In the dispersion curves of a p-type crystal, there are unoccupied states at the top of the valence bands (holes). These holes are regarded as 'charged particles' in solid state physics and have their effective masses (with the sign opposite to that of electron carriers). In general, there is no gap between the occupied- and unoccupied states. Therefore any excitation, as thermal excitation mentioned by Ravi, will cause the motion of the holes. Naturally, there are different ways to describe the charge carriers. The band energy of intermediate electron which will be able to move depends on the excitation, as well.  

Holes and electrons are both quasi-particles in this picture, Bijit. If semiconductor is purely p-type, then all electrons are bounded.

     Dear Bijit,

     Let me try to answer you in a very concrete form. A material of p-type if an atom from fills the hole with one electron, this means that it will leave a new hole adjacent to it. This process shall continue as in a domino and the the hole will be moving in the opposite direction to the electronic current.

    Actually the atoms are remaining fixed in the lattice, although there is an illusion that the holes are physically moving.

    One physical example is a crystalline silicon lattice having each atom sharing all of its four valence electrons among their neighbouring atoms. If you dope it with boron which only offers three electrons, thus you have a p-semiconductor that there are holes allowed for the transport as I have said previously. 

     I hope that this can help.

Thanks to all of you sir for your efforts. I would like to apologize for being too vague regarding the query. My main question is whether the electron associated with hole movement will be flowing with  valence band or conduction band energy. As I know electrons in valence band energy levels are not allowed to conduct current.

   Dear Bijit,

   I think that your question was answered.

1. Holes are on doped semiconductors just on the upper part of the valence band. And although they are below the Fermi level, there electrons in valence band which can be easily be excited to the holes allowing them to be "mobile" in the valence band.

2. Their motion is in fact ficticious and an illution because the electrons are the ones which hopping from hole to hole. Thus speaking about the bonding energy of holes is only a metaphore.

3. I think that your question is interesting and no trivial because the motion of the holes is quite tricky and below an allowed band permitted for the associated electrons in the transport.

4. Notice that your question does not enter in a quasiparticle, excton, allowed in certain semiconductors where the electron-hole are coupled during certain time life.

Thank you sir Daniel Baldomir again. As you have said " there electrons in valence band which can be easily be excited to the holes allowing them to be "mobile" in the valence band ..........because the electrons are the ones which hopping from hole to hole.", so I can assume that flow of hole is due to the movement of electron and naturally these electrons will flow with conduction band energy.

"Now my question is if a hole movement is meant by merging of an electron from covalent bond with a hole thus leaving another hole behind, then what is the energy of this intermediate electron? Will it have energy in conduction band or valence band?"

The answer is easy: It is the energy of the valence band, because from the wavefunction(s) of this band both the hole and the corresponding electrons (occupying binding states!) are derived.

Also, given that hole conduction is the dominating conduction mechanism, this means that the thermal energy is too small to provide a siginifcant density of electrons in the conduction band, thus ruling out the other option mentioned in your question.

Note, however, that with this kind of reasoning about valence-band electrons it will be very difficult to understand p-type conductivity. So, this brings me to the question why you are interested in this aspect at all; what is your deeper question behind?

    Dear Bijit,

    Dear Bijit,

   No, the electrons that you move are below the Fermi level but belong to the valence band (we are in a semiconductor). And in fact, it appers an oposite current to this motion of the holes that such electrons which leave.

   The difficulty is to understand that the holes are only ficticious sources moving in the valence band where the electrons are more bonding than in the conduction band.

    I hope that I have explained me enough clear.

I daresay that the difficulty is to accept that for all properties related to p-type conductivity the holes are "more real" than the valence-band electrons. Just to mention one practical, classical example: Have a look at the Hall effect! How to explain the different sign of the Hall voltage for n-type and p-type conductivity?

   Dear Jean-Martin,

   Hall effect (classical) only give us a positive or negative potential in the voltmeter, which tell us the direction of the Hall electric field. Thus in one case the magnetic field has curved positive (holes) or negative(electrons) carriers. Nothing more!

   What is clear is that we haven't real positive charges as could be positrons and we need to explain how the positive charges are produced with covalent bond interaction in lattice. This was what I have understood in the question. Say, not to accept the existance of positive charges (holes) but to understand how they are possible to survive if we have only electrons as carriers. Obviously the Hall effect only tell us that such oposite currents exists but no what is their origin.

     Dear Jean-Martin,

   Perhaps it would simpler to use the formula of Lorentz force

          F = -e v x B

In Hall effect e is always the same and corresponds to the charge of the electron, BUT the velocity v for the holes is changed of sign, due to be in the valence band and only allow a very short lenght motion for the electron (hopping). Thus the drift electric field only gives a hole that is collectively occuped and which produces an oposite current, or velocity, changing then the electric polarizability in Hall effect.

       F= -e (-v) x B= +e v x B

This is what I call ficticious holes as positive electric charges. Notice once this is understood, the question of if there is a Coulombic interaction between holes or its energy is out of consideration. 

Dear Bijit,

The usual simplistic view of hole conduction is the absence of an electron in the valence band that allows the appearance of a vacant state to move through electron states. However, that is a simple depiction. What is actually happening is a motion in momentum space, where holes are moving in all directions in momentum space. Without an applied voltage, this works out to be a net of zero. With an applied bias, the solution is non-zero because electrons are influenced  by the electric field. Note that electrons are still moving in all directions, its just that more move in the direction of the applied force.

The motion of electrons in the valence band has a different effect than electron motion in the conduction band, mainly because the band curvature in the E-k diagrams are opposite for the conduction band minimum and the valence band maxima (two bands, heavy and light). Hence, electrons have opposite signs of effective mass in the different bands. It is the differences in these occupied levels that give rise to the positive and negative nature of the charge carriers. The electrons in the valence band must have energies that comply with the band curvature at a given momentum, as do the representative energy states (empty or not).

For a better view of this issue, I suggest that you read the excellent introduction by R.F. Pierret, Advanced Semiconductor Fundamentals, Vol. VI, Addison-Wesley, 1989, section 3.2, pp. 57-73. There will see why the concept of the hole works.

I hope that this helps.

Douglas

     Dear Douglas,

     You have forgotten that the effective velocity of the holes is opposite to the electrons that they left in the first Brillouin zone (if you want to use the concavity or convexity of the Bloch bands). This datail is fundamental for understanding the Hall effect that we have previously discussed.

Dear Daniel, why do you think the direction of the effective hole velocity was not considered? Remember how to shift from the "electron picture" to the "hole picture": What happens to the effective mass, what happens to the momentum?

     Dear Jan,

   I was answering to Douglas who was showing the physical features of the holes quasiparticles and only has told him that he has forgotten to say that their effective velocity is opposite to the electrons that they leave. This is fundamental for understanding their dynamics, over all when you apply simultaneously two fields, electric and magnetic, as it happens in the Hall effect that you have discussed.

  If you want I can remember the main ideal characteristics of the holes:

  1. The momenta are

         p= - (h bar) k

  2. As the hole band effective mass is proportional to its curvature, thus mass of the hole is equal and opposite to that of the missing electron in BZ (at least for an intrinsic semiconductor and no in general).

  3. The energy of the hole is equal and opposite to that of the missing electron.

  4. The electric charge is positive and opposite to the one of the electron.

Electron may have any energy, because phonons and photons take part in merging proces.

Dear Daniel,

Thank you for the note. Indeed, I did not explicitly mention that the velocity of holes is opposite of electrons, although it is implied by the change in sign for the effective masses. This is should be obvious because the charge carrier mobilities are functions of the effective masses.

Regardless, I appreciate that you make that point clear.

   By the way, we are assuming that the conduction band is equal to the valence band, just inverted in energy with respect to the Fermi level. Obviously this is not the real case and the mass is a parameter which has to be calculated through the density of states instead of the band. For instance, in germanium, the effective mass of the electron is only 0.041 the mass of the electron in vacuum and the hole 0.28 times, i.e. around 7 times higher. This tell us that the kinetic energy is very different for electrons and holes just considering the mass. On the other hand, the mass depends also very strongly of the direction of the crystal that it moves (anisotropy). This means that the mass needs to have a detailed electronic structure of the semiconductor. Obviously all these details are not the same for the effective velocity, although it shares many features.

   In any case, these are details which carry us far of the original question. I have understood it for creating an excess of holes, for instance by substitutional impurities that have fewer valence electrons per atom than the lattice matrix. Thus a bond with the neighbors is incomplete and cab be imagined as a hole weakly bound to the impurity atom. For instance, elements as (B, Al, Ga) are donors for semiconductors as Si and Ge which in the next column of the periodic table (III and IV respectively). The energy and the electronic bond are quite complex in a concrete case, but the basic physical ideas I think that have been explained.

  What do you think Bijit?

