Tentamen GM 2015

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Tentamen GM 2015
Voorblad bij tentamen – Gecondenseerde Materie 3CGX1
(in te vullen door de examinator)
Tentamen/vakcode: 3CGX1
Aantal deelnemers: 122 ingeschreven
Datum: 22 januari 2016
Begintijd: 13:30
Eindtijd: 16:30
schriftelijk / notebook (*)
open vragen / meerkeuzevragen (*)
Aantal pagina’s: 5
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3CGX1 Condensed Matter 2014/2015
Final Exam, 22 January 2015
General: Motivate your answers. Mention the equations you use, define parameters, and
include intermediate step in your calculations.
Possibly useful equations and physical constants:
1
.
exp[( E   ) / k BT ]  1
1
Bose-Einstein distribution function: n( E ) 
.
exp[ E / k BT ]  1
Electron charge:
e = 1.60 · 10-19 C ~ 10-19 C
Electron mass:
m = 9.1 · 10-31 kg ~ 10-30 kg
Planck constant:
h = 6.6 · 10-34 J·s ~ 10-33 J·s
ħ = 1.05 · 10-34 J·s ~ 10-34 J·s
Boltzmann constant: kB = 1.38 · 10-23 J/K ~ 10-23 J/K
Speed of light:
c = 3.0 · 108 m/s
Fermi-Dirac distribution function:
f (E) 
Final scoring:
Maximum number of points: 70. Final mark is scored points / 7.
________________________________________________________________________
Exercise 1 – A body-centered cubic crystal (total 25 points)
Figure 1a
Figure 1b
Figure 1c
We consider a ‘body centered cubic’ (BCC) crystal based on a conventional cubic cell
with dimensions a  a  a , with one atom A at each corner and one atom A at the center
of the cube, as indicated in Figure 1a.
a) How many atoms does the conventional unit cell contain? [1 point]
 

b) Give the vector expression fort the primitive lattice vectors a1 , a 2 and a3 of the


primitive (so not the conventional) lattice, choosing a1 along the x-axis and a 2
along the y-axis. Sketch the three vectors in a figure. [3 points]
c) How many atoms does the primitive unit cell contain? [1 point]
1

d) Calculate the reciprocal lattice vector b3 . [3 points]
e) Give the Miller indices of the gray plane sketched in Fig. 1b. Base the indices on
the conventional lattice vectors. Explain your answer. [2 points]
An X-ray diffraction experiment is performed on the BCC crystal depicted in Fig. 1a. The
incident wave vector is along the positive x-as, and the scattered wave vector is along the
positive z-axis.
f) Derive an expression for the largest wavelength at which a diffraction peak is
observed. [3 points]
Next we consider another crystal, in which the atom at the center of each cube is
substituted by an atom B, as sketched in Fig. 1c. Atom A has a nuclear charge Z = p, and
 
atom B has Z = 2p. We recall that the structure factor reads S   f j exp(iG  r ) .
j
g) Find the ratio of the [100] and [200] diffracted intensities. [3 points]
Finally we return to the original BCC crystal of Fig. 1a, for which we are going to
calculate the electronic band structure within the tight-binding approximation. We
include one s-orbital per A atom. The corresponding atomic wave function is denoted
 

s j   (r  r j ) for atom j on position r j . De Hamiltonian H is described by matrix
elements  0  s j H s j
and t  s j H s j ' , where j and j’ are indices of nearest-
neighbor atoms. The crystal contains N atoms. Note that the matrix element t is negative.


h) Express the wave function with wave number k ,  k (r ) , within the tight-binding
 
 
approximation in terms of k , r , the atomic wave functions  (r  r j ) and N.
Take care of proper normalization. [1 point]
i) Derive the band structure of the BCC crystal within the tight-binding
approximation, i.e., find an expression for the energy as a function of k x , k y , and
k z . Show sufficient intermediate steps in your derivation, and write your final
result as a real expression, so eliminating complex factors such as e ikx . [6 points]

