Ovrec’s Laboratory Rules

December 5, 2008 by Ovrec
  1. Don’t take any initiative if you are alone. 
     
  2. Perform laboratory work only when another person is present.
     
  3. Be quiet!

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Special Relativity Sketchbook

February 19, 2009 by Ovrec

\displaystyle \beta =\frac{v}{c}

\displaystyle \gamma= (1-\beta^2)^{-1/2}

\displaystyle d\tau = \frac{dt}{\gamma}

\displaystyle \mathbf p =\gamma m \mathbf v = \frac{m \mathbf v}{\sqrt{1-\displaystyle \frac{v^2}{c^2}}}

\displaystyle E= \gamma m c^2 = \frac{m c^2}{\sqrt{1- \displaystyle \frac{v^2}{c^2}}}

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Sketches about Superconductivity II

February 10, 2009 by Ovrec
From: Kittel – Quantum theory of Solids Ch. 7 & 8

Electron-phonon interaction

Conduction electrons sense in various ways any deformation of the ideal periodic lattice of positive ion cores. Even the zero-point motion of phonons has its effect on the conduction electrons. The chief effects of the coupling of electrons and phonons are:   

  1. To scatter electrons from one state \displaystyle \mathbf k to another \mathbf k' , leading to electrical resistivity. 
  2. To cause the absorption (or creation) of phonons: the interaction of conduction electrons and phonons is an important source of attenuation of ultrasonic waves in metals.
  3. To cause an attractive interaction between two electrons; this interaction is important for superconductivity and results from the virtual emission and absorption of a phonon.
  4. The electron will always carry with it a lattice polarization field. The composite particle, electron + phonon field, is called polaron;  it has a larger effective mass than electron in the unperturbed lattice.

cooper

We recall what is called the Bardeen and Shockley deformation potential method. Within this theory the energy is expanded into a Taylor series in powers of a quantity characterizing the strength of the lattice strain. The expansion is truncated after the first power of this parameter, which renders the theory linear. Neglecting terms of the second order is equivalent to the effective masses being unchanged by the induced strain.

We denote the vector displacement operator in a solid with:

\displaystyle \mathbf R= \mathbf r'- \mathbf r  

where: \displaystyle \mathbf r' is the position of an atom after the deformation.

After some calculations we have:

\displaystyle \mathbf R(\mathbf r)= \sum_{\mathbf k} \mathbf e_{\mathbf k \mu}(2\rho \omega_{\mathbf k \mu})^{-1/2}(a_{\mathbf k \mu}e^{i\mathbf k \cdot \mathbf r}-a^+_{\mathbf k \mu}e^{-i \mathbf k \cdot \mathbf r})

where \mathbf e_{\mathbf k \mu} is a unit vector in the direction of the polarization of the phonon.

Taking the gradient we have the dilation operator:

\displaystyle \mathbf \Lambda(\mathbf r)= \frac{\partial R_{\mu}}{\partial x_{\mu}}= i\sum_{\mathbf k} (2\rho \omega_{\mathbf k l})^{-1/2}(a_{\mathbf k l}e^{i\mathbf k \cdot \mathbf r}-a^+_{\mathbf k l}e^{-i \mathbf k \cdot \mathbf r})

Following Bardeen and Shockley, we suppose that the deformation induces a change in the energy C_1 \Lambda(\mathbf r):

\epsilon(\mathbf k, \mathbf r)= \epsilon_0(\mathbf k)+ C_1 \Lambda(\mathbf r)

So C_1 \Lambda(\mathbf r) represent the perturbation in the energy.

We are interested to the matrix elements of C_1 \Lambda(\mathbf r) between the unperturbed one-electron Bloch states \vert \mathbf k \rangle and \vert \mathbf k' \rangle:

\displaystyle H'=\int d^3 x\Psi^+(\mathbf r)C_1\Lambda(\mathbf r)\Psi(\mathbf r)

\displaystyle = \sum_{\mathbf k' \mathbf k}c_{\mathbf k'}^+c_{\mathbf k} \langle \mathbf k'\vert C_1 \Lambda\vert \mathbf k\rangle

\displaystyle \Psi(\mathbf r)=\sum_{\mathbf k}c_{\mathbf k}u_{\mathbf k}(\mathbf r)e^{i \mathbf k \cdot \mathbf r}=\sum_{\mathbf k}c_{\mathbf k} \vert \mathbf k \rangle

Remember that:

\mathbf k -\mathbf k' \pm \mathbf q = 0 → Normal (N) process

\mathbf k -\mathbf k' \pm \mathbf q = \mathbf G → Umklapp (U) process

After some calculations, the deformation potential perturbation becomes:

\displaystyle H' = iC_1 \sum_{\mathbf k \mathbf q}(2\rho\omega_{\mathbf q})^{-1/2}\vert\mathbf q\vert c_{\mathbf k+\mathbf q}^+c_{\mathbf k}(a_{\mathbf q} -a_{-\mathbf q}^+)

The existence of the electron-phonon coupling \displaystyle H' means that an electron in a state \mathbf k with no phonons excited cannot be an exact eigenstate of the system, but there will be always a cloud of virtual phonons accompanying the electron. The composite particle, electron + lattice deformation, is called polaron (this term is most often used for an electron plus the cloud of virtual optical phonons in ion crystals).

The phonon cloud changes the energy of the electron.

We can write the perturbation as:

\displaystyle H' = i \sum_{\mathbf k \mathbf q}D_{\mathbf q}c_{\mathbf k+\mathbf q}^+c_{\mathbf k}(a_{\mathbf q} -a_{-\mathbf q}^+)

and taking D as a constant:

\displaystyle H' = i D\sum_{\mathbf k \mathbf q}c_{\mathbf k+\mathbf q}^+c_{\mathbf k}(a_{\mathbf q} -a_{-\mathbf q}^+)

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Sketches about Superconductivity I

February 10, 2009 by Ovrec
From:
Tinkham – Introduction to Superconductivity (2004) Dover
Kittel – Introduction to Solid State Physics
Ziman – Priciples of the Theory of Solids 

London’s Penetration Depth:

\displaystyle \lambda_L = \sqrt{\frac{mc^2}{4\pi n e^2}}

Careful comparison of the rf penetration depths of samples in the normal and superconducting state have shown that the actual superconducting penetration depths are always larger than the predicted London’s one The qualitative explanation of this excess required the introduction of an additional concept buy Pippard: the Coherence Length \xi_0 .

The Coherence length is a measure of the distance within which the superconducting electron concentration cannot change drastically in a spatially-varying magnetic field. Also Coherence length is a measure of the range over which we should average \mathbf A to obtain \mathbf j.

The assumption in the derivation of the London equation is that the superconducting wavefunction is “rigid” and unaffected by the application of a magnetic field. This is obviously not correct because we know that a strong enough magnetic field can destroy superconductivity. We need to calculate the change in the wavefunction produced by the perturbation \mathbf A(\mathbf r) acting on the superconducting ground state. 

 

London equation is a local one. Pippard introduced the coherence length while proposing a non local generalization of the London equation. This was done in analogy to Chamber’s nonlocal generalization of Ohm’s law (for the anomalous skin effect).

\displaystyle \mathbf j(\mathbf r)=\int \Gamma(\mathbf r - \mathbf r')\mathbf A(\mathbf r')d \mathbf r'

where \Gamma (\mathbf r) is the Fourier transform of \displaystyle \Gamma (\mathbf q)= \frac{1}{\lambda_L^2} that in the London theory is considered as a constant. 

The range of \Gamma(\mathbf r) will be the Coherence length \xi_0.

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Dynamics of electrons in a periodic potential with an applied electromagnetic field

February 3, 2009 by Ovrec

Bloch’s waves packets.
Remember that the \mathbf k that appears in the argument of the exponential part of a Bloch’s function is called pseudo-wavevector (so, multiplied by \hbar, it will be the pseudo-momentum) because the Bloch’s function also depends on other wavevectors that are contained in the periodic term u_{\mathbf k}(\mathbf r).

