Lecture 2: 𝑘 ∙ 𝑝 theory and spin-orbit coupling Dimi Culcer UNSW

Outline of this lecture 𝑘 ∙ 𝑝 theory in semiconductors An approximation to the band-structure at small wave vectors The simplest model of the conduction band – effective mass Spin-orbit interaction Dirac equation Non-relativistic electrons → Pauli equation Effective spin-orbit interaction in crystals Rashba spin-orbit coupling Berry curvature in the Rashba model Anomalous Hall effect in the Rashba model

𝑘 ∙ 𝑝 theory This is an approximation especially useful in semiconductors In semiconductors carrier densities are typically low The Fermi wave vector is much smaller than the size of the Brillouin zone The idea of 𝑘 ∙ 𝑝 theory Suppose we know the energies and wave functions at some point in 𝑘 -space This is usually the bottom of the conduction band We can find an approximate dispersion near this point The Hamiltonian of a crystal is 𝐻 = 𝑃 2 2𝑚 +𝑈 𝑅 Electrons are described by Bloch wave-functions 𝜓 𝑛 𝑘 = 𝑒 −𝑖 𝑘 ∙ 𝑟 𝑢 𝑛 𝑘 Consider translation by a Bravais lattice vector 𝑅 𝑈 𝑟 + 𝑅 =𝑈 𝑟 𝜓 𝑛 𝑘 ( 𝑟 + 𝑅 ) = 𝑒 −𝑖 𝑘 ∙ 𝑅 𝜓 𝑛 𝑘 ( 𝑟 ) 𝑢 𝑛 𝑘 ( 𝑟 + 𝑅 ) = 𝑢 𝑛 𝑘 ( 𝑟 )

𝑘 ∙ 𝑝 theory Let’s write the Schrodinger equation in terms of 𝑢 ℏ 2 2𝑚 − 𝛻 2 −2𝑖 𝑘 ∙ 𝛻 + 𝑘 2 𝑢 +𝑈 𝑢 = 𝜀 𝑢 (1) Regard the LHS as some Hamiltonian corresponding to a vector 𝑘 : ℋ 𝑘 Assume we have solved (1) and know all the u’s and 𝜀’s at 𝑘 Now increase 𝑘 → 𝑘 + 𝛿 𝑘 𝛿ℋ 𝑘 =− ℏ 2 2𝑚 2𝑖𝛿 𝑘 ∙ 𝛻 −2𝛿 𝑘 ∙ 𝑘 − 𝛿𝑘 2 The first-order change in the energy is given by 𝛿𝜀 𝑛 𝑘 (1) = ℏ 2 𝑚 𝑢 𝑛 𝑘 𝛿 𝑘 ∙( 𝑘 −𝑖 𝛻 ) 𝑢 𝑛 𝑘 → ℏ 𝑚 𝜓 𝑛 𝑘 𝛿 𝑘 ∙ 𝑃 𝜓 𝑛 𝑘 See M. Marder Condensed Matter Physics, Ch. 16

𝑘 ∙ 𝑝 theory Often the first-order term in 𝛿 𝑘 vanishes Continue perturbation theory to second order 𝛿𝜀 𝑛 𝑘 (2) = ℏ 2 𝛿𝑘 2 2𝑚 + ℏ 2 𝑚 2 𝑛′≠𝑛 𝜓 𝑛 𝑘 𝛿 𝑘 ∙ 𝑃 𝜓 𝑛′ 𝑘 2 𝜀 𝑛 𝑘 − 𝜀 𝑛 ′𝑘 This looks very much like the free electron dispersion ∝ 𝛿𝑘 2 Except the mass is different – the electrons act as if they have an effective mass This can be very different from the free electron mass For example GaAs conduction band electron effective mass = 0.0665 m This is the case in the conduction band of many semiconductors The energy dispersion can be approximated as parabolic but with an effective mass What kind of non-parabolic terms are important? Remember this is still 2-fold degenerate due to spin

Spin-orbit: 1-Dirac equation Describes quantum mechanics of relativistic particles Has the form of a continuity equation Requirements

Dirac equation 𝐻= 𝑚𝕀 𝜎 ∙ 𝑝 𝜎 ∙ 𝑝 −𝑚𝕀 One representation of the Dirac matrices is 𝛼 = 0 𝜎 𝜎 0 β= 𝕀 0 0 −𝕀 So the Dirac Hamiltonian is 𝐻= 𝑚𝕀 𝜎 ∙ 𝑝 𝜎 ∙ 𝑝 −𝑚𝕀 Notice this is all abstract There is no mention of spin or angular momentum

Dirac equation: Solutions First think of a particle at rest

Dirac equation: non-relativistic limit Finite but small momentum: decompose 𝜋 = 𝑝 +𝑒 𝐴 Positive-energy solutions

