Download Azbelj#Kaner cyclotron resonance - E16

Azbel’-Kaner cyclotron resonance
March 1, 2005
Abstract
The purpose of this laboratory experiment is to theoretically and
experimentally understand the phenomenon of cyclotron resonance
and to measure the dynamics of electrons in the condensed matter.
Because the electrons are moving wave-packets in a periodic potential,
their e¤ective mass (and consequently their dynamics) may di¤er from
that of free electrons. In metals and semimetals at least one band is
partially …lled and contributes to the conductivity. The surface with
the highest …lled energy is the Fermi surface. The shape and symmetry
of this surface plays an important role for the properties of a metal.
The basic ideas can be found in Charles Kittel, Introduction to Solid
State Physics. This laboratory experiment is recommended in the
second term of studying solid state physics (6th semester).
Contributors: F. Koch, K. Oettinger, P. Christmann, and H. E.
Porteanu. Translation corrected by E. Clark.
Supervisor Dr. H. E. Porteanu, Tel: 089 289 12333,
Physik-Department E16, O¢ ce 2274, e-mail: [email protected]
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2
Contents
1 Introduction
1.1 Classical description . . . . . . . . . . . . . . . . . . . . . . .
1.2 Quantum mechanical description . . . . . . . . . . . . . . . .
1.3 Relation to the Fermi surface . . . . . . . . . . . . . . . . . .
3
3
5
5
2 Bismuth
8
3 Tasks
10
4 Experimental arrangement
10
4.1 The microwave system . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Sample and resonator . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Magnetic …eld . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Running the experiments
5.1 Preparation . . . . . . .
5.2 Measurements . . . . . .
5.3 Data processing . . . . .
5.4 Measurement program .
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1
1.1
Introduction
Classical description
In order to explain the principle of the cyclotron resonance in metals and
semimetals we consider …rst the case of a free electron gas under the in‡uence
~ The electrons that have a speed
of an external applied magnetic …eld, B.
component, v, perpendicular on the direction of the magnetic …eld, are forced
~ The motion
to move on a circular orbit due to the Lorentz force, F~L = e~v B.
equation is:
m0 v! c = evB
(1)
or
eB
! c mo
() B =
(2)
mo
e
where m0 is the electron mass. The radial frequency, ! c , is called cyclotron
frequency.
One can get information about the properties of rotating electrons, if
the system is excited with an external driving force. For electrons in condensed matter high frequency electromagnetic radiation is the ideal method
of excitation.
The metallic case di¤ers from the case of semiconductors in that that the
electromagnetic radiation is screened within a very thin penetration depth
(skindepth, few micrometers) . The rotating electrons can therefore be
excited only within this skindepth.
~ parallel to the surface (see
One chose the Azbel’-Kaner geometry, i.e. B
Fig. 1), so one get the following e¤ect. We consider electrons for which
the orbits cross the skinlayer, like in Fig. 1. We consider …rst that the
scattering time is long enough that the rotating electrons can return back
many times in the skin area. Then there is a resonant interaction with the
electromagnetic …eld, when the alternating …eld has the same phase at each
cross of electrons. This occurs if the alternating …eld, having a frequency !,
oscillates n-times while the electron completes one orbit. We require then:
!c=
! = n !c;
n = 1; 2; :::
(3)
Usually one keeps the frequency constant and changes the magnetic …eld.
The values of the magnetic …eld ful…lling resonance conditions are:
Bn =
1 ! m0
;
n
e
i.e. the resonance magnetic …elds are periodic in 1=B.
(4)
4
1 INTRODUCTION
E
quantum mechanical view
EFermi
surface
v
z0=Rc
Landau levels
z
B
Rc
FL=e(v×Β), e<0
jµW
x
classical motion
Skindepth δ
Figure 1: The Azbel’-Kaner cyclotron motion of a free electron.
In order to have a high scattering time , one performs the Azbel’-Kaner
cyclotron resonance (AKCR) experiment at low temperatures (liquid helium). In these conditions one avoids the scattering of electrons on phonons.
However, very pure samples have, at this temperature, scattering times in
the range of 10 9 s. Therefore, one has to work in the microwave range so
the electrons can cross the skin depth area at least once (!
2 ).
1.2 Quantum mechanical description
1.2
5
Quantum mechanical description
The Schrödinger equation for a free electron (charge e < 0) under the in‡uence of a magnetic …eld is:
1
P~
2m0
~
eA
2
(5)
=E :
~ in direction y, i.e. the
One uses the Landau gauge for the magnetic …eld B
~ = (Bz; 0; 0). In this way one can separate the Schrödinger
vector potential A
equation with = eikx x eiky y (z) and get:
!
