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J. Humlíček FKL II
6. Cyclotron resonance of electrons and holes
A remainder of the evolution of semiconductor physics...
Pure crystals of Ge have been obtained and transistor discovered by Bardeen, Brattain, and Shockley at Bell labs in 1947.
Rudimentary bandstructure calculations appeared in early 1950s:
Herman and Callaway, Phys. Rev. 1953
Conduction band minimum of Ge located in the  direction of BZ.
Let us follow the idea of examining the electron structure of a semiconductor around the minima of conduction bands and maxima of valence
bands. Instructive reading in a short paper by Shockley, 1953.
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The motion of (quasi)electrons and holes in magnetic field will be approximated using parabolic dispersion of bands (i.e., constant effective
masses); for simplicity, we will start with the following isotropic band:
2
2
*
( ) .
2

k
E k
m
(6.1)
The classical equation of motion in the static magnetic field of the intensity H and harmonic electric field of the amplitude E0 and frequency 
reads
*
0 ,

  
    
   
i tdv v v H
m e E e
dt c (6.2)
where  is the average time between successive collisions ("scattering time", counteracting the drift movement due to the external field). Assume
the orientation of the magnetic field along z, and linearly polarized electromagnetic wave along x. The harmonic time dependence of the drift
velocity is described by the amplitudes along x and y; from Eq. (6.2), they are related by
*
*
1
,
1
.




 
   
 
 
   
 
y z
x x
x z
y
ev H
m i v eE
c
ev H
m i v
c
(6.3)
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The amplitude of the velocity in the direction of electric field is
 * 2 2 2
1
,
1 2

   


  
x
x
c
eE i
v
m i
(6.4)
where
* *
z z
c
eH eB
m c m c
     (6.5)
is cyclotron frequency. In non-magnetic materials, the magnetic induction (flux density) B is nearly equal to the intensity H. The alternate sign in
Eq. (6.5) selects the direction of rotational movement, which is opposite for electrons and holes.
The above equations are written in the cgs units, frequently used in this context; in SI units, the cyclotron frequency is
*
.z
c
eB
m
   (6.5)SI
A summary of the typical values of important quantities:
the (microwave) frequencies of 24 GHz band mean
151 9 rad/s; =2 / =42 ps; =c =1.3 cm; 0.095 meV.c c c vac c cE T T     
The resonant magnetic field intensity is
860 Oe=6.8 5 A/m.H E
The horizontal component of the field in Brno is about 0.2 Oersted (16 A/m).
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We can also estimate the radius, Rc, of the cyclotron orbit. Involved is the mean thermal velocity,
*
0*
8
4 6 cm/s pro 0.1 , 4 K
kT
v E m m T
m
   
which implies
300 nm .c
c
v T
R

 
Note: the regular cyclotron orbiting is superposed on the chaotic thermal movement.
The drift velocity of charged (quasi)particles is related to a current density; the latter is also of harmonic time dependence, its component along x
is
 
2
2* 2
c
1/
( ) = ,
1/
i t i t
x x x
Ne i
j t Nev e E e
m i
  
  


 
(6.6)
where N is the carrier density. According Eq. (6.4), the amplitude of the drift velocity is proportional to the amplitude of the electric field;
consequently, we can introduce (complex) conductivity as the proportionality factor between the current density and electric field intensity:
 0 2 2 2
1
,
1 2

 
   

 
  
x
x c
j i
E i
(6.7)
where
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2
0 *

 
Ne
m
(6.8)
is the dc (=0) conductivity.
Energy transferred from the alternating electric field is proportional to the real part of conductivity (it appears at the Joule heat resulting from the
damping of the carrier motion, related to the relaxation time ). Its frequency dependence is described conveniently using dimensionless
quantities in the following relation:
 
 
2 2
22 2 2
0
Re 1
, , .
1 4

  

 
  
  
c
c c
c
r r
r r
r r r
(6.9)
Using this result, we can assess the chances of observing the resonant absorption of energy for  c, see the following figure.
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Frequency dependence of the absorbed energy; from Dresselhaus, Kip, Kittel, Phys. Rev. 1955.
Given the ranges of frequencies  and c, the pronounced maxima of absorbed energy occur for a small damping of the free carrier movement,
i.e., a large relaxation time . The latter is proportional to the mobility , which is a suitable characteristics of the electric transport in
semiconductors (remember very broad range of carrier densities N in semiconductors):
0 *
, .

   
e
Ne
m
(6.10)
Mobility is the mean drift velocity in the electric field of unit intensity; it is usually measured in the units of (cm/s)/(V/cm) = cm2
/Vs. In order to
obtain well resolved cyclotron resonances, the condition   1 has to be fulfilled,
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*
.


e
m
(6.11)
Taking the convenient microwave frequencies (e.g., f = /2  24 GHz), rather large values of mobility are required, amplified by the inverse
proportionality to the effective mass }usually significantly lower than the free electron mass:
20
*
11000 cm /Vs. 
m
m
(6.12)
Using the dimensionless quantities of Eq. 6.9, the requirement of observable cyclotron resonance can be also put in the following form,
1 . cr r (6.13)
The real part of conductivity is then equal to the dc value halved. The reason is obvious: we have assumed the linear polarization of the
alternating field, which is a superposition of two orthogonal circular polarization states. One of them (that of the same sense of the cyclotron
orbit), the other with the opposite phase and no energy dissipation. In the circularly polarized field of the same phase, the resonant absorption
would be the same as in the dc field.
In actual cyclotron resonance experiments, the frequencvy of the (microwave) field is kept constant and the cyclotron frequency is variaqble
using variable magnetic field.
Proper concentration of free carriers is important. It can be produced by optical excitation (chopped light from a tungsten lamp, focused on the
sample), which can also produce reference signals for lock-in (phase sensitive) detection of the microwave absorption.
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Typical profile of the absorbed energy:
Ge, frequency 24 GHz, temperature 4 K.
From Dresselhaus, Kip, Kittel, Phys. Rev. 1955.
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Typical profile of the absorbed energy:
Si, frequency 24 GHz, temperature 4 K.
Static magnetic field in the (100) plane,
inclined 30 deg towards (100).
From Dresselhaus, Kip, Kittel, Phys. Rev.
1955.
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Experimental setup for measurements with circularly polarized microwave field; from
Dresselhaus, Kip, Kittel, Phys. Rev. 1955.
Changing orientation of single-crystalline sample allows studies of the directional dependence of effective masses.
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Effective masses of electrons in Ge; from Dresselhaus, Kip, Kittel, Phys. Rev. 1955. Different
symbols are results of independent measurements.
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Effective masses of electrons in Si; from Dresselhaus, Kip, Kittel, Phys. Rev. 1955. Different
symbols are results of independent measurements.
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Effective masses of holes in Ge; from Dresselhaus, Kip, Kittel, Phys. Rev. 1955. Different
symbols are results of independent measurements.
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Effective masses of holews in Si; from Dresselhaus, Kip, Kittel, Phys. Rev. 1955.
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Summary of results from Dresselhaus, Kip, Kittel, Phys. Rev. 1955: