Why Does Kittel's Equation for Fermi Level Include ln(me/mh)?

[DrBrainDead]

Homework Statement

In chapter 8, page 207 Kittel derives an equation for the fermi level in an intrinsic conductor:
$$\mu$$=½E_g + $$\frac{3}{4}$$k_BT*ln(m_e/m_h)

How am I to understand the ln(m_e/m_h) part? Earlier he states that the effective mass is proportional to the curvature of the energy band, and hence m_e = - m_h.. thus it would be ln(-1) ?

 

[olgranpappy]

Homework Statement

In chapter 8, page 207 Kittel derives an equation for the fermi level in an intrinsic conductor:
$$\mu$$=½E_g + $$\frac{3}{4}$$k_BT*ln(m_e/m_h)

How am I to understand the ln(m_e/m_h) part? Earlier he states that the effective mass is proportional to the curvature of the energy band, and hence m_e = - m_h.. thus it would be ln(-1) ?

No. First of all, that is absurd since ln(-1) is imaginary...

In my version of Kittel he gives an example right after the equation in which m_e=m_h=m. I.e., it is apparent from context that the masses m_e and m_h are taken as positive quantities here.

 

[DrBrainDead]

Naturally the masses will have to be of the same sign for anything to make sense; what I'm asking is why? The masses than enter the ln(x) are the effective masses, right? How is the equation then to be understood? Is the mass of the hole just assumed to be positive, or...?

 

[olgranpappy]

Naturally the masses will have to be of the same sign for anything to make sense; what I'm asking is why? The masses than enter the ln(x) are the effective masses, right? How is the equation then to be understood? Is the mass of the hole just assumed to be positive, or...?

The density of states for the electron (or holes) depends on the effective mass. The flatter the curvature of the parabola (i.e., E(k) near it's minimum) the greater the density of states... but that is (by definition) the same as saying that the greater the effective mass the greater the density of states.

The density of states of the electron and hole comes into the calculation of the total number of excited electrons and holes (N_e and N_h). That's how the terms m_e and m_h come into the calculation. The term m_h is in this calculation defined to be the negative of the curvature of the valence band near the band maximum.

Kittel gives explicit expressions for N_e and N_h and says that by setting them equal to each other one arrives at the equation given in your original post. Have you reproduced these calculations of N_e and N_h yourself yet?