2 2
h
k
1
y
P 2 + (hkx eBz)2 (z) = E
(z):
(6)
2m0 z
2m0
This is nothing else as the Schrödinger equation of a harmonic oscillator
x
with the center at z0 = hk
' Rc . The wavefunctions are then the known
eB
functions of the harmonic oscillator and the energy quantization is given by:
E =E
h2 ky2
=
2m0
+
1
2
h! c ;
or
E=
+
1
2
h! c +
= 0; 1; 2; :::
h2 ky2
2m0
(7)
(8)
This energy scheme is shown in Fig. 1, where it is supposed ky = 0.
Transitions between occupied and unoccupied levels may happen, if the
Fermi energy (at T = 0 K) lies between the both levels and the excitation
energy, h!, corresponds exactly to the distance between the two levels. One
gets:
!=
!c;
(9)
where
is the di¤erence between the quantum numbers of the levels. With
this we …nd the classical results (3) and (4) again.
1.3
Relation to the Fermi surface
Because the Fermi energy of a metal is very big compared with the microwave
energy, h!, only electrons sitting on the Fermi surface can participate to the
resonance. Therefore with an AKCR experiment one measures the properties
of the Fermi surface.
Until now we considered only free electrons, i.e. obeying the quadratic
dispersion relation E = h2 k 2 =2m , where m was equal to m0 . The Fermi
6
1 INTRODUCTION
surface is, in this simple case, a sphere. In general the crystal lattice has a
considerable in‡uence on the dispersion relation and on the e¤ective mass
m . Additional to this, the shape of the cyclotron orbit and the cyclotron
frequencies are considerable di¤erent from the case of free electrons.
The general form of the cyclotron frequency can be derivated from the
equation of motion of a wave packet described in the k-space:
d~k
F~ = h
= e~v
dt
~
B:
(10)
~
B:
(11)
~ k E (~v , group velocity):
With h~v = r
e ~
d~k
= 2r
kE
dt
h
~
~ and
From the last equation one can see that ddtk must be perpendicular on B
~ k E. With this, one determines the motion of an electron on a certain
on r
orbit as a cross section in the Fermi surface with a plane perpendicular on
~ One then gets:
the magnetic …eld B.
dk?
eB ~
= 2 r
kE
dt
h
(12)
:
?
One integrates over the whole orbit to get the period of motion:
2
h2
=
T =
!c
eB
I
From this we get the result ! c = eB=mc
as:
I
h2
mc =
2
dk?
~ kE
r
:
(13)
?
where the cyclotron mass is de…ned
dk?
~ kE
r
:
(14)
?
Using geometrical considerations (see the elementary area in Fig. 2) this
integral can be transformed as follow:
h2 @A
mc =
;
2 @E
(15)
where A is the area of the cross section. The geometry of the orbit on the
Fermi surface is shown in Fig. 2.
1.3 Relation to the Fermi surface
r
a re a A ⊥ B
r
k
7
r
B
r
B
r-space
k-space
dt =
h 2 dk⊥
r
eB ∇ k E ⊥
h2
1
r
eB ∇ k E
( )
k⊥ (t +rdt )
B
( )
⊥
dk ⊥
k⊥ (t )
area A
Figure 2: Geometry of the cyclotron orbit.
We estimate now some typical orders of magnitudes for parameters entering in the AKCR in metals. For ! = 2
1010 s 1 (10 GHz) and mc = m0
one …nds the magnetic …eld, B, for the fundamental (n = 1) resonance:
jB1 j =
!m0
(2
=
jej
1010 s 1 ) (9; 1 10
j 1; 6 10 19 j C
31
kg)
0; 35 T = 3; 5
103 G
At this …eld one gets, with kF = 1010 m 1 (typical cyclotron radius in k~ k E):
space), the cyclotron radius in the real space (derivated from h~v = r
Rc =
hkF
(1; 054
=
eB1
( 1; 6
10
10
34
19
Js) (1010 m 1 )
C) ( 0; 35 T)
2
10
5
m
8
2 BISMUTH
The skindepth, , is typically 10
than Rc .
2
7
m, therefore one hundred times smaller
Bismuth
For this experiment we chose the semimetal Bismuth for two reasons:
1. The magnetic …elds required in the microwave range are relatively small
for Bi .
2. The Fermi surface of Bi is relatively simple.
The element Bismuth crystallizes in a structure with rhomboedric symmetry. In this structure one recognizes one trigonal axis with threefold symmetry, three binary axes and three bisectrix axes. The trigonal, binary and
bisectrix axes are oriented perpendicular to each other. With two atoms per
elementary cell and …ve valence electrons per atom Bi has an big enough
number of valence electrons to …ll …ve energy bands. Therefore Bi should
in fact be a dielectric, but because of partially band overlapping Bi is a
semimetal with the same number of electrons and holes.
