Bands of energy

1) Evolution of the model of the gas of Firm:

The model of the gas of Firm neglects 2 important phenomena:

to)       reasonable ? that is an interaction electron-electron, it is associated upgrades them ignoto but constant.

b)       It is reasonable that there is an interaction with the Ionian ones of the reticulum, is associated upgrades them periodic in tightened analogy with the periodic nature with which atoms they are introduced in the reticulum in how much must be held account that the electron endures one strongly repulsion from part of the nuclei in virtue of the principle of exclusion of Pauli, since in correspondence of the nuclei all the states are occupied.

 

2) Equation of wave for an electron in upgrades them periodic:

We consider a linear reticulum in which the atoms are distance to you of to between of they. In the equation of Schroedinger they are replaced:

with)       the series of Fourier of ythe (x) added on the single ones k you concur yourself from the conditions with the contour .

b)       the series of Fourier of upgrades them periodic u(x) added on the G carriers of the mutual reticulum .

Obtaining from which having place K' =G K but being a dumb index can also be risostituire with K therefore placing it has uguagliando the coefficients of the 2 series and carrying all to 1° the member is had that is said system of equations centers them.

 

3) Theorem of Block:

It asserts that the autofunzioni of the equation of Schroedinger for upgrade them periodic have the shape that is are the product of a flat wave for upgrade having them the same regularity of the reticulum.

We had assumed for ythe (x) a shape of the type.

From the system of equations they centers looks at that the k allowed for the wave function, they are not a continuous one but they are of the shape where k is chosen to case between the k is concurred from the conditions with the contour. She has herself: where it is upgrades them periodic since the series carried out on the carriers of the mutual reticulum, is invariant for translation of the same one.

 

4) Considerations on the impulse for the electron of Block:

It verification the relation to the base of the theory of Schroedinger in fact does not have itself: therefore k is not the moment of the electron of Block and comes therefore called nearly moment or crystalline moment.

 

5) Model of nearly free electrons:

Considering 2 equations drawn from the system of equations centers them that they only differ fora carrier G ₁ of the mutual reticulum, ha is noticed that not there are summary the relative ones to the energy upgrades them because in the model of nearly free electrons it is assumed that the kinetic energy is much greater one of the energy upgrades them, we could neglect it completely but we hold 2 terms in order to facilitate the calculations, assumes moreover that equal function upgrades them periodic is one therefore . The previous system has solutions single if the determining one is 0, is found that the roots are , to this point if it is chosen as index of the autofunzioni coincides with the edge of 1ª the zone, obtains and therefore e . Replacing in the departure equations is determined and and from they the y that they turn out to be 2 standing waves one of type type cosine for which has that the density of probability to find electrons is maximum in correspondence of the Ionian ones positi to you and therefore it diminishes the energy, while the other is of type breast and the density of probability to find electrons is maximum between two Ionian ones therefore is the solution to higher energy.

 

6) Gap of energy:

It is the band of forbidden energies that it gushes from the equation of the electron of Block. We have inasmuch as on the zone edge e is had therefore the gap is worth 2u₁ . You notice yourself that in the model of the Firm gas of u₁ = 0 and therefore the gap not is and a continuous parabola is had.

 

7) Number of states in one band:

We had already inasmuch as applying the conditions to the contour of Karmann one finds that the n° of k compatible contained in 1ª the zone it is just equal N that is to the atom number of which the linear chain is made up, but the electrons are fermions therefore us can be 2 electrons with same k but spin the opposite one, therefore to the inside of 1ª the zone 2N they have been available.

 

8) Determination of the metals with the theory of the bands:

Everyone of the N atoms of a metal can supply an electron which in total will be N and therefore they can only fill up the half of 1ª the composed band from 2N be free, these electrons are therefore free to absorb energy and to pass to states to elevated energy more. In a generalized manner all are metals the atoms with valence an uneven electron number.

 

9) Determination of the insulators with the theory of the bands:

Everyone of the N atoms of an insulator can supply two electrons which in total will be 2N and therefore they completely fill up 1ª the composed band from 2N be free, since the free states are separate to you from the top of the elevated full band from a gap, these electrons cannot absorb therefore to transport energy. In a generalized manner all are insulators the atoms with a n° equal of valence electrons.

 

10) Determination of the semiconductors with the theory of the bands:

The N semiconductor atoms has medium valence 4 therefore in total is 4N atoms and therefore the first 2 bands will be completely full like for an insulator, from which but they distinguish in how much the gap between the 2ª and 3ª the band is much small, therefore already to ambient temperature electrons in conduction band are had.

 

11) Outline of the reduced zone:

It is a procedure that consists in making that the carrier crystalline moment k index of the function of Block belongs to 1ª the zone of Brillouin, that it is always possible adding an opportune G carrier of the mutual reticulum.