Dirac equation: non-relativistic limit

Pauli equation This is the Pauli equation It applies to non-relativistic electrons Now 𝜎 can be interpreted as spin The next relativistic correction gives the spin-orbit interaction Given on next slide Proof takes more time Spin appears after you separate particles from antiparticles The hierarchy of energy scales is Mass – kinetic – Zeeman – spin-orbit Key message Spin-orbit is a relativistic effect

Spin-orbit interaction We have gone from the Dirac equation to the Pauli equation This requires perturbation theory So far we stopped at the leading order In the next order of perturbation theory another correction This is called the spin-orbit interaction For a free particle the effect is very small ∝ 1 𝑐 2 usually neglected For a spherically symmetric (central) potential 𝐻 𝑠𝑜 = ℏ 4 𝑚 2 𝑐 2 𝑟 𝜕𝑉 𝜕𝑟 𝜎 ∙ 𝐿 𝐿 = 𝑟 × 𝑝 In this case it has the form L.S hence the name spin-orbit Spin-orbit coupling evidently increases with atomic number

Spin-orbit in crystals In general the full Hamiltonian contains a term V is the full potential acting on the electron 𝑉= 𝑉 𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 + 𝑉 𝑐𝑜𝑛𝑓𝑖𝑛𝑒𝑚𝑒𝑛𝑡 + 𝑉 𝑖𝑚𝑝𝑢𝑟𝑖𝑡𝑖𝑒𝑠 In a crystal this will contain the periodic potential In a heterostructure the confinement potential is also present Spin-orbit in crystals typically treated in 𝑘 ∙ 𝑝 theory Periodic part: add 𝐻 𝑠𝑜 = ℏ 4 𝑚 2 𝑐 2 𝜎 ∙ 𝛻 𝑉 𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 × 𝑝 to the crystal Hamiltonian This will contribute to 𝛿ℋ 𝑘 and we can do perturbation theory as before Typically this contributes to linear order in 𝑘 Usually we want the effective spin-orbit interaction for a particular band We will focus on the conduction band of semiconductors Typically use symmetry arguments to determine the form of the spin-orbit Remember the conduction band looks very much like free electrons So expect spin-orbit to look like free electrons

Spin-orbit and inversion symmetry Free-particle spin-orbit very small But in a crystal interband projections can make it much larger Spin-1/2 electrons in diamond & zincblende lattices Conduction band s-like No angular momentum so no magnetic moment Hence spin-orbit for electrons is generally weak SO strong when NO inversion symmetry Holes more complicated but not discussed here Inversion asymmetry can come from the underlying crystal Bulk inversion asymmetry Crystal has no centre of inversion e.g. Si vs GaAs Si has a diamond lattice which almost has inversion symmetry Diamond is two interlocking fcc lattices GaAs like diamond but the two lattices have different atoms So no centre of inversion anymore Allowed SO terms odd in k: Dresselhaus (Winkler, 2003)

Rashba spin-orbit coupling Inversion asymmetry from the confinement potential Quantum well in E-field // z Or a 2DEG in a triangular well Terms odd in k allowed Generally called Rashba terms How do we understand these? Add the confinement potential whose gradient is // z 𝐻 𝑠𝑜 = ℏ 4 𝑚 2 𝑐 2 𝜎 ∙ 𝛻 𝑉 𝑐𝑜𝑛𝑓𝑖𝑛𝑒𝑚𝑒𝑛𝑡 × 𝑝 From Winkler (2003)

Rashba spin-orbit coupling Remember conduction band similar to free electrons The effective spin-orbit has the form 𝐻 𝑠𝑜 =(𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) 𝜎 ∙ 𝛻 𝑉 𝑐𝑜𝑛𝑓𝑖𝑛𝑒𝑚𝑒𝑛𝑡 × 𝑝 Where the constant is material-specific What is the gradient? 𝛻 𝑉 𝑐𝑜𝑛𝑓𝑖𝑛𝑒𝑚𝑒𝑛𝑡 = 𝜕 𝑉 𝑐𝑜𝑛𝑓𝑖𝑛𝑒𝑚𝑒𝑛𝑡 𝜕𝑧 𝑧 =− 𝐸 𝑧 𝑧 Lump the electric field Ez into the overall constant which we call 𝛼 𝛼 also incorporates an ℏ So we can write the effective spin-orbit as 𝐻 𝑠𝑜 =−𝛼 𝜎 ∙ 𝑧 × 𝑘 =𝛼 𝜎 ∙ 𝑘 × 𝑧 =𝛼 𝑧 ∙ 𝜎 × 𝑘

