Effective Mass of an Electron and Concept of Hole

EFFECTIVE MASS OF AN ELECTRON AND CONCEPT OF HOLE

Definition: Effective mass of an electron is the mass of the electron when it is accelerated in a periodic potential and is denoted by m*.

Explanation: When an electron of mass 'm' (9.11 x 10^-31 Kg) placed in a periodic potential and if it is accelerated with the help of an electric (or) magnetic field, then the mass of the electron will not be a constant, rather it varies with respect to the field applied. That varying mass is called as effective mass (m*).

Expression: To study the effect of electric field on the motion of an electron in one dimensional periodic potential, let us consider the Brillouin Zone which contains only one electron (1^st Brillouin Zone) of charge 'e' in the state k, placed in an external field 'E'. Due to the field applied the electron gains a group velocity (V_g) [Quantum mechanically] and therefore the acceleration changes.

The group velocity with which the electron can travel is

……(1)

where k → wave vector

ω→ Angular velocity of the electron

We know ω= 2πv

Since E = hv] …….(2)

Substituting equation (2) in equation (1) we get

group velocity 

............(3)

If the field (E) is applied to the electron for a time say dt seconds then

Change in field (or) Work done = Force × distance

dE = Force × Velocity × Time

dE = eEv_g dt [Since Force = eE] .....(4)

Substituting equation (3) in equation (4) we get

............(5)

We know Acceleration 

Substituting for v_g from equation (3) we get

............(6)

Substituting equation (5) in equation (6) we get

Acceleration

(or)

(or)............(7)

Equation (7) resembles with the newtons force equation

(i.e) F= eE= m* a ……(8)

where m* is the effective mass of the electron.

Comparing equation (7) and (8), we can write

............(9)

Equation (9) represents the effective mass of an electron in a periodic

potential, which depends on d²E/dk²

Special Cases

Case (i) If d²E/dk² is+ve, then effective mass m* is also + ve.

Case (ii) If d²E/dk² is - ve, then effective mass m* is also -ve.

Case (iii) If d²E/dk² is 0, then effective mass m* becomes ∞.

Thus we can say that it is not so the effective mass (m*) should always be greater than real mass (m), it may also have negative value.

Negative effective mass (or) Concept of hole

To show that the effective mass has negative value, let us take the Energy - wave vector (E-k) curve of a single electron in a periodic potential i.e., consider the 1st Brillouin Zone (allowed energy band) alone as shown in Fig. 1.24.

In the E-k curve, the band (1^st Brillouin Zone) can be divided into two bands viz, upper band and lower band with respect to a point (P) called as Point of inflection.

Note: Point of inflection is a point in the curve from which the curve changes from concave upward to concave downward and viceversa as shown in Fig. 1.25.

From the E-k curve (fig. 1.24) we can say that

(i) In the Lower band the value of d²E/dk² is a decreasing function (Indicated by arrow mark) from the point of inflection.

d²E/dk² is +ve. and hence m* should be + ve in the lower band.

If a plot is made between m* and k for various values of we get the curve as shown in Fig.1.26. In which we can see that m* has +ve curve.

 (ii) In the Upper band of E-k curve (Fig. 1.24) the value of d²E/dk² is found to be an increasing function from the point of inflection.

d²E/dk² is -ve and hence m* should also be -ve in the upper band.

If a plot is made between m and k (Fig. 1.26) we can see that, if d²E/dk²is -ve, the effective mass (m*) has - ve value.

(iii) At the point of inflection d²E/dk² =0 [Fig. 1.24] and hence in m*-k plot, effective mass goes to ∞ as shown in Fig. 1.26.

Physically speaking we can say that, In the upper band [Fig. 1.24], the electron has negative effective mass.

Hole: The electron with the negative effective mass is called Hole, in other words the electron in the upper band which behaves as a positively charged particle is called hole. It has the same mass as that of an electron but with positive charge.

Conclusion:

If a single electron is taken in a one dimensional periodic potential, we get the 1^st Brillouin Zone (i.e.,) only one allowed energy band.

If that energy band is divided into two bands (i.e) upper band and lower band. The electron is found to exist with positive effective mass in the lower band and with negative effective mass (hole) in the upperband as shown in Fig. 1.27.

Therefore, the advantage of the concept of hole is, for a nearly filled band with 'n' number of empty states as shown in Fig. 1.28 'n' number of holes (empty states) arises.

In other words we can say that the presence of hole is attributed to an empty state, for an electron to be filled. Thus, based on the hole concept several phenomenon like Thomson effect, Hall effect etc are well explained.