Question

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(a) The crystal structures of Si is shown below. The side length of the unit cell is 0.54 nm. с (i) Based on the structure of this unit cell, calculate the number of atoms in 1 cm of silicon. (ii) Explain qualitatively what you would expect to happen to the dimensions of this unit cell if the temperature should increase ? (iii) How would such a temperature increase affect the bonding in the Si and how would this then affect that bandgap of Si ? (iii) How would such a temperature increase affect the bonding in the Si and how would this then affect that bandgap of Si ? (iv) Is your answer in (iii) consistent with the changes in carrier concentration expected in Si with increasing temperature ?

Answer 1

a)

i)

8 atoms are placed at corner. Each of them has contribution of 1/8

excluding 8 corner atoms, there are total 10 atom we can see. 6 atom has been placed at each face of the cube. Each of them has contribution of 1/2.

Remaining (10-6)= 4 atoms are inside the cell. Each of them has contribution of 1.

So total number of atoms in 1cm³ is = 8*1/8 + 6*1/2 + 4 = 1+3+4 = 8 (ans.)

iii) Temperature effects on bonding :-

Silicon cell is a n-type semiconductor. It contains with covalent bonding and 1 free electron for each silicon atom. The present of large no. of electrons increase recombination rate and decrease the hole concentration below the intrinsic level. Thus here the no. of free electrons far exceeds the no. of holes.The donor's extra electron is weakly bound and needs very little energy, thermal or otherwise, to break the bond and become a free electron. As temperature is increased, more and more of these bonds are broken, until all the donors are ionized, producing an increase in electron concentration.

  Temperature effects on band gap :-  We know,

EG(T) = E.0) (T+

where EG(T) is the band gap energy, E_G(0) is the limiting value of the band gap at 0 K and a and b are constants.

This relation shows that as temperature increases, the band gap energy decreases linearly because the crystal lattice expands and the interatomic bonds are weakened. Weaker bonds means less energy is needed to break a bond and get an electron in the conduction band.

iv) Temperature effects on career concentration :-

Yes, answer iii) is consistent with the changes in career concentration expected in silicon with increasing temperature.

For an electron-hole pair to be created in an intrinsic semiconductor, a bond must be broken in the lattice, and this requires energy. An electron in the valence band must gain enough energy to jump to the conduction band and leave a hole behind.  n_i represents the intrinsic carrier concentration, or we can see it as the number of bonds broken in an intrinsic semiconductor.

As the temperature is increased, the number of broken bonds (carriers) increases because there is more thermal energy available so more and more electrons gain enough energy to break free. Each electron that makes it to the conduction band leaves behind a hole in the valence band and there is an increase in both the electron and hole concentration. As the temperature is decreased, electrons do not receive enough energy to break a bond and remain in the valence band. If electrons are in the conduction band they will quickly lose energy and fall back to the valence band, annihilating a hole. Therefore, lowering the temperature causes a decrease in the intrinsic carrier concentration, while raising the temperature causes an increase in intrinsic carrier concentration.

ii) Dimension of unit cell remains the same if the temperature is increased.