The traditional (not the Bloch’s one) wave packet is the superposition of plane waves (→ Fourier components) having nearly the same wavevector centered in k_0.
The construction of the wave packet needs to assure two conditions:

  1. To assure a definite wavevector for the packet, the wavevectors of the different Fourier components must be nearly the same.
  2. To assure a definite position the packet must not be too much broad.

The two “extentions” (one in the Fourier’s space and the other in the real space) are related by the uncertainty relation.

We have similar considerations about time and energy.

We suppose that the Bloch’wave packet is constructed by Bloch’s states belonging to one only energy band. This assumption works well in semiconductors because bands are well separated.

\displaystyle \psi_{\mathbf k_0}(\mathbf r, t)=\sum_{\mathbf k} a_{\mathbf k_0}(\mathbf k) u_{\mathbf k}(\mathbf r) e^{i[\mathbf k \cdot \mathbf r - \frac{\epsilon(\mathbf k)}{\hbar}t]}

Where a_{\mathbf k_0}(\mathbf k) are functions strongly peaked around \mathbf k_0.

In order to know the true momentum of the Bloch’s packet we express the u_{\mathbf k}(\mathbf r) as a Fourier series in terms of the reciprocal lattice vectors:

\displaystyle u_{\mathbf k}(\mathbf r) = \sum_{\mathbf G} b_{\mathbf k}(\mathbf G) e^{i\mathbf G \cdot \mathbf r}

\displaystyle \psi_{\mathbf k_0}(\mathbf r, t) = \sum_{\mathbf {G, k}} b_{\mathbf k}(\mathbf G) a_{\mathbf k_0}(\mathbf k) e^{i[(\mathbf k + \mathbf G) \cdot \mathbf r - \frac{\epsilon(\mathbf k)}{\hbar}t]}

\displaystyle \mathbf k' = \mathbf k + \mathbf G

\displaystyle \psi_{\mathbf k_0}(\mathbf r, t) = \sum_{\mathbf {G, k'}} C(\mathbf k') e^{i[\mathbf k' \cdot \mathbf r - \frac{\epsilon(\mathbf k')}{\hbar}t]}

Considering the group velocity of the Bloch packet, we can note that:

\displaystyle v(\mathbf k_0 + \mathbf G) = \frac{1}{\hbar} \frac{\partial \epsilon(\mathbf k')}{\partial \mathbf k'}\Big \vert_{\mathbf k_0 + \mathbf G} = \frac{1}{\hbar} \frac{\partial \epsilon(\mathbf k')}{\partial \mathbf k'}\Big \vert_{\mathbf k_0}

Note that the group velocity of the wave packet can be negative, in particular, in the parabolic band approximation (or effective mass approximation), the group velocity at k is zero and above, for wavevectors included between this value and \mathbf k + \mathbf G, it is negative.

 

Effective mass theorem for e^- in a periodic pot. with external e.m. field.
Hamiltonian for an electron in a periodic crystal potential with an applied electromagnetic field:

\displaystyle \hat H = \frac{\big(\mathbf p - q \mathbf A(\mathbf r, t)\big)^2}{2m}+V_{CR}(\mathbf r)+q \phi(\mathbf r, t)

Gauge transformations:

\displaystyle \mathbf A \rightarrow \mathbf A'= \mathbf A +\mathbf{\nabla} \Lambda
\displaystyle \phi \rightarrow \phi'= \phi -\frac{\partial \Lambda}{\partial t}

The gauge transformation can act only on the phase of the wave packet:

\displaystyle \psi \rightarrow \psi'= \psi e^{i\frac{q}{\hbar}\Lambda}

We make some assumptions:

  1. Only one energy band contribute to the Bloch’s wave packet (single band approximation?)
  2. The potential doesn’t varies appreciably whithin the extension of the packet.
  3. \Lambda = -\mathbf r \cdot \mathbf A(\langle \mathbf r \rangle, t)

with these assumptions we have:

\mathbf A'\approx 0

so:

\displaystyle i\hbar \frac{\partial \psi'}{\partial t}= \big[ \epsilon(-i\mathbf{\nabla})+q\phi'(\mathbf r, t)\big]\psi'

After some elaborations we find:

\displaystyle i\hbar \frac{\partial \psi}{\partial t}= \big[ \epsilon(-i\mathbf{\nabla} - \frac{q}{\hbar}\mathbf A(\mathbf r, t))+q\phi(\mathbf r, t)\big]\psi

 

Envelope function
We write \psi' in terms of the Bloch’s functions as in the begining but neglecting the time dependence.

