#1
A bulk sample has the reflection coefficient R = 0.36 for the electromagnetic radiation with the
wavelength λ = 100 μm. A film of the same material with the thickness d = 1 mm has the transmission
coefficient T = 0.17. Calculate the absorption coefficient for this material.
#2
The electron affinity of GaAs is χ ≈ 4 eV and the gap Eg ≈ 1.4 eV. The work functions for some
metals are: Al – 4.1 eV, Ag – 5.1eV, Au – 5.0 eV, Cu – 4.7 eV.
Which of these metals form Ohmic contacts on GaAs
a) of p-type?
b) of n-type?
c) intrinsic?
Justify and illustrate your reasoning by graphics.
#3
a) Ferrites have the general formula MO · Fe2O3, where M is a divalent element such as Fe, Cu, Mn,
Zn, and Ni. Néel has shown that the ferrimagnetism observed in
magnetite (Fe3O4) and many other ferrites is due to the fact that the
magnetic moments of the cations in the octahedral and tetrahedral
sites are antiparallel. In the unit cell of magnetite, the 8 occupied
tetrahedral sites are filled with trivalent ions and the 16 occupied
octahedral sites are equally divided between di- and trivalent ions.
Fe has an atomic number 26, with atomic configuration [Ar]
6 2
3d 4s .
i) Explain quantitatively the values of the magnetic moments
of Fe2+ and Fe3+ ions often observed in ferrites (and other oxides)
given in the table.
ii) Using values from the table, calculate the net magnetic
moment (per formula unit) expected for nickel ferrite.
iii) Calculate the saturation magnetization, Ms, for nickel
ferrite, which has a unit cell lattice parameter of a = 0.834 nm.
iv) Explain the (conditions for) the observation of a ferrite
with Fe partially replaced by Ni ions, that presents a magnetization of 4.6x105 A/m.
#4
Iron, which has an atomic number 26 and atomic configuration [Ar] 3d64s2, is an element present in
many magnetic materials and the magnetic moment of Fe3+ ions is 5 Bohr magnetons. One example are
rare-earth iron garnets (RIG) with general chemical composition R3Fe5O12 which are ferrimagnetic
materials. In RIG, the five Fe3+ ions occupy two octahedral and three tetrahedral sites. The magnetic
moments for the R3+ and the Fe3+ ions positioned in the octahedral sites are oriented parallel to one
another and antiparallel to the Fe3+ ions positioned in the tetrahedral sites. Consider a RIG in which each
cubic unit cell consists of eight formula (R3Fe5O12) units and has an edge length of 1.2376 nm. The
magnetic moment (in Bohr magneton) associated with each rare earth R3+ ion is 1.75.
i) Determine the total magnetic moment of this RIG per formula unit in Bohr magnetons.
ii) Determine the saturation magnetization for this ferrimagnetic RIG in SI units: A/m.
iii) Compare and explain briefly the differences and similarities between ferromagnetic and
ferrimagnetic materials.
#5
Calculate the absolute value of the linear momenta of the electron and hole created by a photon with
the energy hν = 0.31 eV in a semiconductor with the gap Eg = 0.3 eV. Admit the direct band structure and
parabolic dispersion relations for the electron and hole energy vs. momentum in the corresponding bands.
me* = mh* = 0.4m0. Compare the obtained value with the momentum of the photon (the refraction index of
the material is n = 4).
#6
The free electron concentration in an intrinsic semiconductor at the temperature T = 400 K is ni =
1.38×1015 cm–3. It is known that the electron and hole effective masses are approximately equal and that
the gap width varies with temperature according to the expression: Eg = (E0 – b·T), where E0 = 0.785 eV,
b = 4×10–4 eV/K. Calculate the electron effective mass for this material.
#7
The table below presents characteristics of magnetic nanoparticles: Tc= Curie temperature; K
(uniaxial effective anisotropy constant at room temperature; 300K)
i) If you wanted to use nanoparticles from the table below for magnetic recording explain for each
case if you would consider or reject it and select your best choice (from magnetic considerations).
Assume cubic shape nanoparticles and a microscopic “attempt” time, of the order 10-9seconds
ii) Determine the blocking temperature for nanoparticles B (for conventional measurement time 10s)
and sketch:
a) the temperature dependence of the magnetization measured at low fields.
b) the magnetic cycle M vs H above and below the blocking temperature.
#8
A magnetic tunnel junction with identical ferromagnetic electrodes presents a resistance ratio Rap/Rp
of 400%.
a) Use the Julliere approximation to find the spin polarization P of the ferromagnet.
b) What will be the TMR (magnetoresistance) if one of the electrodes is replaced by cobalt? (P(Co)=45%)
#9
The electron mobility measured in Ge (m*n = 0,1 m0) a T= 300K equals 4800 cm2/(V·s)
Calculate:
a)
b)
c)
d)
e)
The relaxation time of the linear momentum;
The electron drift velocity in the electric field E=6V/cm;
The thermal velocityof the electrons;
The mean free path of the electrons;
The electric field strength in which an electron acquires energy equal to its thermal energy
#10
Consider compensated n-type silicon at T=300K, with the conductivity σ = 16 (Ω·cm)-1 and the
acceptor concentration NA=1017 cm-3. Calculate the respective donor concentration.
#11
Characterize the solids (metals, insulators, etc.) having the following energy band structures (F is the
Fermi level):
#12
Consider a sample of a paramagnetic Ho compound with Ho3+ ions concentration (number of ions
per unit volume, n=1027m-3). Given the
values in the table and assuming only the
magnetic contribution from Ho3+ ions:
a) At sufficiently high
emperatures/low magnetic field,
the sample’s magnetization is
proportional to the magnetic field.
Describe theoretically the
magnetic susceptibility an indicate
explicitly the quantitative
dependence with temperature (in
SI units).
b) For a sample of a paramagnetic Gd. At sufficiently high magnetic field/low temperature, the
sample’s magnetization is not proportional to the magnetic field. Sketch the plot of the
magnetization of this paramagnetic material as a function of the magnetic field, indicating
explicity the maximum value of the magnitization (in SI units) that can be achieved. Discuss an
important magnetic consequence of having L=0 when considering Gd compounds.