x and k y. When, k F = k BZ, the band gap opens up and when k F > k BZ the Fermi surface spills over from the first BZ to the second BZ. For real solids, with 3D lattice the same process happens, though it is harder to visualize this in 3D. Figure 10 shows the Fermi surface of Cu. There is some distortion in the Fermi sphere, in the [111] direction, where the Fermi surface is closest to the BZ boundary. Apart, from that the rest of the surface is a sphere. Thus, Cu is one of the elements where the FEG model works well. Al, on the other hand, has a much more complex Fermi surface, see figure 11. Al, has a valency 3, and hence the Fermi surface extends from the first BZ, into the second BZ, and also parts of the third BZ. This makes electron conductivity in Al, more complex than Cu. 3 NFEG and DOS Real solids can have a complex Fermi surface, which are hard to visualize. One way to compare various solids, is to look at the density of states (DOS) function and its deviation from the FEG model. In the FEG model, DOS, g(e), is a monotonically increasing function of energy i.e. g(e) E. In the NFEG model, this can no longer be true since there are forbidden energy gaps and as the energy nears the BZ boundary there is electron scattering. Then, g(e) decreases near the BZ boundary and should reach zero at the BZ boundary. This is shown in figure 12. It is possible to show the difference between the different metals, using the density of states plot and the Fermi energy. Figure 13 shows the DOS function for alkali metals, valency of one. The 12

13 Figure 10: The Fermi surface of Cu. Most of the Fermi surface is a sphere, lies within the first BZ. There is some distortion along the [111] direction when k F comes close to the BZ boundary. For Cu, the FEG model works well since this distortion is minimal. Source Figure 11: The Fermi surface of Al. Al, has a valency of three and the Fermi surface encloses the first BZ, second BZ, and parts of the third BZ. Source tschoy/r2d2/fermi/fermi.html 13

14 Figure 12: Change in DOS near the BZ boundary. There is a peak near the boundary and then g(e) starts to decrease, due to Bragg scattering. At the boundary g(e) goes to zero. Fermi surface is nearly a sphere and is not distorted by the lattice. E F is located within the first BZ, which is half full. If we have a material with valency two, then there can be two possibilities, as shown in figure 14. If there is an overlap between the energy bands corresponding to the first and second BZ, then we get a metal. The Fermi level, lies in the region of the overlap, and there are continuous energy states. On the other hand, if there is no overlap, i.e. there is a forbidden energy gap, then the first BZ is full and the second is empty, and we get an insulator/semiconductor, depending on the band gap value. Typically, if the band gap is less than 3 ev, it is called a semiconductor and if the band gap is more than 3 ev, we get an insulator. If we have a trivalent metal, then the first BZ is completely full and the second BZ is partially full, so we get a metal, see figure 15. If there is a very small overlap, between the first and second BZs, then the number of states available for conduction will be very small. These types of materials are called semi-metals. Thus, the electrical properties of metals depend on the shape of the Fermi surface and how it is affected by the Brillouin zone. This depends on the crystal structure and the valency of the material and the energy overlap between the different Brillouin zones. 4 Experimental measurement tools There are various techniques for measuring the density of states, Fermi surface distribution, Fermi energy, and effective mass. These are used in con- 14

15 Figure 13: DOS function, g(e), for a monovalent metal like alkali metals. The band is half full in this case. Figure 14: DOS function, g(e), for a bivalent material. (a) If there is an overlap between the first and second BZ, then it is a metal. (b) If there is no overlap of the band, then we get an insulator. Figure 15: DOS function, g(e), for a trivalent material. completely full and the second BZ is partially full. The first BZ is 15

16 Figure 16: Schematic of the Hall measurement setup. Electrons traveling in the x direction are deflected by a magnetic field applied in the z direction. This creates an electric field in the y direction. By measuring the created field and knowing the current and magnetic field, it is possible to calculate the electron density. junction with calculation techniques like tight binding approximations, and pseudopotential methods to arrive at a complete picture of the electronic structure of a material and its role on the transport properties. In this section, we will look at some commonly used measurement techniques, and the information obtained from them. 4.1 Hall measurements The phenomenon behind the Hall measurement, is the deflection of a charged particle in the presence of a magnetic field. This deflection is due to a force, called the Lorentz force, and is mathematically represented by F = q ( v B) (11) where, q is the electrical charge, v is the velocity and B is the magnetic field. An external potential is applied to a solid, as shown in figure 16. Charge 16