Rashba spin-orbit coupling Characterised by SO constant α Depends on material parameters as well as on vertical electric field Ez In principle it can be tuned by tuning the vertical (gate) electric field Like a Zeeman field that depends on wave vector Integrates to zero over FS – true for all forms of SO because of time-reversal From Winkler (2003)

Spin-orbit interaction Consider the conduction band Remember it was 2-fold degenerate because of spin The Hamiltonian is really a 2x2 matrix In the degenerate case this matrix is diagonal The spin-orbit interaction splits the degeneracy So the 2x2 matrix has off-diagonal terms Spin-degenerate conduction band Spin-split conduction band Fermi energy Fermi energy 𝐻= ℏ 2 𝑘 2 2 𝑚 ∗ 𝐻= ℏ 2 𝑘 2 2 𝑚 ∗ +𝛼( 𝜎 𝑥 𝑘 𝑦 − 𝜎 𝑦 𝑘 𝑥 )

Spin-orbit and Berry curvature Spin-orbit bands frequently have finite Berry curvature For Rashba to get a finite curvature we also need a perpendicular magnetisation Here the magnetisation is called h0 Consider a generalized Rashba model

The Berry curvature is really a monopole at k = 0 . The width is given by h0. As h0 decreases the curvature gets sharper and larger, it approaches a 𝛿 – function.

Anomalous Hall effect Recall the intrinsic term in the conductivity has the form 𝑓 𝑒𝑞 𝑒 𝐸 × Ω ℏ 2 At T=0 the Fermi-Dirac distribution is just a Heaviside function So it only changes the limits of integration The Hall conductivity due to the lower band takes the form 𝜎 𝑥𝑦 =− 𝑒 2 ℎ 𝑑 2 𝑘 2𝜋 Ω ↓ 𝑧 So it is DIRECTLY proportional to the Berry phase If 𝑘 𝐹 → ∞ the Berry phase tends to 𝜋 and the lower band contributes − 𝑒 2 2ℎ

Anomalous Hall effect However this is completely unrealistic We assumed only the lower band is occupied This is clearly the case only for a narrow range of k Remember the upper band has the opposite Ω It makes the opposite contribution to 𝜎 𝑥𝑦 So at large kF the two almost cancel out What is the best we can hope for? Wurtzite structures – linear Rashba Conduction band of CdSe Valence band of CdSe

Another problem with Rashba AHE We have not taken into account disorder at all Phys. Rev. Lett. 97, 046604 (2006); Phys. Rev. B 76, 235312 (2007) At large kF disorder kills the Rashba AHE completely Calculation complicated [Phys. Rev. B 96, 035106 (2017)] but can get qualitative understanding Recall the non-equilibrium correction to the distribution 𝑔= 𝑒 𝐸 𝜏 ℏ ∙ 𝜕 𝑓 𝑒𝑞 𝜕 𝑘 𝑐 ∝𝜏 𝜏 is needed to keep the Fermi surface near equilibrium We introduced 𝜏 phenomenologically but it can be calculated from Fermi’s Golden Rule At T=0 scattering mostly off impurities and 𝜏∝ 𝑛 𝑖 −1 , where 𝑛 𝑖 is the impurity density Our transport theory assumes 𝜀 𝐹 𝜏 ℏ ≫1 ⇒ ℏ 𝜀 𝐹 𝜏 ≪1 The small parameter is ℏ 𝜀 𝐹 𝜏 ∝ 𝑛 𝑖 Our theory is really an expansion in 𝑛 𝑖 But the expansion starts at order 𝑛 𝑖 −1 , so the next term in the expansion is ∝ 𝑛 𝑖 0 It appears to be independent of disorder although of course it is not This term kills the Berry curvature contribution to AHE in the Rashba model

Rashba AHE is not dissipationless There is also 𝜎 𝑥𝑥 = 𝑛 𝑒 2 𝜏 𝑚 Very similar to what we calculated in the previous lecture So what shall we do? Zoom into Rashba to get SO dominant Open gap so only lower band is occupied Put Fermi energy in the gap EF εF

Summary 𝑘 ∙ 𝑝 theory Spin-orbit interaction Rashba spin-orbit coupling The conduction band of many semiconductors can be approximated as parabolic But the electrons have an effective mass rather than the real mass Spin-orbit interaction It comes when going from the Dirac equation to the Pauli equation In crystals electrons experience an effective spin-orbit interaction For electrons in the conduction band this is usually odd in wave vector Rashba spin-orbit coupling Most common in asymmetric heterostructures Linear in wave vector Berry curvature gives a contribution to the anomalous Hall effect Under certain circumstances this can be half-quantised

Supplement: Pauli equation

Klein-Gordon equation Apply the Dirac Hamiltonian operator twice (Klein-Gordon) Derive 3 conditions for the matrices in the Dirac equation