Dear Mr. Liton,

welcome!

The energy gap of material is inversely proportional to lattice constant ,d,and to the the dielectric constant, epsilon. The dielectric constant itself is proportional to density of atoms  N in the solid material and their polarizability, alpha. That is

Eg is proportional to  1/ epsilon *d = consatnt/ N alpha * d,

d decreases with compression in all direction or it decreases with compression in the direction of the compression and increases increases on the normal direction. So, the the change in the Eg with pressure depends on the direction of the measurement. N also increases with compression. So, we have to competing factors N and d. Therefore the different behavior of the materials with pressure.

Best wishes 

In the same spirit of the previous answers Prof. M N H Liton, let's consider the gas of electrons and hole quasiparticles at room temperature in a semiconductor as an ideal gas, that is, P V = n R T, n is the number of quasiparticles, let's fix T as a room T0, then P ~ n and n ~ e - ( Δ E_G /K_B T_0), it can be seen that the Δ EG ~ - Ln P, that is, if P increases the bandgap Δ EG reduces. I suspect that at very low temperatures, this simple reasoning cannot take place in semiconductors, I also suspect that this is for a hydrostatic cell pressure, not for uniaxial or biaxial strains, and I also suspect that doping with impurities will change this.

Pedro L. Contreras E. and Abdelhalim abdelnaby Zekry

Thank you very much for these valuable information.

My pleasure, Prof. M N H Liton.

It depends which kind of pressure hydrostatic or uniaxial (biaxial). It also depends on the type of material, more particularly on crystal symmetry of the material.

- If it is hydrostatic (homogeneous) pressure, then the symmetry of the material (crystal) is conserved in the elastic (small magnitude) range, and in this case, hydrostatic pressure induces a homogeneous (in real space) decrease of the lattice constant of the crystal which in turn induces changes of the energy band structure (energy band structure = variation of the energy as a function of wavevector k : E=f(k) along different k-space directions (001), (111), (011), ... in the Brillouin zone (BZ)). It results in turn in a variation of different energy minima occuring at k=(000), k=001), k=(111), ...as a function of pressure. This variation depends on the type of point k in the BZ.

At k=(000), the energy minimum increases rapidly with pressure with respect to the valence band, i.e. the energy gap increases with high positive pressure coefficient.

At k=(111), the energy minimum increases with pressure but less rapidly than in k=(000). i.e. the energy gap increases with smaller but still positive pressure coefficient.

At k=(100), decreases slowly with pressure with respect to the valence band, i.e. the energy gap decreases with a small and negative pressure coefficient.

Therefore, the sign (positive or negative) of the pressure coefficient depends essentially on the symmetry of the conduction minimum states (000), (001), (111), ..

The magnitude of the pressure coefficient depends essentially on the effective mass of electrons in a given minimum. The smaller the effective mass the larger the hydrostatic pressure coefficient. in (000) minimum the effective mass is small and this is why the hydrostatic pressure coefficient is large. In other minima (111) and (001), the effective mass is large and therefore the hydrostatic pressure coefficient is small.

Very interesting information, Dear Prof. Abderrahmane Kadri. Is there any formalism (references) on how to classify the changes in energy ϵm (k) (m - the energy band) as a function of the strain or stress for conduction states along with different k directions?

Is that formulation only for semiconductors, or for normal metals as well? Thank you in advance.

Dear MNH Liton and Professor Contreras,

You raised very interesting issues, and I am very pleased to add my contribution to this exciting discussion about pressure effects.

1. The effects of hydrostatic pressure (HP) are very strong in semiconductors (SC) because they are quantum in nature and they affect first of all the crystal lattice of the SC and to a much lesser extent the electron and hole gases as such.

2. By affecting the crystal lattice, HP affects the heart of energy band structure which is the ''lattice periodic potential'' Vc. This perturbation of Vc is dealt with within the ''deformation potential approximation'' Vdp (for small enough perturbations in the elastic approximation) i.e. the SC under HP is now charaterized by a new ''quasi-periodic potential'' which can be roughly written as V'c=Vc+Vdp.