\displaystyle \psi= e^{-i\frac{q}{\hbar}\Lambda} \psi'(\mathbf r, t)= e^{-i\frac{q}{\hbar}\Lambda} \sum_{\mathbf k}a_{\mathbf k_0(t)}(\mathbf k,t)\psi_{\mathbf k}(\mathbf r)

Always supposing that the periodic part of each Bloch’s components is weakly varying so that u_{\mathbf k}(\mathbf r) \approx u_{\mathbf k_0}(\mathbf r)

The envelope function now is:

 \displaystyle F(\mathbf r,t)=e^{-i\frac{q}{\hbar}\Lambda}\sum_{\mathbf k}a_{\mathbf k_0(t)}(\mathbf k,t)e^{i\mathbf k \cdot \mathbf r}

And the equation of the motion for the envelope function is found to be:

\displaystyle i\hbar \frac{\partial F}{\partial t}= \big[ \epsilon(-i\mathbf{\nabla} - \frac{q}{\hbar}\mathbf A(\mathbf r, t))+q\phi(\mathbf r, t)\big] F

where we can define the effective hamiltonian:

\displaystyle \hat H_{eff} =\epsilon(-i\mathbf{\nabla} - \frac{q}{\hbar}\mathbf A(\mathbf r, t))+q\phi(\mathbf r, t)

 

to be continued…

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Parabolic Band Approximation or Effective Mass Approximation

January 22, 2009 by Ovrec

If \mathbf k is near a central minimum of a band, it is possible to approximate the energy (band) \epsilon(\mathbf k ) with a parabolic form:

\displaystyle \epsilon(\mathbf k ) \approx \frac{\hbar^2 k^2}{2m^*}

m^* is the effective mass of the electron (← HERE is the approximation of the effect of the periodic potential) .

Rewriting the Schrödinger equation:  

\displaystyle \frac{\hbar^2 k^2}{2m^*}\psi_{n \mathbf k}(\mathbf r)=\epsilon_n(\mathbf k )\psi_{n \mathbf k}(\mathbf r)

What Solutions?

Plane waves → \psi_{n\mathbf k}(\mathbf r)\propto e^{i \mathbf k \cdot \mathbf r}

Where’s the periodic envelope? 

The Bloch’s theorem states: a solution of the Schrödinger equation for an electron in a periodic potential has the form of product between a plane wave and a periodic function having the periodicity of the direct lattice.

We must apply the periodicity condition:

\displaystyle \epsilon(\mathbf k + \mathbf G ) \approx \frac{\hbar^2 k^2}{2m^*}

to be continued…

Some links:

Fourier transforms and Blochs theorem

Double Sums and Index Shuffling

Periodic Potentials

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Ideas about Energy Bands

January 22, 2009 by Ovrec

The crystal potential in a solid has definite periodicity.

\displaystyle - \frac{\hbar^2 }{2m} \nabla^2 \psi( \mathbf r) +V_{cr}( \mathbf r)\psi( \mathbf r)= \epsilon \psi( \mathbf r)

This is the Schröd. eq. in the one electron approximation: each orbital describes a single electron subjected to the potential of the ion cores and the average potential of the other electrons.

THEOREM: Having the crystal potential the periodicity of the direct lattice, it can be written as a Fourier series in terms of the reciprocal lattice vectors.

\displaystyle V_{cr}( \mathbf r)= \sum_{ \mathbf G} U( \mathbf G) e^{i \mathbf G \cdot \mathbf r}

the series is constituted of an infinite (\mathbf G) number of Fourier’s components.