j) What is the energy at k  0 ? Explain why this could have been expected. [2
points]
Exercise 2 – Varia (total 25 points)
Lattice waves in a diatomic crystal (10 points)
We consider a one-dimensional (1D) diatomic crystal, consisting of atoms A and B with
masses of M and 3M, resp. The distance between neighboring atoms is d, see Fig. 2a. We
describe longitudinal lattice waves by means of a mass-spring model with spring constant
f. The displacement with respect to the equilibrium positions of atoms A and B in unit
cell j is denoted by uj en vj, resp., see Fig. 2b.
2
Figure 2
a) Give the equations of motion for atom A and B in unit cell j. [3 points]
b) Consider the snapshot of a phonon mode in Figure 2c. Give the wavenumber K of
this phonon mode and indicate whether it belongs to the acoustic branch or the
optical branch. Motivate your answer briefly. [2 points]
The general solution for a lattice wave with wave vector K is given by
u j (t )  u exp(2ijKd ) exp(it ) and v j (t )  v exp(2ijKd ) exp(it ) .
c) Derive expressions for the angular frequency  belonging to the two solutions at
the edge of the BZ. Hint: rather than solving the equations of a) in general, there
is a short-cut making life much easier! [5 points]
Heat capacity: (10 points)
A researcher has discovered a new quasi-particle, dubbed ‘trion’, in a cubic piece of a 3D
material with dimensions L  L  L . Trions behave as bosons, and fulfill an energy

dispersion relation  (k )  C k 3 , where C is a constant. There is only one trion state per

k -point. Use periodic boundary conditions.
d) Derive an expression for states D( ) of trions in terms of given parameters and
physical constants. [4 points]
e) Derive an expression for the total energy U (T ) in the trion system in terms of
given parameters and physical constants in the low temperature limit. You may

xn
use the standard integral 
dx  Gn . [5 points]
exp( x)  1
0
f) Derive an expression for the heat capacity of this trion system in terms of given
parameters and physical constants. [1 point]
Hall effect: (2 points)
The Hall effect is measured in a strip of Si, doped by an element with an electron
configuration s2p1 in the outermost shell. A current is sent through the strip in the positive
x-direction in the presence of a magnetic field in the positive z-direction. We use a
definition in which a positive Hall voltage corresponds to the case that positive charge
accumulates in the positive y-direction, orthogonal to the strip.
g) Explain whether the measured Hall voltage is positive or negative. [2 points]
3
Magnetism: (3 points)
We consider the magnetic properties of (i) the noble gas neon (Ne), (ii) a Ni2+ containing
salt (Ni2+ has a 3d8 configuration) and (iii) metallic nickel measured below the
temperature at which the magnetic susceptibility diverges. The results are shown in Fig. 3
(figures are not to scale).
Figure 3a
Figure 3b
Figure 3c
h) Explain for each of the figures (3a, 3b, 3c), whether they correspond to Ne, to the
Ni2+ containing salt, or to the Ni metal. Briefly explain your answer. [3 points]
Exercise 3 – Free electrons and semiconductors (total 20
points)
We consider a one-dimensional (1D) crystal of length L, a simple lattice with one atom
per unit cell and lattice constant a. First we describe the electrons within the free-electron
approximation.
a) Sketch the free-electron bandstructure in the reduced zone scheme (within the
first Brillouin zone). Consider at least the two bands of lowest energy at each kvalue. Give the value of k en E k of both bands at the center and at the edge of the
Brillouin zone. [5 points]
Next we introduce a finite electrostatic interaction between the electrons and the positive
ions in the lattice. The ions are at position x j  ja , where j is an integer. The resulting
electronic bandstructure is sketched in Fig. 4.
Figure 4
4
Consider the wave functions of four different electronic states in this crystal:
A:  A  C cos(x / a ) ,
B:  D  C sin(x / a ) ,
C:  B  C sin( 2x / a ) ,
D:  C  C , where in all cases C is a normalization constant.
b) Indicate for all four wave functions (A – D) whether it represents a state in the
lowest, middle or highest band, and give the respective k-values within the first
Brillouin zone. [5 points]
The material turns out to be an intrinsic semiconductor. The second band is (almost)
completely filled and the third band is (almost) completely empty. De band gap of
magnitude E0 separates those two bands, as indicated in Fig. 4
c) How many electrons does each of the atoms in this crystal contribute to the
bandstructure of Fig. 4? Briefly explain your answer. [2 points]
The dispersion of the second band, i.e. the energy as a function of wavenumber k, is
approximately equal to  (k )  W  W cos(ka) .
d) Express the effective mass of both electrons as well as holes in this band near
k  0 in terms of given parameters and physical constants. [3 points]
The density of states at the top of the second (filled) band equals D( )  D0 , whereas
D( )  23 D0 at the bottom of the third (empty) band. Note that in both cases D( ) is
assumed to be independent of energy!
e) Derive an expression for the chemical potential  as a function of temperature for
this intrinsic semiconductor in terms of given parameters and physical constants.
Do so for the limit that  / k BT  1 and ( E0   ) / k BT  1 , so that you can use a
simple exponential approximation to the Fermi-Dirac distribution function. [5
points]
END OF EXAM
5

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