3. This is the origin of the changes in energy band structure of the SC under HP. Because they are quantum in nature and thightly related to the periodicity of Vc', they depend strongly on the symmetries of quantum electronic states in different points of the Brillouin zone (000), (001) (111) 011), ... For a given semiconductor e.g. Si or GaAs or whatever else, symmetry considerations are to be taken into account through ''group theory'' which is the theory of symmetries of crystals and their influence on electronic properties of a SC.

4. Looking here to the SC as just an electron or hole gases does not help, because electron-electron (or more generally particle-particle) is not the dominant effect at all. The dominant effect is the crystal lattice perturbation induced by pressure.

5. In SC it is the energy band structure which controls the electronic properties and more particularly electron n and hole p densities not the reverse.

6. In the case of metal, because of high electron densities (Avogadro density) and because the crystal lattice is fully screened by electrons, HP will affect predominantly electron-electron interactions. It is the reverse effect with respect to SC's. These HP effects are less spectacular in metals than in SC, but they may change fundamentally metallic properties as conductivity, magnetic interactions, and if HP is high enough then phase transitions can be induced.

7. In the case of unixial (US) or biaxial (BS) strains, the situation is different from HP fundamentally because the symmetry is now broken. Indeed US and BS strains induce a complicated mechanical perturbation which in most cases can be devided into two components:

- a hydrostatic (HP) component (symmetry conserving) and

- a shear strain (ShS) component (symmetry breaking)

- The first HP component do execatly the same thing as explained before i.e. shifting energy states globally with respect to each other and more particularly shifting conduction band with respect to valence band and conserving symmetry and properies of these bands.

- The second ShS component (symmetry breaking) will induce strong changes of electronic states by modifying these electronic interactions resulting in mixing of states and energy degeneracy liftings. In SC, these effects will affect predominantly the valence band (VB) and may have strong consequences on energy states at the top of VB. For example ShS will induce degeneracy liftings of Heavy holes, Light holes and Spin-orbite split holes inducing energy changes as well as nature changes of electronic states at the top of the VB.

8. At low temperatures, the effects are alomost the same except that thermal broadening (kT) is now smaller magnifying quantum effects, and thermo-mechanical properties of the SC may have an influence as thermally induced changes of elastic constants of the material, and consequently changes in Bulk modulus and compressibility properties of the material.

I'll give you some references as soon as possible.

Best regards,

Professor A. Kadri

Thank you so much for your detailed answer concerning the effects of the pressure on the electronic spectrum of semiconductors, Dear Prof. Abderrahmane Kadri.

I always have been intriguing on how the pressure affects the physical properties in normal metals, (following the idea of the Lifshitz electronic phase transition)

It is very nice to have the possibility to compare, how the electronic spectrum is changed in these two classes of solid-state materials.

In addition, the uniaxial stress is believed to break time-reversal symmetry in the superconducting transition in Sr2RuO4.

For me, a poor chemist, it seems plausible that increasing pressure results in more overlap between the electronic clouds of the neighboring molecules, resulting in narrower band gap. At least in semiconducting polymers this explanation is suggested for such observed effects. It is quite possible that in crystalline materials the situation is more complicated.

Gyorgy Banhegyi

Yes in semiconducting (organic) polymers which are macromolecular solids, the situation is different from that of crystalline mineral solids in that the interatomic interactions at the basis of neighboring atoms bondings are different.

In semiconductors these are essentially covalent interactions with sometimes a zest of ionic contribution resulting in the characteristics energy band structure of such solids with an energy bandgap separating bonding (valence band) with respect to antibonding (conduction band) states,

In the case of semiconducting organic (polymers) macromolecular materials the intra-molecular as well as inter-molecular interactions build up to form what we call LUMO and HOMO states (bands) separated by an energy bandgap. LUMO and HOMO states are equivalent to conduction and valence band states of their mineral semiconductors counterparts but they are not too much the same.

In the case of semiconducting polymers hydrostatic pressure (HP) will affect intermolecular and intramolecular interactions at the basis of HOMO and LUMO bands. Of course HP will affect first of all wavefunctions (orbitals) overlap of neighboring atomic and molecular states inducing changes in the energy band structure of the polymer increasing or decreasing the energy bandgap depending on the nature and symmetries of the states involved.

Best regards,

Professor A. Kadri