The coeficients U_{\mathbf G} are real (U_{\mathbf G}=U^*_{\mathbf G}) because we have made some assumptions:

  • V_{cr}(\mathbf r) must be real, so → U_{-\mathbf G}=U^*_{\mathbf G}
  • V_{cr}(\mathbf r) must be even, so → U_{\mathbf G}=U_{-\mathbf G}

Also the wavefuction can be written as a Fourier series but we don’t make any assumption about its periodicity:

\displaystyle \psi( \mathbf r)= \sum_{ \mathbf k} C( \mathbf k) e^{i \mathbf k \cdot \mathbf r}

here the \mathbf k are all which satisfy the periodic boundary cond. \displaystyle \frac{2\pi}{L}n.

The presence of the crystal periodic potential changes the Fourier space for the electrons. This can be viewed substituting the Fourier series for the potential and for the wave function in the Schrödinger equation. What emerges is that: because of the orthogonality property on the Fourier components, not all the values for the wavevector are allowed in the Fourier series for the wavefunction.

Making the mentioned substitution, the Schrödinger equation (that is differential equation) will be transformed in an infinite set of algebraical homogeneous equations with an infinite number of terms, in which the unknowns are the coefficients C(\mathbf k) of the Fourier components. This set is called central equation:

\displaystyle (\frac{\hbar^2 k^2 }{2m}- \epsilon) C( \mathbf k)+\sum_{ \mathbf G} U(\mathbf G) C( \mathbf k -\mathbf G)=0

  • We have an independent equation for each C(\mathbf k) → one equation for each \mathbf k. So the set has an infinite number of equations. 
  • Each equation has an infinite number of terms because it is possible to find an infinite number of \mathbf G.
  • We can cut the number of terms by cutting the Fourier series of the potential.
  • The central equation connects a given Fourier coefficient C(\mathbf k) with the infinite number of other Fourier coefficients for which the wave vector differs from \mathbf k by a reciprocal lattice vector \mathbf G.
  • Each momentum component \mathbf k is mixed with all the other components differing from it by a reciprocal lattice vector. There is not conservation of the momentum (the momentum is conserved up to a reciprocal lattice vector). →The electron changes its momentum many time during the travel in the crystal.

Solving the central equation, gives a definite wavefunction \psi_{\mathbf k} for any chosen \mathbf k as a sum over just C(\mathbf k - \mathbf G) coefficients.

\displaystyle \psi_{\mathbf k}(\mathbf r)=\sum_{\mathbf G}C(\mathbf k - \mathbf G)e^{i(\mathbf k - \mathbf G)\cdot \mathbf r}

We can write the solution as:

\displaystyle \psi_{\mathbf k}(\mathbf r)=\sum_{\mathbf G}C(\mathbf k +\mathbf G)e^{i \mathbf G\cdot \mathbf r}e^{i \mathbf k \cdot \mathbf r}

\displaystyle u_{\mathbf k}(\mathbf r)=\sum_{\mathbf G}C(\mathbf k +\mathbf G)e^{i \mathbf G\cdot \mathbf r}

\displaystyle u_{\mathbf k}(\mathbf r)=u_{\mathbf k}(\mathbf r+\mathbf T)

 

\displaystyle \psi_{\mathbf k}(\mathbf r)=u_{\mathbf k}(\mathbf r)e^{i \mathbf k \cdot \mathbf r}

from which the Bloch’s Therorem.

Somenthing good from wikipedia:
The plane wave wavevector (or Bloch wavevector) k (multiplied by the reduced Planck’s constant, this is the particle’s crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone. For a given wavevector and potential, there are a number of solutions, indexed by n, to Schrödinger’s equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wavevectors, it is called a (complete) band gap. The band structure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within the independent electron approximation.

A corollary of this result is that the Bloch wavevector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the group velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that electrical resistance in a crystalline conductor only results from imperfections and finite size which break the periodicity and interaction with phonons.

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New Universities

December 18, 2008 by Ovrec

I read the article written by Manuel Castels a Spanish sociologist, speaking about the Spain Universities and making some social considerations on the actual role of the University in the modern era and how it should be in a modern Country.
The article has been taken from the site of the “Internazionale” a popular Italian weekly magazine reporting articles taken from foreign journals and magazines.