 

12) Outline of the repeated zone:

It is obtained repeating space along the axis of the k, how much has been obtained for the k of 1ª the zone, one is obtained in such a way function andperiodic _K.

 

13) Outline of the extensive zone:

It is had that for every zone it comes brought back the corrected value of the energy of the states available, without that is to compress all to the inside of 1ª the zone.

 

14) Like designing the surface of Firm for free electrons:

They design 1ª, 2ª, 3ª the zone with the procedure of Wiegner Seitz, then considers the circumference of Firm which contains 1ª the zone entire, for the other zones other is not made that to characterize the intersection with the circumference of Firm and traslare these areas to the inside of 1ª the zone so that they create with connected.

 

15) Like designing the surface of Firm for nearly free electrons:

The surface of Firm is gained before for free electrons and then it is necessary to hold account that to the zone edge the v_g it is cancelled and therefore the gradient of the energy regarding k is worth 0 that is us cannot be of the discontinuities, therefore is necessary to dull the angles.

 

16) Effect of an electric field and on the electron of Block:

Applying to a fieldin a direction, integrating the is had therefore k increases in the time in opposite direction to until to that, reached on the zone edge it endures Umklapp and ricompare from the other side of 1ª the zone that is with k a negative, is that is come to create one alternating current, that in truth not verification fondamentamente for the presence of two causes which render upgrade them not periodic to the contrary of how much are asserted in the hypothesis of Block:

to)       imperfections in the reticulum like gaps or impurità.

b)       to 0K the reticulum it is only firm while to greater temperatures, the reticulum is in vibration and in average the regularity of the same reticulum is only had.

 

17) Motion of an electron in presence of a magnetic field:

The electrons cover the lines of Firm in fact have and of the rest therefore the force is always orthogonal to the but it is orthogonal to the superficial ones to constant energy as exactly the lines of Firm along which after all the motion of the electron is carried out. To second about the distance back electron is spoken about orbit of type or type gap. Type orbits are had gap when the top of a full band possesses of the empty states.

 

18) Gaps:

Detention remaining that the only bearers of load is the electrons and has loads negative, the hypothesis of their existence is much profit in order to explain the following situation:

There is a completely full band unless one be free (gap) in top to it, applying a field and_z in the direction of the k increasing it is had that the electrons (..e implicitly the gap) are moved on the left being . Considering a successive moment to the application of the field it is had that the gap has been moved, and its speed is increased (..corrisponde to the tangent to the band). Such result can be motivated or assigning to the gap it loads negative and a negative mass, or in agreement to the classic mechanics, a positive mass and loads positive.

 

19) Second law of dynamics for free electrons:

Part from the determination of the job completed from the field and on an electron in the time dt moreover and therefore uguagliando the dand has therefore apparently is respected 2ª the law of dynamics, apparently in how much k is not the impulse of the electron of Block since it does not respect .

 

20) effective Mass:

It is a measure of the inertia that an electron of Block introduces when to it an external force is applied, is gained deriving regarding the time the v_g has: and therefore is had where m^* are the effective mass that therefore is inversely proporziona them to the curving of the curve of dispersion in the space of the k, in particular in correspondence to the zone edge the curving are negative therefore it are also the effective mass, that it wants to say that applying a force directed towards the zone edge, the motion happens in opposite sense, that happens for via of the reflection to the Bragg. In analogous way if it is approached to us the edge of zone from the advanced band the curving is positive and with it the effective mass, that it is small therefore is like if a recoil effect facilitates the electron to go away from the zone edge.

 

21) Resistività of a metal:

The resistività r of a metal is function is of the temperature that of the imperfections of the reticulum, in particular is from chargeing to this last residual resistività r₀ that is had also when the temperature comes down to 0 K. It is had that the resistività leaves from value r₀ grows like T⁵ for T < < q_D in order then to grow like T to temperatures T < < q_D .

It is important however to notice that for a metal the resistività grows to growing of the temperature while for a semiconductor the resistività diminishes to growing of the temperature.

 

22) Explanation of the linear course of the resistività to the high temperatures:

It is necessary to make reference to the traslata sphere of Firm and to the modalities in order to bring back it in the position it begins them once that the electric field does not act more. The resistività increases for via of hits of Umklapp type of the fononi with electrons, the fononi useful they are those with k similar to k_{the Debye} and their number linearly increases with the temperature and it the resistività.

 

23) Explanation of the course of type T⁵ of the resistività in function of the temperature:

With decreasing of the temperature, it is observed from the function of Bose-Einstein that the number of the fononi able to give place to hits of Umklapp type decreases in exponential way, but becomes the number of the fononi meaningful that as a result of happened to you hits with k more small succeed to obtain the same result.