Experiment 5Bench 5MZU-PE / KPS-5Metal

Electronic Band Structure

Kronig-Penney Band Synthesiser

Raise the periodic potential until gaps open, fill the bands, and test the sample with a conduction lamp.

Duration: 3 hours
Kittel: Chapter 7
Topics: Bloch theoremKronig-PenneyEnergy bands

Band diagram — reduced zone

P = 6.0 · a = 3.00 Å · E₀ = 0.423 eV

Running

ENERGY versus CRYSTAL MOMENTUM

Band 1Band 2Band 3Band 4Forbidden gap

Conduction test

5 V through the specimen and a 1000 Ω lamp

Running

Ammeter

Conductivity σ

S/m

Classification

From the band filling and the gap at E_F

Specimen behaves as

metal

Gap at E_F

eVOPEN

Fermi level

Band edges

Band	Bottom / eV	Top / eV	Gap above / eV
1	2.408	4.177	6.035
2	10.212	16.711	7.854
3	24.565	37.602	8.768
4	46.370	66.849	9.254

Band width (1st)

1.770eV

Electrons / cell

1.00

Band filling

50 %

Band synthesiser

MZU-PE / KPS-5

Ready

Barrier strength P6.0

P = 0 is a free electron. Winding P up deepens the periodic potential and prises the gaps open.

Filling

Electrons per cell1.00

Specimen

Lattice constant a3.00 Å

Temperature300 K

E₀ = ħ²/2ma² = 0.4233 eV at this lattice constant.

Key Equations

\cos ka = P\,\frac{\sin\alpha a}{\alpha a} + \cos\alpha a

with $\alpha = \sqrt{2mE}/\hbar$ . Whenever the right-hand side exceeds 1 in magnitude there is no real $k$ , and that range of energies is forbidden — a band gap.

E_0 = \frac{\hbar^{2}}{2ma^{2}}

Each band holds two electrons per unit cell. An odd number of electrons per cell therefore leaves a half-filled band and the specimen conducts; an even number fills the band exactly and conduction depends entirely on the size of the gap above it.

Procedure

0/4

AFrom free electrons to bandsSet P = 0 and note the shape of the lowest band — the free-electron parabola folded into the zone. Wind P up in steps of 2 and record the first band gap each time.
BFilling and classificationWith P = 6, set 1, 2, 3 and 4 electrons per cell in turn. Record where E_F lands, what the classification lamp says, and what the ammeter reads.
CMake a semiconductor and an insulatorWith 2 electrons per cell, find the barrier strength that gives a gap near 1.1 eV (silicon) and then one above 5 eV (diamond). Compare the lamp in each case.
DTemperature dependenceLeave a 1 eV gap set and sweep the temperature from 100 K to 800 K. Record the current at each step. Then set 1 electron per cell (a metal) and repeat.

Table 5 — bands and conduction

0 entries

#	P	a/ Å	e⁻/cell	Gap at E_F/ eV	E_F/ eV	Classification	T/ K	σ/ S/m	Current/ A
No readings yet — set the controls, then press “Record reading”.

Analysis & Reflection

Analysis questions

Plot the first band gap against P from Task A. Is the relationship linear? What is happening physically as P increases?
The bands flatten as P grows. Since $v_g = \frac{1}{\hbar}\frac{dE}{dk}$ , what does a flatter band mean for how easily an electron moves through the crystal?
Why does the group velocity go to zero at the zone boundary, whatever the value of P?
Sodium has one valence electron per cell and is a metal; magnesium has two and is still a metal. Using the band diagram, suggest what must be true of magnesium's bands for that to happen.
From Task D, describe how the current changes with temperature for the 1 eV gap and for the metal. Explain the opposite signs.

Physics problems

Show that $E_0 = \hbar^2/2ma^2$ is about 0.42 eV for $a = 3$ Å, and confirm it against the readout on the panel.
State the difference between an insulator and a semiconductor in terms of band structure alone. Is there a sharp boundary?
Intrinsic carrier density goes as $n_i \propto e^{-E_g/2k_BT}$ . By what factor does it change between 300 K and 400 K for a 1.1 eV gap? Check your answer against the ammeter.
Explain the physical meaning of Bloch's theorem, and why it makes k a good quantum number in a periodic potential even though momentum is not conserved.