The Fermi surface for holes is an ellipsoid, with the long axis aligned
parallel to the trigonal axis. The cyclotron mass for holes is higher than that
of electrons, so that in this experiment one cannot observe the cyclotron
resonance of holes. In the following we will therefore consider only electrons.
The Fermi surface for electrons consists of three equal big ellipsoids with
a shape like a cigar. The long axis of the ellipsoid lies in the plane determined
by the trigonal and bisectrix; their tilting angle is 6 200 . A representative
picture is given in Fig. 3.
The Fermi surface of an electron represented in the main axes coordinates
(Fig. 3 bottom) and has the following dispersion relation:
k12
k22
k32
E
=
+
+
2m1 2m2 2m3
h2
(16)
The masses m1 , m2 and m3 are determined from the cyclotron resonance
measurements of Edelman and Khaikin:
m1 = 0; 0058 m0
m2 = 1; 28 m0
m3 = 0; 011 m0
(17)
9
r
g1 [100]
C3 (kz)
trigonal axis
r
g 2 [010 ]
Γ
C1 (kx)
binary axis
r
g 3 [001]
C2 (ky)
bisectrix axis
2, 3, C2, C3
3
are in the same plane
C3 (kz)
trigonal
2
Φ=6°20'
C2 (ky)
bisektrix
C1 (kx) || 1
binary
1, 2, 3: semiaxes of the
ellipsoids
Figure 3: Fermi surface for electrons in Bi and their orientation relative to
the binary (C1 ), bisectr. (C2 ) and trig. (C3 ) axis.
From the de-Haas-van-Alphen data the maximal values of the momenta
along the main axes are determined:
k1max = 5; 16
k2max = 8; 84
k3max = 6; 78
107 m
107 m
107 m
1
1
(18)
1
The length of the cigar is therefore approx. 15 times bigger than its
thickness.
If one applies a magnetic …eld along one of the main axis, one …nds:
p
mjc = mi mk ;
(19)
10
4 EXPERIMENTAL ARRANGEMENT
p
i.e. for B parallel to axis 2 one gets m2c = m1 m3 etc. Beside this, it is
apparent that the cyclotron mass is independent of the momentum value,
kB , along the direction of the magnetic …eld. All the electrons on an ellipsoid
have the same cyclotron frequency and the same resonance …eld and therefore
all contribute to the resonance line. Additional informations about Bi can
be found in Fig. 4.
References
[1] M. Ya. Azbel’und E. A. Kaner, J. Phys. Chem. Solids 6, 113 (1958).
[2] J. F. Koch, Atomic Energy Review 12, 675 (1974) and the references
therein.
[3] V. S. Edelman and M. S. Khaikin, Sov. Phys. JETP 22, 77 (1966).
3
Tasks
1. How big are the quantum numbers (equation (7)) for the resonant
electrons at microwave frequencies in a typical metal?
2. Derivate from equation (14) the cyclotron mass for a free electron!
3. (a) Derivate the results (19)!
(b) Show that the cyclotron mass for the Fermi surface (16) is independent of kB !
p
(c) Show that the cyclotron mass increases like m1 m3 = cos , if the
magnetic …eld is rotated by an angle from the axis 2.
(d) At which magnetic …eld do you expect the fundamental resonance
(n = 1) for a microwave frequency of 16 GHz with B parallel to
the long axis (2) of an ellipsoid?
4
Experimental arrangement
The whole experimental arrangement is presented in the Fig. 5. In the
following we give a summary description of the elementary components.
11
z trigonal axis
crystal lattice (quasi-fcc)
α=57°16' < 60°
a=4.74 Å,
α'=87°34' < 90°
a'=6.57 Å,
band structure
conduction band
1.0 valence band
1.5
E (eV)
Basis: 2 Bi
0.5
atoms at 5.61 Å
0.474 × b. diag. 0
t2
-0.5
t3 a'
a a t
α' α a 1 a'
a'
-1.0
Fermi level
-2.0
x
binary axis
-2.5
y
bisectrix axis
kz trigonal axis
reciprocal lattice (quasi-bcc) r r
r
g
-1.5
r
− g1
L
r
g
ky
bisectrix
axis
2
kz trigonal axis
first Brillouin zone
T
Λ
kx
binary
axis
L
Φ=6°20'
Γ
bisectrix
axis
L
X
Γ
kx
binary axis
trigonal axis
T
X
t j × tk
r
gi = 2π r r r
ti ⋅ (t j × tk )
3
Γ
T
KΣ Γ
σ
T
L
Fermi surface
of electrons
and holes
Γ
L
ky
bisectrix L
X
axis
L
Figure 4: Crystal structure, …rst Brillouin zone, band structure and Fermi
surface of Bismuth.
