Semiclassical Model of Electron Dynamics 8.1 Description of the Semiclassical Model …physics.bu.edu/~okctsui/PY543/6_notes_ Electron Dynamics... · 2013-03-25 · Semiclassical

Semiclassical Model of Electron Dynamics

by OKC Tsui based on A&M 1

8.1 Description of the Semiclassical Model Electrons in crystalline solids assume Bloch wave functions. The semiclassical model deals with the dynamics of Bloch electrons. Drude assumed that electrons collide with the fixed ions. This picture cannot account for very long mean free paths that had been found in metals, as well as their temperature dependence. On the other hand, the Bloch theory would have predicted infinite conductivity since the mean velocity of a Bloch state, v(k) = (1/ħ)(/k), is nonvanishing. This can be traced to the fact that Bloch states are stationary solutions to the Schrödinger equation incorporating the full crystal potential. So, the interaction between the electron and the fixed periodic array of ions has been fully accounted for and the ions can no longer be sources of scattering. However, no real solid is a perfect crystal. Furthermore, there are always impurities, missing ions or other imperfections that can scatter electrons. In fact, it is these that limit the conductivity of metals at very low temperatures. At high temperatures, there are thermally excited lattice vibrations producing deviations from the perfect crystal structure, which can scatter electrons and limit conductivity. While the above reveals that Drude’s picture of electron-ion scattering is inappropriate, by substituting the scattering events by the realistic ones (defect and phonons), his approach for formulating the electron dynamics is still valid. The formulation presented below describes the motion of the Bloch electrons in between collisions. The semiclassical model predicts how, in the absence of collisions, the position r and wave vector k of an electron evolve in the presence of externally appled electric and magnetic fields, assuming knowledge of the electron’s band structure, n(k), which supposedly fully accounted for the crystal field. In the course of time, with the presence of external electric and magnetic fields E(r,t) and H(r,t), the position, wave vector, and band index are taken to evolve according to the following rules:

1. The band index n is a constant of motion. The semiclassical model ignores the possibility of “interband transition”.

2. The time evolution of the position and wave vector of an electron with band index n are determined by the equations of motion:

.)(1

)(k

kkv

dt

rd nn

(8.1a)

)].,()(),([ trBkvtrEedt

kd

(8.1b)

We shall discuss the origin of these equations below.

3. The wave vector of an electron is only defined to within an additive reciprocal lattice vector K. Therefore, all distinctive wave vectors for a single band lie in a single primitive cell of the reciprocal lattice. In thermal equilibrium, the contribution to the electronic density from those electrons in the nth band with wave vectors in the infinitesimal volume element dk in the k-space is given by the usual Fermi distribution:



1]/))(exp[(

4/

4))((

3

3

Tkk

kdkdkf

Bn

n

(8.1c)

Points to Note: - One should always bear in mind that very large fields (electric or magnetic) may

cause interband transitions. You are referred to A&M Ch. 12 for the discussion. - At or near equilibrium, bands with all energies many kBT above the Fermi energy

will be unoccupied. As for those bands many kBT below the Fermi energy, we will later see that they can be igored for consideration of electronic transport properties since they are completely filled. Therefore, one often only needs to consider energy bands within a range ~kBT about the Fermi energy. This greatly reduce the number of bands needed to be considered.

For free electrons:

m

k

dt

rd

(8.2a)

)].,()(),([ trBkvtrEedt

kd

(8.2b)

The equations of motion (8.1) for Bloch electrons within each band are the same as those of free electrons (8.2) if we adopt ħ2k2/m for the electron energy n(k). However, while ħk is the momention of the free electrons (and so eqn. 8.2b is anologous to Newton’s second law), it is not for the Bloch electrons. Instead, ħk is the crystal momentum, concerned in the momentum selection rule for scattering. To appreciate eqn. 8.1b, notice that the total rate of change of an electron’s momentum, p, is given by the total force acting on the electron, including the crystal field. Since the RHS of eqn. 8.1b accounts for the forces due to the external E and B fields (without the crystal field), the equation implies that the rate of change of p – ħk is exactly accountable for by the interactions of the electron with the crystal field. Below is the proof: The Bloch wave function is: k = The expectation value of the momentum of an electron in state k is: p = <k|-iħ|k> = (8.2c) Consider the change in p produced by the external fields:

p = (8.2d) It shows that p has a piece coming from the plane wave components k + G in the original wave function. When the electron state changes by k, the amplitudes of the



plane wave components k + G change as well. Physically, this corresponds to the electron being reflected by the lattice and thereby causing those amplitudes to alter. If an incident electron with plane wave component of momentum ħk is reflected with momentum ħ(k + G), the lattice acquires the momentum -ħG. The momentum transfer to the lattice, plat, when the state k goes over to k+G is:

(8.2e) In eqn. 8.2e, use has been made of the fact that the portion of the plane wave component k + G being reflected during the state change k is: Substitute eqn. 8.2e in 8.2d, we find for the total momentum change of the system (electron + lattice), ptot = pplat = ħk as implied by eqn. (8.2b). 8.2 Semiclassical Equation of Motion in a DC Applied Electric Field The eqn. of motion, 8.1a, states that the velocity of a semiclassical Bloch electron is the group velocity of the underlying wave packet, which is perhaps readily comprehensible. But, eqn. 8.1b may not be as straightforward to justify. We reason it by considering conservation of energy for an electron traversing in the field E = , whereby

n(k(t)) e(r(t)) = constant (8.3) in the motion. The time derivative of this equation is:

rek

kn

= 0. (8.4)

By Eqn. 8.1a, Eqn. 8.4 becomes:

vn(k) [ħdk/dt e] = 0. (8.5) Since vn(k) is generally non-zero, we must have

ħdk/dt = e = eE, (8.6)

which is eqn. 8.1b, without a magnetic field. Next, observe that eqn. 8.6 is not the only condition that gives rise to energy conservation. For example, (8.5) would still be



satisfied if any term perpendicular to vn(k) is added to the RHS of Eqn. 8.6. It turns out the only appropriate term that may be added is evn(k) × B. This proves the second eqn. of motion. In most of the following discussions, we shall take the electronic equilibrium distribution function to be that appropriate to zero temperature. In metals, finite temperature effects will have negligible influence on the properties discussed below. The spirit of the following analysis is similar to that of the analysis discussed for the Drude model transport properties. That is, we shall describe collisions in terms of a simple relaxation-time approximation, and focus mostly on the motion of electrons between collisions as determined by the semiclassical equations of motion (8.1a and b). Filled Bands Are Inert

A filled band is one in which all the energies lie below the Fermi energy. Such bands cannot contribute to an electric or thermal current. To see this, notice that an infinitesimal phase space volume element dk about the point k will contribute dk/43 electrons per unit volume, all with velocity v(k)=(1/ħ)((k)/k) to the current. Summing this over all k in the Brillouin zone, we find that the total contribution to the electric and energy (thermal) current densities from a filled band are:

(8.7)

Both of these are zero since the integral over any primitive cell of the gradient of a periodic function must vanish, and (k) is periodic. Therefore, only partially filled bands need to be considered in calculating the electronic properties of a solid. This explains why Drude’s assignment to each atom of a number of conduction electrons equal to its valence had been successful. Clearly, a solid in which all bands are completely filled or empty will be an electrical insulator. Since the number of levels in each band is twice the number of primitive cells in the crystal (due to the two degenerate spin states of electrons), all bands can be filled or empty only in solids with an even number of electrons per primitive cell. 8.3 Semiclassical Motion in an Applied DC Electric Field

In a uniform dc electric field, the solution to the semiclassical equation of motion for k (8.1b) is:

.)0()(

tEektk (8.8)

Therefore, in time t, every electron changes its wave vector by the same amount. Accordingly,



.)0())((

tEekvtkv (8.9)

If the band is completely filled, this constant shift in the wave vector of all the electrons can have no effect on the electric current. This is in contrast with the free electron case, where v is proportional to k, and would thus grow linearly with time. Fig. 8.1 illustrates a typical plot of v(k). It is noteworthy that it decreases with increasing k near the two edges of the Brillouin zone. In other words, the electrons of those k states actually decelerate with the externally applied field E. This extraordinary behavior is a consequence of the additional force exerted by the crystal field, which, though is no longer explicit in the semiclassical model, lies buried in it through the dispersion relation, (k). Physically, as an electron approaches a Bragg plane, the external electric field moves it toward levels in which it is increasingly likely to be Bragg-reflected back in the opposite direction. For example, it is just in the vicinity of Bragg planes that the plane-wave levels with different wave vectors are most strongly mixed in the nearly free electron approximation. This effect leads to the curious observation that electrons that are close enough to the zone boundary have been found to behave like “holes”.

