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

2. 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

3. 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 )

4. 𝑘 ∙ 𝑝 theory Let’s write the Schrodinger equation in terms of 𝑢
ℏ 2 2𝑚 − 𝛻 2 −2𝑖 𝑘 ∙ 𝛻 + 𝑘 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

5. 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 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 = 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

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

7. Dirac equation 𝐻= 𝑚𝕀 𝜎 ∙ 𝑝 𝜎 ∙ 𝑝 −𝑚𝕀
One representation of the Dirac matrices is
𝛼 = 0 𝜎 𝜎 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

8. Dirac equation: Solutions
First think of a particle at rest

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

10. Dirac equation: non-relativistic limit

11. 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

12. 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

13. 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

14. 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)

15. 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)

16. 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

17. 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)

18. 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 )

19. 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

20. 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.

21. 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

22. 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

23. Another problem with Rashba AHE
We have not taken into account disorder at all
Phys. Rev. Lett. 97, (2006); Phys. Rev. B 76, (2007)
At large kF disorder kills the Rashba AHE completely
Calculation complicated [Phys. Rev. B 96, (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

24. 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

25. 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

26. Supplement: Pauli equation

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