Thank you sir for all your replies. I have an assumption in my mind. In my logic, if a hole breaks a bond and hole movement is achieved, the associated electron energy will be in conduction band as the typical energy band diagram is observed in light of electron energy. So this associated electron will move with conduction band level energy and thus the hole movement will be possible. Please rectify if you think the idea is not correct.

Well, compare the total energy before and after the movement. Why should there be any conduction band energies be involved? Do you think of a "virtual conduction band electron" doing the job (being just temporarily excited)? But why would you need such a complicated picture; what do you think is missing in the simple explanation that just valence band particles are moving?

Jan-Martin Wagner Sir, my question is that whether the electrons in valence band are allowed for current conduction or not.

    Dear Bijit,

    Your question is not simple to answer in a clear form from the real electronic structure, although in most of text books (over all in Solid State Physics) it is assumed in some simple words. It seems that saying words as valence band, conduction band, holes or electrons everything is understood.

   Let me go to more basic things taking the Si as a real example. This is an atom with diamond structure, i.e. with each and every silicon atom having four other silicon atoms as its nearest neighbor atoms.Silicon is a group IV element in the periodic table and has four valence electrons. Every pair of Si atoms share two electrons with quite strong covalent bond of 1.1 eV of energy. THUS WE HAVEN'T FREE ELECTRONS, WHICH ARE ALSO CALLED CONDUCTION ELECTRONS. This is not a metallic structure at all!.

   Coming back to the mechanism of creating holes in the valence band, the densities of thermally generated electrons and holes in semiconductors are generally very small at room temperature given that the thermal energy is only 25 meV and we need 1.1 eV, i.e. more than ten thousand times. How can we overcome this difficulty? By doping, i.e. introducing suitable impurity atoms which below the effective energy to loose electrons (physically changing the Fermi level). As I told you (B, Al, Ga) are elements of the III column of the periodic table which can accept one COVALENT electron of the Si leaving a hole close to the Si. That is, a place where the energy to accept an electron is much lower than the one to move an electron! This is the key idea! Notice that in this case, no hole is created in conjunction with the creation of a conduction electron and therefore we haven't the initial problem of the zero whole current. These are the P-type semiconductors which have a much higher current due to the holes than to the electrons in the conduction band.

   I hope that you can see that the electrons cannot produce currents in the valence band and what are the energy and mechanisms involved.

   Let me to suggest you two texts for going deeper in this issue:

   Adrian P. Sutton, Electronic structure of materials, Clarendon Press-Oxford, 1993.

   N.W.Ashcroft and N.D.Mermin, Solid State Physics, Harcourt College Publishers, 1976.

Thank you sir

    You are welcome.

Current transport in a semiconductor (and in other materials) is commonly treated by the mixed classical/quantum-mechanical Boltzmann transport equation within the relaxation time approximation (cf. http://www.tf.uni-kiel.de/matwis/amat/stat_meth/kap_1/backbone/r_ch5.html). The first-order perturbed Fermi distribution, which is the dominant function for all first-order perturbation effects (ohmic law, particle diffusion and heat transport phenomena), depends on the derivative of the Fermi distribution with respect to the energy. This derivative is mainly zero, except in the "soft zone" around the Fermi energy (where the Fermi distribution goes from 1 to 0). This means that only charge carriers in the energetical vicinity of the Fermi energy can contribute to electrical current flow. In p-doped semiconductors, the Fermi energy is close to the valence band, so only valence band particles are involved in current conduction.

Thanks Jan-Martin Wagner sir. 

Your question was "whether the electrons in valence band are allowed for current conduction or not." So, what is the result, in your eyes? And what is your main conclusion/implication?

My question was whether the electrons associated with hole movement in p-type semiconductor will be having energy in conduction band or valence band. As you have said that electron will follow Fermi- Dirac distribution, correctly the electrons only having energy close to the Fermi level will participate in conduction. In p-type semiconductors, the Fermi level is close to the top of valence band. So the electrons at top of the valence band will be able to conduct current. In this case, I don't think the electrons in conduction band in p-type semiconductors have any role regarding hole movement. Again I would like to thank you for  the pin-point direction needed in this discussion.

      Dear Bijit,

      I thought that your question:

     What is the energy of an electron before merging with hole?

was to understand how the energies involved could allow the motion of one hole, obviously in the valence band and close to the Fermi level. In any case if you want know how the transport is allowed, I must say that your summary is very poor and far of what happens in the real semiconductors. I attach you some complementary information for your kind consideration.