17 carriers, in this case electrons, travel in the negative x direction, under the influence of the potential. An external magnetic field is now applied in the positive z direction. Using equation 11, this produces a deflection in the negative y direction. v = v xˆx B = B zˆ(z) q = e Electron has negative charge Electrons are moving in negative x direction Magnetic field in the postive z direction F = ev x B z ŷ (12) This deflection causes an electric field to be built up in the material, see figure 16, that prevents further deflection, so that a steady state is reached. This electric field, E H, is called the Hall field and is given by ee H = ev x B Z (13) The current density (J x ), i.e. current per unit area, can be related to the velocity of the electrons, v x, and the electron density, n, i.e. no. of electrons per unit volume. J x = nev x (14) By convention, J x, is in the opposite direction to the flow of electrons. Using equations 13 and 14 it is possible to write ee H = J x n B z E H = 1 ne J x B z (15) E H J x B z = 1 ne = R H This quantity, R H, is called the Hall coefficient and can be measured experimentally, from the applied magnetic field, measured current and developed Hall field. This is then used to calculate the electron density, which can be used to calculate the valency. Some values for different metals are listed in table 1. Most metals have negative R H, since the charge carriers are electrons and the values tabulate well with the known valency. For semiconductors, R H can be negative or positive. This depends on the type of charge carriers, electrons or holes. Hall measurements do not provide information about the density of states or the Fermi surface. 17

18 Table 1: Hall coefficients and calculated electron densities and valencies Metals R H (exp) n Atomic density valency (10 11 m 3 A 1 s 1 ) (10 28 m 3 ) (10 28 m 3 ) Ag Au Cu Al Mg Na Figure 17: (a) Schematic of the photo emission process. The kinetic energy of the emitted electrons depends on their binding energy and the intensity depends on the density of occupied states (b) Experimental UPS spectrum from Cu, compared to the calculated DOS. 4.2 Ultraviolet photoelectron spectroscopy Photoelectric effect refers to the emission of electrons from a solid, by shining light above a certain threshold energy (the work function). In X-ray photoelectron spectroscopy (XPS), electrons are removed from the core levels and this provides chemical information about the samples. If UV light is used (energy of a few ev ) electrons are removed from the valence level. This is shown in figure 17. The kinetic energy of the emitted electron depends on its binding energy. The general equation is KE = hν (φ + E F E b ) (16) where φ is the work function, E F is the Fermi energy and E b is the binding energy, with respect to the bottom of the band (taken as reference). The 18

19 Figure 18: Angle resolved UPS, showing the resolution of the emitted electron wavevector into different components, parallel and perpendicular to the surface. intensity is proportional to the number of electrons present at that energy level, at the given temperature. This is given by the density of filled states, which is the product of the density of states and Fermi function, n(e) = g(e) f(e). Thus, the UPS spectrum provides information on the density of states. Figure 17(b) compares the experimental UPS spectrum from Cu with a calculated one. The overall shape of the two spectra match, though the energy resolution of the detector prevents seeing the finer details in the calculated spectrum. Instead of a polycrystalline sample, if measurements are done on a single crystal with a specific known orientation and if a semi-hemispherical electron detector is used, which is capable of measuring both energy and the angle of emission, then it is possible to get information on the magnitude and direction of the electron wavevector, k. This technique is called Angle resolved UPS (ARUPS). It is based on the fact that the photo electric effect conserves both energy and momentum. Consider light of energy hν incident on a surface, emitting an electron of energy E k, as shown in figure 18. The momentum components can be obtained in terms of angle θ and φ. k z = 1 2mEk cos θ k x = 1 2mEk sin θ cos φ (17) k y = 1 2mEk sin θ sin φ 19

20 Figure 19: (a) Movement of a free electron in a magnetic field (b) Movement of a nearly free electron in a magnetic field. In both cases, the field causes a deflection but not a change in energy so that the electron moves in a constant energy surface. By measuring, E vs. k along the different direction and at different energies it is possible to construct the three dimensional energy surface for a material. 4.3 Cyclotron resonance Cyclotron resonance is used to measure the electron effective mass at the Fermi energy. This is also based on the concept of Lorentz force, i.e. deflection of a charged particle in a magnetic field. Consider a free electron gas, moving with velocity v in a plane, and a magnetic field perpendicular to the field. The Lorentz force changes the direction of the electron but does not alter the total energy, so that the effect of the magnetic field is to make the electron move in a circle, see figure 19. The Lorentz force is related to the change in electron velocity, so that using equation 11 e( v B) = d k dt (18) If v and B are perpendicular, then the change of k is in the direction not the magnitude. Then rearranging the above equation gives T = dt = dt = eb evb dk 1 dk v(k) (19) 20

21 T is the time of revolution, and is related to the cyclic integral over the constant energy surface. Measurement of T, provides information on the electron effective mass, m e. This can be seen by considering the FEG equation, where Fermi velocity is a constant and independent of direction, and v F = k F /m e. Then, the cyclic integral in equation 19 becomes T = ebv F (2πk F ) = 2πm e eb (20) Rearranging, the cyclotron frequency ω c is given by ω c = 2π T = eb m e (21) Thus, measuring the cyclotron frequency gives the electron effective mass at the Fermi energy. 21