Energy Gap of a Semiconductor

8/3/2019 Energy Gap of a Semiconductor

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Determination Of TheEnergy Gap InSemiconductors

Technical University of Cluj-NapocaPhysics Depratment



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THEORETICAL APPROACH

Among all materials, the semiconductors, at the room temperature present an electricalconductivity between 10-10 –102 1 )cm( −⋅Ω , which is smaller than the metals conductivity and

greater than that of dielectrics. Due to this characteristic the semiconductors are tremendouslyimportant in technology. Semiconductor devices, electronic components made of semiconductor materials, are essential in modern electrical devices. Examples range from computers to cellular phones to digital audio players. Silicon is used to create most semiconductors commercially, butdozens of other materials are used as well.

The quantum theory led to the conclusion that electrons in crystals can only have valueswithin permitted bands, separated by energy gaps of forbidden values. The way that electronscan occupy the energetic levels of an allowed band depends on the thermal state of the crystal.At 0 K, the electrons will occupy the permitted levels, starting at lower energy levels, up to acertain band, partially or totally filled with electrons. The highest energy level, which still contains

electrons, is called the Fermi level . The energetic bands are conventionally represented like infigure 1 below. The valence band (VB) is the highest energy band, totally or partially occupiedwith electrons. The conduction band (CB) represents the first allowed but completely free bandof energy. The band which separates the highest level of VB from the lowest level of CB iscalled the forbidden band (band gap, energy gap Eg). The width of the gap is determined by thetype of the chemical bonds within the crystal and further determines the conductioncharacteristics of the solid:

Eg = 0 if the solid is a conductor (like carbon and metals)

Eg < 2 eV if the substance is a semiconductor and

Eg > 2 eV for an insulator (or dielectric material).The ionic bond which is formed within dielectrics like NaCl is much stronger than the

covalent bond found in pure semiconductor crystals, like Si and Ge. The band gap is larger indielectrics than in semiconductors. The band gap (Eg) represents the minimum amount of energy necessary for an electron to pass from the VB into the CB: Eg = EC -EV.a) Intrinsic semiconductors have no impurities and thus, no additional energy levels in theforbidden band (Fig. 3).When increasing the temperature, the thermal energy of the electrons in VB will increase andthe total energy of some e

-attain values within the conduction band. These electrons leave

behind, in VB, empty places called holes, which are considered to be positive charges, withrespect to the negative charge of the electrons. This is the way that the e-hole pairs arethermally generated. If a potential difference is set across the heated semiconductor, this will

produce a controlled motion of thecharge carriers (electrons travel to

the positive pole and holes travel tothe negative pole), resulting in acertain electric conductivity of thesemiconductor. While increasing thetemperature, this conductivity further increases, in contrast with the metalsbehavior under the same conditions.b) Extrinsic (or doped)semiconductors have someadditional allowed energy levels,Fig. 1 Energy levels and energy bands in materials



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between CB and VB. These additional energy levels aredue to the impurities or to the lattice defects, present in

the semiconductor crystal. The conductivity of semiconductors is characteristically a function of thetemperature. Three ranges are distinguished:- at low temperatures we have extrinsic conduction(range I), i.e. as the temperature rises, charge carriersare activated from the impurities.- at moderate temperatures (range II) we talk of impuritydepletion, since a further temperature rise no longer produces activation of impurities.-at high temperatures (range III) it is intrinsic conductionwhich finally predominates: charge carriers areadditionally transferred by thermal excitation from VB to

CB. Temperature dependence is now essentiallydescribed by an exponential function:

kT2

E

0

g

e−

⋅σ=σ (1)

where Eg = the energy gap, k = Boltzmann’s constantand T = the absolute temperature. The logarithm of thisequation:

kT2

Eg

0 −σ=σ lnln (2)

is with σ ln= y andT

x310

= . A linear equation of the

type y = a + bx wherekT2

E b

g−= is the slope of the straight line.

