SOLID STATE PHYSICS PART III Magnetic Properties of Solids M. S. Dresselhaus 1 Contents 1 Review of Topics in Angular Momentum 1 1.1 Introduction . 1 1.1.1 Angular Momentum, De¯nitions and Commutation Relations . 1 1.1.2 Angular momentum Eigenvalues . 3 1.2 Angular Momentum in Wave Mechanics . 7 1.3 Magnetic Moment and Orbital Angular Momentum . 9 1.4 Spin Angular Momentum . 10 1.5 The Spin-Orbit Interaction . 11 1.5.1 Solution of SchrÄodinger's Equation for Free Atoms . 12 1.6 Vector Model for Angular Momentum . 15 2 Magnetic E®ects in Free Atoms 19 2.1 The Zeeman E®ect . 19 2.2 The Hyper¯ne Interaction . 20 2.3 Addition of Angular Momenta . 21 2.3.1 Hund's Rule . 21 2.3.2 Electronic Con¯gurations . 23 2.4 Clebsch{Gordan Coe±cients . 26 3 Diamagnetism and Paramagnetism of Bound Electrons 31 3.1 Introductory Remarks . 31 3.2 The Hamiltonian for Magnetic Interactions for Bound Electrons . 34 3.3 Diamagnetism of Bound Electrons . 35 3.4 Paramagnetism of Bound Electrons . 38 3.5 Angular Momentum States in Paramagnetic Ions . 45 3.6 Paramagnetic Ions and Crystal Field Theory . 46 3.7 Quenching of Orbital Angular Momentum . 47 4 Paramagnetism and Diamagnetism of Nearly Free Electrons 50 4.1 Introduction . 50 4.2 Pauli Paramagnetism . 50 4.3 Introduction to Landau Diamagnetism . 53 4.4 Quantized Magnetic Energy Levels in 3D . 54 4.4.1 Degeneracy of the Magnetic Energy Levels in kx . 55 4.4.2 Dispersion of the Magnetic Energy Levels Along the Magnetic Field 56 4.4.3 Band Parameters Describing the Magnetic Energy Levels . 59 2 4.5 The Magnetic Susceptibility for Conduction Electrons . 61 5 Magneto-Oscillatory and Other E®ects Associated with Landau Levels 62 5.1 Overview of Landau Level E®ects . 62 5.2 Quantum Oscillatory Magnetic Phenomena . 65 5.3 Selection Rules for Landau Level Transitions . 68 5.4 Landau Level Quantization for Large Quantum Numbers . 69 6 The Quantum Hall E®ect (QHE) 71 6.1 Introduction to Quantum Hall E®ect . 71 6.2 Basic Relations for 2D Hall Resistance . 71 6.3 The 2D Electron Gas . 75 6.4 E®ect of Edge Channels . 79 6.5 Applications of the Quantized Hall E®ect . 83 6.6 Fractional Quantum Hall E®ect (FQHE) . 83 7 Magnetic Ordering 89 7.1 Introduction . 89 7.2 The Weiss Molecular Field Model . 90 7.3 The Spontaneous Magnetization . 91 7.4 The Exchange Interaction . 97 7.5 Antiferromagnetism and Ferrimagnetism . 103 7.6 Spin Waves . 108 7.7 Anisotropy Energy . 115 7.8 Magnetic Domains . 117 8 Magnetic Devices 121 8.1 Introduction . 121 8.2 Permanent Magnets . 121 8.3 Transformers . 122 8.3.1 Magnetic Ampli¯ers . 122 8.4 Data Storage . 123 8.4.1 Magnetic Core . 123 8.4.2 Data Storage: Magnetic Tape . 123 8.4.3 Magnetic Bubbles . 127 8.4.4 Magneto-optical Storage . 129 8.5 Faraday E®ect . 131 8.6 Magnetic Multilayer Structures . 134 8.6.1 Introduction . 134 8.6.2 Metallic Magnetic Multilayers . 135 8.6.3 Insulating Magnetic Multilayers . 137 8.6.4 References . 137 3 Chapter 1 Review of Topics in Angular Momentum References Sakurai, Modern Quantum Mechanics, Chapter 3. ² Schi®, Quantum Mechanics, Chapter 7. ² Shankar, Principles of Quantum Mechanics, Chapter 12. ² Jackson, Classical Electrodynamics, 2nd Ed., Chapter 5. ² 1.1 Introduction In this chapter we review some topics in quantum mechanics that we will apply to our discussion of magnetism. The major topic under discussion here is angular momentum. 1.1.1 Angular Momentum, De¯nitions and Commutation Relations In classical mechanics, as in quantum mechanics, angular momentum is de¯ned by L~ = ~r p:~ (1.1) £ Since the position operator ~r and momentum operator p~ do not commute, the components of the angular momentum do not commute. We note ¯rst that the position and momentum operators do not commute: ¹h @ ¹h @ (xp p x)f(~r) = x f(~r) [xf(~r)] = i¹hf(~r): (1.2) x ¡ x i @x ¡ i @x The result of Eq. 1.2 is conveniently written in terms of the commutator de¯ned by [r ; p ] j k ´ r p p r using the relation j k ¡ k j [rj; pk] = i¹h±jk; (1.3) where ±jk is a delta function having the value unity if j = k, and zero otherwise. Equation 1.3 says that di®erent components of ~r and p~ commute, and it is only the same components of 1 ~r and p~ that fail to commute. If we now apply these commutation relations to the angular momentum we get: [L ; L ]=L L L L = (yp zp )(zp xp ) (zp xp )(yp zp ) x y x y ¡ y x z ¡ y x ¡ z ¡ x ¡ z z ¡ y (1.4) = i¹hyp + i¹hxp = i¹hL : ¡ x y z Similarly, the commutation relations for the other components are: [Ly; Lz] = i¹hLx [Lz; Lx] = i¹hLy: (1.5) The commutation relations in Eqs. 1.4 and 1.5 are conveniently summarized in the symbolic statement L~ L~ = i¹hL:~ (1.6) £ These commutation relations are basic to the properties of the angular momentum in quan- tum mechanics. Since no two components of the angular momentum commute, it is not possible to ¯nd a representation that simultaneously diagonalizes more than one component of L~ . That is, there is no wavefunction that is both an eigenfunction of Lx and Ly, for if there were, we could then write Lxª = `xª (1.7) and Lyª = `yª: (1.8) Equations 1.7 and 1.8 would then imply that LxLyª = Lx`yª = `x`yª (1.9) and also LyLxª = `y`xª (1.10) so that if Lx and Ly could be diagonalized simultaneously, then Lx and Ly would have to commute. However, we know that they do not commute; therefore, they cannot be simultaneously diagonalized. On the other hand, all three components of angular momentum, Lx, Ly, and Lz commute with L2 where 2 2 2 2 L = Lx + Ly + Lz: (1.11) For example [L ; L2] = L L2 L2 L + L L2 L2L z z x ¡ x z z y ¡ y z = L L L + i¹hL L L L L + i¹hL L (1.12) x z x y x ¡ x z x x y +L L L i¹hL L L L L i¹hL L = 0 y z y ¡ x y ¡ y z y ¡ y x 2 2 and similarly for [Lx; L ] = 0 and [Ly; L ] = 0. Since Lx, Ly and Lz do not commute with each other, it is convenient to select one component (e.g., Lz) as the component that is simultaneously diagonalized with L2. 2 It is convenient to introduce raising and lowering operators L = Lx iLy (1.13) § § so that we can write 2 2 1 L = Lz + (L+L + L L+): (1.14) 2 ¡ ¡ From Eq. 1.13 we know that L+ and L are non-hermitian operators, because Lx and ¡ Ly are both Hermitian operators and have real eigenvalues. Since Lx and Ly individually commute with L2, we have the commutation relations: 2 2 [L ; L+] = 0 and [L ; L ] = 0: (1.15) ¡ Furthermore, L+ does not commute with Lz but rather [L ; L ] = [L ; (L + iL )] = i¹h(L iL ) = ¹hL (1.16) z + z x y y ¡ x + and likewise [Lz; L ] = ¹hL : (1.17) ¡ ¡ ¡ By the same procedure we obtain [L+; L ] = [(Lx + iLy); (Lx iLy)] = 2¹hLz (1.18) ¡ ¡ and 1 i Lx = (L+ + L ) Ly = (L+ L ) (1.19) 2 ¡ ¡2 ¡ ¡ 1.1.2 Angular momentum Eigenvalues We will now use the commutation relations in 1.1.1 to ¯nd the eigenvalues of the angular x momentum matrices. Let us choose as our representation, one that diagonalizes both Lz and L2 and we will use quantum numbers m and ` to designate the representation. For, example, if ª is an eigenfunction of Lz we can write Lzª = m¹hª (1.20) where the eigenvalue of Lz is m¹h. From Eq. 1.20 we can then write the matrix element of L in the m` representation as z j i m` L m` = m¹h (1.21) h j zj i where m is a dimensionless real integer denoting the magnitude of Lz, while ` is the maxi- mum value of m and ¹h has the dimension of angular momentum. Physically, we can think of m as telling how many units of angular momentum there are along the z direction. Since Lz has only diagonal matrix elements, the above relation can be written more generally as m0`0 L m` = m¹h± 0 ± 0 : (1.22) h j zj i mm `` 2 We will now compute the matrix elements of L+, L , and L in the m` representation. ¡ j i 3 From the commutation relation 2 [L ; L+] = 0 (1.23) we obtain 2 2 m0`0 L L L L m00`00 = 0 (1.24) h j + ¡ + j i for all m` states. But L2 is diagonal in the m` representation by hypothesis and therefore j i j i is speci¯ed by some function of `, the quantum number associated with L2. We thus obtain 2 2 L m00`00 = (L ) 00 m00`00 (1.25) j i ` j i 2 2 where we have written the eigenvalue of the operator L using the notation (L )`00 . We will now show that 2 2 (L )`00 = ¹h `00(`00 + 1): (1.26) From Eqs. 1.24 and 1.25 we write 2 2 [(L ) 0 (L ) 00 ] m0`0 L m00`00 = 0: (1.27) ` ¡ ` h j +j i For Eq.