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Remembering Square Wells…

December 16, 2008 by Ovrec
Quantum effects → V(x) must show an appreciable relative variation over a distance of the order of a wavelength.

Square Potentials → discontinuities of the first kind.

Schrödinger equation → \psi''+(\epsilon - U)\psi = 0

\epsilon > U_i → linear combination of imaginary exponentials:

Ae^{ik_ix}+ Be^{ik_ix}

\epsilon < U_i → linear combination of real exponentials:

Fe^{\kappa_ix}+ Ge^{\kappa_ix}

The parameters of these combinations are fixed by the condition of continuity of the wavefunction and of its first derivative at the points of discontinuity of the potential.

Eingenfunction → solution bounded everywhere (at both x=-\infty and x=+\infty).

Energy \epsilon lower than the potential over the entire interval (-\infty; +\infty) →No solutions

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Molecular Nanomagnets

December 12, 2008 by Ovrec
From D. Gatteschi, R. Sessoli, J. Villain book: Molecular Nanomagnets (2006)

The Spin Hamiltonian approach is widely used in various spectroscopies.

The Spin Hamiltonian approach eliminates all the orbital coordinates needed to describe the system and replaces them with spin coordinates, taking advantage of the symmetry properties of the system. An example of these approximations coming from the symmetry is the quencing of orbital angular momentum of the magnetic bricks.

The system with orbitally non-degenerate ground states are usually well treated with the Spin Hamiltonian approach.

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Interaction (Intermediate) Representation

December 11, 2008 by Ovrec
From A. Messiah’s Quantum Mechanics p.321 and 722 ff. , Dover (1999)

Any unitary transformation of the vectors and the observables of the Schrödinger (or Heisenberg) representations defines a new representation.

Any problem of quantum mechanics essentially consists of the determination of the properties of the unitary operator U(t,t_0). *

In the intermediate representation, we suppose that the Hamiltonian H can be put in the form:

\displaystyle H(t)=H_0(t)+V(t)

where H_0(t) is the Hamiltonian of a Schrödinger equation whose solution is know.

Let U_0(t,t_0) be the evolution operator corresponding to H_0. We obtain vectors and observables in the interaction representation by applying the unitary tranformation U_0^{\dagger}(t,t_0) to the Schrödinger ones.

\displaystyle \left|\psi_{\mathrm{I}}(t)\right>=U_0^{\dagger}(t,t_0)\left|\psi_S(t)\right>

\displaystyle A_{\mathrm{I}}(t)=U_0^{\dagger}(t,t_0)A_S U_0(t,t_0)

The evolution operator for the states in the intermediate representation is:

\displaystyle U_{\mathrm{I}}(t, t_0)=U_0^{\dagger}(t,t_0)U(t,t_0) 

The time dependence of U_{\mathrm{I}} is determined by:

\displaystyle V_{\mathrm{I}}(t)=U_0^{\dagger}(t,t_0) V(t) U_0(t,t_0)

The vectors in the intermediate representation move in time satisfying the Schrödinger equation:

\displaystyle i\hbar\frac{\mathrm{d}}{\mathrm{d}t}\left|\psi_{\mathrm{I}}(t)\right>=V_{\mathrm{I}}(t)\left|\psi_{\mathrm{I}}(t)\right>

On the other hand, the physical quantities are represented by moving observables that are subject to the Heisenberg equation of motion written with the “unperturbed” Hamiltonian H^{0}_{\mathrm{I}}(t):

…

The evolution operator in the interaction representation is:

→ \displaystyle U_{\mathrm{I}}(t,t_0)=1-i \hbar^{-1} \int_{t_0}^t V_{\mathrm{I}}(t') U_{\mathrm{I}}(t',t_0) dt'

This Integral equation can be solved by iteration.

To fit with the introduction of the  C. Kittel’s book Quantum Theory of Solids and Abrikosov’s Methods of Quantum Field Theory in Statistical Physics, we note that, if H_0 is time independent and putting t_0=0 we have:

\displaystyle U_0(0,t)=e^{-iH_0/\hbar}

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