Holes Here we shall provide a detailed account for how the transport properties of electrons in some cases can be described by that of positive charges called “holes”. By Eqn. (8.7), the contribution of all the electrons in a given band to the current density is:

occupied

3)(

4kv

kdej

(8.10)

Fig. 8.1 (k) and v(k) vs. k (or vs. time, via Eqn.8.8) in one dimension (or three dimensions, in a direction parallel to a reciprocal lattice vector that determines one of the first zone faces.)



where the integral is over all the occupied levels in the band. But the integral of eqn. 8.10 over the entire band should be zero. So,

,)(4

)(4

)(4

0unoccupied

3occupied

3zone

3 kvkd

kvkd

kvkd

(8.11)

Hence we can equally well write Eqn. 8.10 as:

.)(4

)(unoccupied

3 kvkd

ej

(8.12)

It follows that the current produced by occupying a specified set of levels with electrons is precisely the same as the current that would be produced if (i) the specified levels were unoccupied AND (ii) all the other levels in the band were occupied but with particles of charge +e. Such fictitious particles of positive charge occupying the levels unoccupied by the electrons are called holes. When one chooses to regard a current as being carried by positive holes rather than by negative electrons, one should regard those states occupied by electrons to be unoccupied by holes, and vice versa. However, for any given band, one should never adopt both pictures (hole and electron) for the charge carriers. Nevertheless, one may regard some bands by the electron picture, and the other bands by the hole picture, for one’s convenience. Under an applied electric field, the unoccupied levels in a band evolve precisely as if they were occupied by real electrons (of charge –e). That is, if there are applied E and B fields, the motion of both electrons and holes is governed by the same equation:

.1

)(

Hv

cEek

(8.13)

This is because eqn. (8.13) describes how the occupied orbitals evolve with time; and the unoccupied orbitals have to evolve in the same manner because a newly occupied orbital is necessarily accompanied by a newly emptied orbital, etc., which requires the unoccupied states to evolve in the same manner do the occupied ones. (see pp. 226-227 of A&M.) Given eqn. 8.13, how can holes be distinguished from electrons?

In classical treatment, the RHS of eqn. (8.13) is the force acting on a charged particle due to E and B, and would have been set equal to the mass of that particle times dv/dt. So, if dv/dt // dk/dt, the electron orbit would resemble that of a free particle with negative charge. On the other hand, if dv/dt is // -dk/dt, the electron orbit would resemble that of a free particle of positive charge. It turns out, it is more often the case that dv/dt is directed opposite to dk/dt when the k orbital is unoccupied. This may be perceived from the following: At equilibrium or near equilibrium (which is the condition found in most cases of interest for electron transports), the unoccupied levels usually lie near the top of the band. If the band energy (k) has its maximum value at k0, say, then if k is sufficiently close to k0, we may expand (k) about k0. The linear term in (k k0) vanishes because k0 is a maximum point. If we assume that k0 is a point with sufficently high symmetry, then



,)()()( 200 kkAkk

(8.14)

where A is a positive constant. Rewrite A as:

.*2

2

Am

(8.15)

Hence, for levels near k0, we have

,*

)(1)( 0

m

kk

kkv

(8.16)

So, ,*

)( km

kvdt

da

(8.17)

This shows that the acceleration, a, of states near the top of a band is opposite to k

. It follows from eqn. (8.13) that acceleration of these states are opposite to the electric force, making these electrons behave like positive-charges. By regarding those near-top electrons as particles with charge +e, Eqn. 8.17 shows that they move as if they have an effective mass of +m*. We can demonstrate the above point in a more general way. Consider:

,)(1)(1)(1 2

ij

jji

ii i

i kkk

kk

k

k

dt

dk

k

k

dt

dkv

dt

dkak

(8.18)

Then a is opposite to k

if

). any vector(for 0)(2

ijj

jii kk

k (8.19)

Suppose the local maximum of (k) is at k0. In the vicinity of k0, one can expand (k) as:

ijj

jii kk

kkk

)(

2

1)()(

2

0

. (8.20)