For the extrinsic semiconductors, the forbidden band is much decreased in the presenceof either the acceptor (EA) or the donor (ED) levels respectively. Therefore, a smaller gain in the

energy of the electrons will create charge carriers: e- in CBfor crystals doped with donors and h+ (holes) in the crystaldoped with acceptor impurities. Since the conductivity of metals follows an exponential law, as written above, itfollows that the resistivity of semiconductors decreases withincreasing temperatures, obeying the law:

kT2

E

0

g

e⋅ρ=ρ (3)where ρ is the resistivity of the semiconductor at T (K), 0 ρ

the semiconductor resistivity at To (K) and Eg is the energygap.

The resistance of the semiconductor, given by:

S

lR ⋅ρ= , (4)

also decreases while increasing the temperature:

Fig. 2 Last occupied energy bands inmetals, semiconductors andisolators.

Fig. 3 Last occupied energy band(valence) and conduction bandseparated by forbidden band,which contains the Fermi level, inintrinsic semiconductors.



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kT2

E

0

g

eR R ⋅= (5)

and by applying ln to this equation, one obtains:

kT

E R R

g

2lnln 0 += . (6)

If the decimal logarithm is further considered, the aboverelationship becomes:

T

10 baR

or kT2

E430R R

3

g

0

⋅+=

⋅+=

lg

.lg lg

(7)

Therefore, a graph showing the dependence of lg (R ) on

T

310

is

represented by a straight line (Fig. 6) whose slope helps findingthe band gap Eg:

( )

( ) ( )

)eV(

T

10

T

10

R lgR lg2.0E

or Jtan43.0

k 2E

k 2

E43.0tan

1

3

2

312

g

g

g

−

−⋅=

α⋅=

⇒⋅=α

(8)

EXPERIMENTAL SET UP AND PROCEDURE

The resistance of semiconductor test-piece is to be measured as a function of temperature. The ln(R ) is to be plotted against the reciprocal of the temperature. A linear plot willbe obtained from whose slope the energy gap of the semiconductor can be determined.

The experimental set-up is as in figure 6. The test piece is connected to a bridge typeohmmeter and is electrically heated. The heating coil is supplied by an alternating voltage outputof an autotransformer. The test-piece is to be warmed up slowly. The maximum permissibletemperature of 110

oC must not be exceeded. The test-piece temperature is determined using a

Fig. 3 Fermi level andenergy gap in extrinsicsemiconductors.

Fig. 4 Electrical scheme. Fig. 5 The dependence of electrical conductivityon the inverse of temperature.



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cromel/Cu thermocouple and a mV-meter (calibrated in oC). An ohmmeter - indicates the valuesof the resistance at different values of t(oC). In figure 4 the symbols represents O - electrical

oven; S - semiconductor test-piece; Th – thermocouple; mV – milivoltmeter (directly indicatesthe temperatures). We know: 1e = 1,6.10-19 C, and 1eV= 1,6.10-19 J.

PROCEDURE

1. One measures the resistance of the semiconductor at the room temperature.2. One plugs the oven and reads the resistance of the semiconductor at each 10

oC increase

of the temperature of the oven. All data are to be recorded in the Data table. The temperaturemust not exceed 1000C !!

3. One plots the dependence of lgR onT

310

, as in

figure 6.4. One calculates the slope of the line and further

estimates the band gap of the test-piecesemiconductor, by using the formulae deduced inthe previous section.5. Computing the relative deviation:

1221

2112g

g

R R

1

R

1

R

1R

T

1

T

1T

TT

T2

E

E

lg lg −⋅

+⋅∆+

+⋅∆+

−∆

⋅=∆

(9)

which was deduced, considering that:

R R R ∆=∆=∆ 21 and C T T T 0

21 10=∆=∆=∆ .

Data table

t

[oC]

T

[K] T

103

[K-1]R

[Ω]

lg(R)

[-]

Eg

[eV] g

g

E

E∆[%]

10

20

30

40

50

60

70

80

90

100

Fig. 6 Linear fit of experimental data andgraphical representation of the Hallconstant as the graph slope.

Energy Gap of a Semiconductor

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