SEMICONDUCTORS Semiconductors Semiconductor devices Electronic Properties Robert M Rose, Lawrence A Shepart, John Wulff Wiley Eastern Limited, New Delhi (1987)

Energy gap in solids In the free electron theory a constant potential was assumed inside the solid. In reality the presence of the positive ion cores gives rise to a varying potential field. In a simple model the potential as in Fig.1 can be assumed (‘a’ is the lattice spacing and ‘w’ is the width of the potential). If ‘w’  0, we get ‘’ functions. The travelling electron wave interacts with this periodic potential (for a crystalline solid). The electron wave can be Bragg diffracted. 1D   = 90o Bragg diffraction from a 1D solid n = 2d Sin n = 2a n2/k = 2a

k → E → Band gap

The Velocity of electrons for the above values of k are zero. These values of k and the corresponding E are forbidden in the solid. The waveform of the electron wave is two standing waves. The standing waves have a periodic variation in amplitude and hence the electron probability density in the crystal. The potential energy of the electron becomes a function of its position (cannot be assumed to be constant (and zero) as was done in the free electron model).

The magnitude of the Energy gap between two bands is the difference in the potential energy of two electron locations. The effective mass of an electron (m*) in a solid could be different from the rest mass (m0). The effective mass can be larger or smaller than the rest mass. Collisions with atoms increases the rest mass, while enhanced propagation in the crystal gives a reduced effective mass. From the concept of group velocity the effective mass can be deduced. It is related to the curvature of the E-k curve. Close to band edges, m* can be negative. K.E of the electron increasing Decreasing velocity of the electron ve effective mass (m*) of the electron E → k → Within a band

If Fermi energy lies in this region then the conductivity would be high

[100] [110] Effective gap E → E → k → k → Effective energy gap → Forbidden gap → Band gap The effective gap for all directions of motion is called the forbidden gap. There is no forbidden gap if the maximum of a band for one direction of motion is higher than the minimum for the higher band for another direction of motion  this happens if the potential energy of the electron is not a strong function of the position in the crystal. [100] [110] Effective gap E → E → k → k →

Divalent metals Monovalent metals Energy band diagram: METALS Monovalent metals: Ag, Cu, Au → 1 e in the outermost orbital  outermost energy band is only half filled Divalent metals: Mg, Be → overlapping conduction and valence bands  they conduct even if the valence band is full Trivalent metals: Al → similar to monovalent metals!!!  outermost energy band is only half filled !!!

Energy band diagram: SEMICONDUCTORS Conduction Band 2-3 eV Valence Band Elements of the 4th column (C, Si, Ge, Sn, Pb) → valence band full but no overlap of valence and conduction bands Diamond → PE as strong function of the position in the crystal  Band gap is 5.4 eV Down the 4th column the outermost orbital is farther away from the nucleus and less bound  the electron is less strong a function of the position in the crystal  reducing band gap down the column

> 3 eV Energy band diagram: INSULATORS Insulator Conduction Band Valence Band

P(E) → E → Eg Intrinsic semiconductors At zero K very high field strengths (~ 1010 V/m) are required to move an electron from the top of the valence band to the bottom of the conduction band  Thermal excitation is an easier route P(E) → E → Eg Eg/2

T > 0 K  Unity in denominator can be ignored ne → Number of electrons promoted across the gap (= no. of holes in the valence band) N → Number of electrons available at the top of the valance band for excitation

Conduction in an intrinsic semiconductor Under applied field the electrons (thermally excited into the conduction band) can move using the vacant sites in the conduction band Holes move in the opposite direction in the valence band The conductivity of a semiconductor depends on the concentration of these charge carriers (ne & nh) Similar to drift velocity of electrons under an applied field in metals in semiconductors the concept of mobility is used to calculate conductivity Mobility Species Mobility (m2 / V / s) at RT Si Ge Electrons 0.14 0.39 Holes 0.05 0.19

Conductivity as a function of temperature Ln()→ 1/T (/K) →

Extrinsic semiconductors The addition of doping elements significantly increases the conductivity of a semiconductor. In fact impurity is accidentally present in semiconductors (even in low concentrations like 1 atom in 1012 atoms), which make it extrinsic. Doping of Si V column element (P, As, Sb) → the extra unbonded electron is practically free (with a radius of motion of ~ 80 Å)  Energy level near the conduction band  n- type semiconductor III column element (Al, Ga, In) → the extra electron for bonding supplied by a neighbouring Si atom → leaves a hole in Si.  Energy level near the valence band  p- type semiconductor

Ionization Energy→ Energy required to promote an electron from the Donor level to conduction band. EIonization < Eg  even at RT large fraction of the donor electrons are exited into the conduction band. n-type EIonization~0.01 eV Eg ~1 eV Donor level (ED) Electrons in the conduction band are the majority charge carriers The fraction of the donor level electrons excited into the conduction band is much larger than the number of electrons excited from the valence band Law of mass action: (ne)conduction band x (nh)valence band = Constant The number of holes is very small in an n-type semiconductor  Number of electrons ≠ Number of holes

p-type Eg Acceptor level (EA) EIonization~0.01 eV At zero K the holes are bound to the dopant atom As T↑ the holes gain thermal energy and break away from the dopant atom  available for conduction The level of the bound holes are called the acceptor level (which can accept and electron) and acceptor level is close to the valance band Holes are the majority charge carriers Intrinsically excited electrons are small in number  Number of electrons ≠ Number of holes

Ionization energies for dopants in Si & Ge (eV) Type Element In Si In Ge n-type P 0.044 0.012 As 0.049 0.013 Sb 0.039 0.010 p-type B 0.045 Al 0.057 Ga 0.065 0.011 In 0.16

+ve slope due to Temperature dependent mobility term Intrinsic slope All dopant atoms have been excited Exhaustion R4 R2 R3 Exponential function R1  (/ Ohm / K) → Extrinsic +ve slope due to Temperature dependent mobility term Slope can be used for the calculation of EIonization 0.02 0.04 0.06 0.08 0.1 10 K 50 K 1/T (/K) →

Semiconductor device  chose the flat region where the conductivity does not change much with temperature Thermistor (for measuring temperature)  maximum sensitivity is required