It follows that Eqn. 8.19 must hold for (k0) to qualify to be a local maximum. The quantity m* in Eqn. 8.15 determining the dynamics of the electrons near band maxima of high symmetry is known as the “hole effective mass”. More generally, one defines an “effective mass tensor” by:

ijj

i

ijjiij

k

kv

k

kv

kk

k

kk

kkM

)(1)(1)(1)(1)(

2

2

2

21

, (8.21)

where the sign is – or + according to whether k is near a band maximum (holes) or minimum (electrons), respectively. By using eqn. (8.21), we can write:



.)(1 kkMdt

kd

k

v

dt

vda

(8.22)

Hence, the equation of motion (8.13) takes the form:

).)(1

()( Hkvc

EeakM

(8.23)

If the pocket of holes (or electrons) is small enough, one can replace the mass tensor by its value at the maximum (minimum), leading to a linear equation only slightly more complicated than that for free particles. Such equations describe quite accurately the dynamics of electrons and holes in semiconductors.

8.4 Semiclassical Motion in a Uniform Magnetic Field

In a uniform magnetic field, the semiclassical equations are

,)(1

)(k

kkvr

(8.24)

.)( Bkvek

(8.25) These equations immediately evident that the component of k along the field B and the electronic energy (k) are both constants of motion. (The latter is because d(k)/dt = F v = e(v B) v = 0.) These conservation laws dictate that electronic orbits in k-space are curves given by the intersection of surfaces of constant energy with planes perpendicular to the magnetic field (Fig. 8.2). The sense in which the orbit is traversed follows by observing that v(k), being ~k, points to higher energies in the k-space. Therefore, closed k-space orbits surrounding levels of higher energies (hole orbits) are traversed in the opposite sense to closed orbits surrounding levels of lower energy (electron orbits). In other words, the sense of orbital motion determines whether the carriers are electron- or hole-like under applied magnetic field.

Fig. 8.2

In the example show, v is pointing away from k = 0 and so the particle is electron-like. Convince yourself that the sense of rotation of the orbit reverses when the particle is hole-like or v is pointing towards k = 0.



The projection of the real space orbit in a plane perpendicular to the field,

)ˆ(ˆ rBBrr , can be found by taking the vector product of both sides of Eqn. 8.25

with a unit vector parallel to the field B. This yields

reBrBBreBkB

))ˆ(ˆ(ˆ , (8.26) which integrates to:

(8.27) -------------------------------------------------------------

Below is a derivation of Eqn. (8.27) and exploration of its physical meaning. )ˆ(ˆ rBB

----------------------------------------------------------------------- The above derivation shows that the projection of the real space orbit in a plane perpendicular to the field is simply the k-space orbit, rotated through 90o about the field direction and scaled by the factor ħ/eB. But it should be noted that orbits in semiclassical generalization need not be closed curves (Fig. 8.4).

BB

2

v (-ħ/e)B k

1

B



It is of interest to express the rate at which the orbit is traversed. Consider an orbit of energy in a particular plane perpendicular to the field B in the k-space (Fig. 8.5a). The time it takes to traverse that portion of the orbit lying between k1 and k2 is

(8.28) By Eqns. 8.24 and 8.25,

k

eBB

k

ek

22 (8.29)

Therefore, Eqn. 8.28 can be rewritten as:

(8.30) where (/k) is the component of /k perpendicular to the field, i.e., its projection in the plane of the orbit. The quantity has the following geometrical meaning: In the plane of the orbit for electron energy , we defined the vector (k) to be one perpendicular to the orbit at point k, and connects the point k to a neighboring orbit in the same plane (i.e., same k) of energy + (Fig. 8.5b). When is very small, we have

B

Fig. 8.3 The projection of the r-space orbit (b) in a plane perpendicular to the field is obtained from the k-space orbit (a) by scaling with the factor ħ/eB and rotation through 90o about the axis determined by B.

B B



(8.31) Since /k is perpendicular to surfaces of constant energy, the vector (/k) is perpendicular to the orbit and B, and hence parallel to (k). So, Eqn. 8.31 can be written as:

(8.32) and Eqn. 8.30 becomes:

(8.33)

Fig. 8.4 Presentation in the repeated-zone scheme of a constant-energy surface with simple cubic symmetry, capable of giving rise to open orbits in suitably oriented magnetic fields. One such orbit is shown for a magnetic field parallel to [101].

Fig. 8.5 The geometry of orbit dynamics. (a) The time of flight between k1 and k1 is given by (ħ2/eB)k1

k2 dk/|(/k)| (b) A section of (a) in a plane perpendicular to B containing the orbits with the same kz. The shaded area is (A1,2/).

B



The integral in (8.33) gives the area of the plane between the two neighboring orbits from k1 to k2 (Fig. 8.5b), A1,2. Hence if we take the limit of (8.33) as 0, we have

. (8.34) Eqn. 8.34 is most often used in cases where the orbit is a simple closed curve, and k1 and k2 are chosen to give a single complete circuit (k1 = k2). The quantity t2 – t1 is then the period T of the orbit. If A is the k-space area enclosed by the orbit in its plane, then Eqn. 8.34 gives

(8.35) Compared to the free electron result, i.e.,

(8.36) it is customary to define a cyclotron effective mass m*(,kz):

(8.37) 8.5 Semiclassical Motion in Perpendicular Uniform Electric and Magnetic Fields When we discussed the Hall effect in the Drude model, we only qualitatively considered the electron motion under crossed E and B fields. (Recall: We balanced the Lorentz force and the electric force in the transverse direction and thereby derived that RH = Ey/(jxB) = 1/ne.) Here, we discuss it in the semi-classical model. When a uniform electric field E is added with a perpendicular uniform magnetic field B with |E| < |B|, Eqn. 8.27 for the projection of the real space orbit in a plane perpendicular to B acquires an additional term

(8.38) where

(8.39) Note that w is the the velocity of the frame of reference in which the electric field vanishes (see Jackson’s book pp. 582-584). It follows that electrons in crossed E and B fields move in a superposition of circular orbits (as if only the B field is present) plus a linear drift at the velocity w in the r-space (see Fig. 8.6).

B

B

B

B

BB



Recall that d/dt = (/k)·(dk/dt) = ħv·[(-e/ħ)(vB + E)] = -ev·E. Since v is in general not perpendicular to E, d(k)/dt 0 and (k) may vary with time. When E and B are perpendicular, one can show that the equation of motion (8.1b) can be written in the form (see below):

(8.40) where

(8.41) One may appreciate the physical origin of these equations by perceiving that Eqn. 8.41 is the energy of a free electron in the reference frame moving with velocity w, and Eqn. 8.40 is the equation of motion an electron would have if only the magnetic field B were present, and the band structure given by . Given Eqns. (8.40) & (8.41), the k-space orbits are given by intersections of surfaces of constant. with planes perpendicular to the magnetic field. Note that in cases where |E| > |B|, the orbit will be hyperbolic and not closed. --------------------------------------------------------------------- Below is a derivation of Eqn. (8.41):

B w

Fig. 8.6 E B drift of charges +e and –e in crossed E and B fields.

B

Bigger radius when v//-E.

Smaller radius when v//E.

(k) (k)



This gives . ------------------------------------------------------------------ 8.6 High-Field Hall Effect and Magnetoresistance (in theor long limit) In this section, we shall analyse the case for crossed E and B field where B is very large of the order of 1 T or more (so |w| = E/B << c), and differs only slightly from (k). The limiting behavior of the current for these cases turn out to be quite different depending on whether (a) all occupied (or all unoccupied) electronic levels lie on orbits that are closed curves or (b) some of the occupied and unoccupied levels lie on orbits that do not close on themseleves, but are “open” in k-space. (a) Cases where all occupied (or all unoccupied) orbits are closed. We shall take the high magnetic field condition to mean that these orbits can traverse many times between successive collisions. In the free electron case, this reduces to the condition c >> 1, where is the relaxation time and c is the cyclotron frequency (see Chapter 1).

Suppose that the period T is small compared with the relaxation time, , for every orbit containing occupied levels. To calculate the current, j, one uses j = ne<v>. Here, <v> is the average velocity of an electron between two collisions, which may be taken to be from t = to t = . By Eqn. 8.38, we have

(8.42) Since we are considering for the case where all the occupied orbits are closed, k = k(0) – k() is bounded in time. So for sufficiently large , the contribution from the drift velocity w dominates the average velocity. This provides the long- limit to the current

(k)

BB

B BE

The triple product would not be E-hat had E been not perpendicular to B.



(8.43)

But if it is the unoccupied levels that all lie on closed orbits, the corresponding result is

(8.44) Eqns. 8.43 and 8.44 suggest that when all relevant orbits are closed, the deflection of the Lorentz force is so effective in preventing electrons from acquiring energy from the electric field that the uniform drift velocity w perpendicular to E gives the dominant contribution to the current. Recall that the definition of the Hall coefficient is the component of the electric field perpendicular to the current, divided by the product of the magnetic field and the current density. We have RH = Etransverse/(jtotB) = E/(newB) = 1/(ne):

(8.45) Note that the use of the symbol R is to indicate that the expression is valid under the condition /T . It is remarkable that Eqn. 8.45 gives an identical result as that found in the free free electron case, which preserves the notion that the high-field Hall coefficient R provides a valuable measure of the electron (or hole) density. It is also remarkable that Eqn. 8.45 allows for the possibility of a positive Hall coefficient. If several bands contribute to the current density and each of them has only closed electron (or hole) orbits, then Eqn. 8.43 or 8.44 holds separately for each band, and the total current density in the high-field limit will be

)ˆ(lim/

BEB

enj eff

T

, (8.46)

where neff is the total density of electrons minus the total density of holes. The corresponding expression for R is

enR

eff

1 . (8.47)

Furthermore, it can be shown that the corrections to the high-field current densities (Eqns. 8.43 & 8.44) are smaller by a factor of order 1/c (Problem 5 of A&M Ch. 12), the transverse magnetoresistance approaches a field-independent constant in the high-field limit. (b) Case 2 Some orbits are open. More specifically, we refer to those cases where there are energy bands near the Fermi surface containing open orbits (Fig. 8.4). Electrons in open orbits are not forced by the magnetic field to undergo a periodic motion along the direction of E. (This is in contrast with the case of closed orbits where the electrons spend equal amount of time traveling along and opposite to E (Fig. 8.6).) Therefore, the

BB

BB



electrons may acquire energy from the E field. In particular, if the unbounded orbit stretches in a real space direction n , one would expect to find a finite contribution to the current directed along n and proportional to the projection of E along n :

(8.48) Fig. 8.7 illustrates the physical origin of how an open orbit may give rise to a net electric current along n . The possibility of a non-vansihing term not parallel to w in the high-field limit is also in accord with the general result of Eqn. (8.42). For an electron traversing in an open orbit, the growth in its wave vector k due to the first term is unbounded, enabling its contribution to j to dominate that of the second term. Because the rate at which the orbit is traversed is proportional to B, one expects this contribution to the average velocity to be independent of B and directed along the real space direction of the open orbit n . To appreciate the limiting behavior implied by Eqn. (8.48) on the high-field magnetoresistance, consider an experiment in which the direction of current flow does not lie along the direction of the open orbit in real space n (which happens when n does not lie parallel to the direction where the circuit is closed, see Fig. 8.8). When the B field is high, by Eqn. (8.48) j can be misaligned from n only if E n = 0. So, if we write

Contribution from projection of open orbit parallel to E

Contribution from components of the open orbit perpendicular to E, and from the closed orbits, if any.

Fig. 8.7

vk

8.41

j, n



the E field in the following form:

(8.49)

where n ’ is a unit vector perpendicular to both n and Ĥ ( n ’= n Ĥ), E(0) constant and E(1) 0 as H . By definition, the magnetoresistance, , is :

(8.50) When j is not parallel to n , E E(0) n ’ in the high-field limit, which gives the corresponding limit of :

(8.51) To find (E(0)/j), subs. Eqn. 8.49 in Eqn. 8.48:

(8.52) Next, take the dot product with n ’on both sides, and use the fact that n ’ n = 0, one has:

(8.53) which gives

(8.54)

Fig. 8.8



Subs. this result in Eqn. 8.51, we find:

(8.55) Since (1) vanishes in the high-field limit, this gives a magnetoresistance that grows without limit with increasing field, and is proportional to the square of the sine of the angle between j and the open orbit n (since n is perpendicular to n ’ in the same plane with j). Therefore, the semiclassical model resolves another anomaly of free electron theory, providing possible mechanisms by which the magnetoresistance can grow without limit with increasing magnetic field.

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Semiclassical Model of Electron Dynamics 8.1 Description of the Semiclassical Model …physics.bu.edu/~okctsui/PY543/6_notes_ Electron Dynamics... · 2013-03-25 · Semiclassical