12
4 EXPERIMENTAL ARRANGEMENT
Scheme of the experiment
adjustable short circuit
"magic Tee"
Gunn oscillator
GaAs-detector
Signal
out
Signal
in
Lock-in-amplifier
waveguide
Osc.
out
fl. nitrogen
fl. helium
Modulation coils
sample and resonator
ADC
in
computer
DAC
out
Audio-amplifier
B
current supply
Helmholtz coils
Figure 5: The measurement system for the cyclotron resonance.
4.1
The microwave system
Microwaves in the frequency range 15 to 16 GHz are produced by a Gunn
oscillator and with the help of a BNC/waveguide-adapter are fed in a rectangular waveguide via a so called ”magic T”. Inside the Gunn oscillator box is
built in a ferrit isolator, which hinders a parasitic interference of the re‡ected
with the emerging wave. The magic T distributes the radiation (Fig. 5) in
the lateral arms. If the power sent to the upper arm is absorbed, then the
power coming to the detector arm depends only on the microwaves re‡ected
from the resonator arm.
With the help of a tunable short circuit one adjusts the length of the
waveguide in order to have an integer multiple of half wavelengths (maximum
amplitude of the standing waves). On such a maximum appears the resonance
of the microwave resonator as a sharp minimum. In resonance conditions it
changes from a re‡ector to an absorber.
A high speed GaAs diodeAs is used as a detector, which gives a voltage
proportional to the microwave power (quadratic dependence on the electric
…eld, not a simple recti…er!). The detected voltage is a measure of the re‡ected microwave power from the resonator arm.
4.1 The microwave system
13
glass cryostat
Helmholtz coils
Modulation coils
Figure 6: The measuring system (top) and the magnetic …eld coils (bottom).
14
4 EXPERIMENTAL ARRANGEMENT
thermometer (carbon resistor)
Bi-sample
microwave resonator
"magic Tee"
microwave detector
Gunn oscillator
Figure 7: The microwave system.
4.2 Sample and resonator
4.2
15
Sample and resonator
Because the metallic surface absorbs relatively little microwave energy, it is
generally not enough to mount the sample as a simple short circuit re‡ector. Instead, the sample is introduced as part of a resonator, so that the
microwaves can re‡ect many times (thousands usually) from the sample.
In the upper part of the resonator (Fig. 5) there is a coupling hole, with
which, microwave power in a certain oscillating mode is introduced in the
resonator. The dimensions of the parallelepipedic resonator are chosen so
the mode H102 (Fig. 8) resonates (has an eigenfrequency) at 16 GHz. The
sample covers the hole in the bottom wall of the resonator. The cut in the
resonator is done in such a way that the surface currents J~ = ne~v , (~v is
the velocity vector of the electrons in the skin layer) that are induced in the
sample ‡ow almost parallel to the long side of the resonator (Fig. 8).
The direction of the currents have an in‡uence on the amplitude of the
observed cyclotron resonance peaks, because the power absorption is propor~ rf . If the sample is once mounted, one cannot
tional to the product J~ E
~ rf or of the sample
change the polarization, anymore, i.e. the direction of E
respectively. However, one can change the direction of the magnetic …eld and
hence, the plane of the cyclotron orbits.
4.3
Magnetic …eld
The magnetic …eld B0 is produced by a pair of Helmholtz coils, which are
fed by a constant current supply. The constant current supply is controlled
through a D/A converter by a computer.
Additional to this slowly changing magnetic …eld is overlapped an alternating …eld, Bm sin(! m t), with the frequency fm = ! m =(2 ) ' 625 Hz with
the help of a pair of modulation coils. The signal from the detector is a measure of the absorbed power, A, in the sample. A is dependent on the total
…eld B0 + Bm sin(! m t). For Bm
B0 one can develop A(B) in a Taylor
series:
@A
Bm sin(! m t)
(20)
A (B) ' A (B0 ) +
@B B0
The lock-in ampli…er selects the frequency component fm , and in this way
the A/D converter will measure a signal proportional to @A=@B. After the
theory of Azbel’and Kaner the peaks of the cyclotron resonance correspond
to zero crosses (in general turning points) of @A=@B, appearing at resonance
…elds Bn (see the equation (4)).
16
5 RUNNING THE EXPERIMENTS
References
[1] H. Heinke und F.W. Grundlach, Taschenbuch der Hochfrequenztechnik
Springer - Verlag, Berlin 1956.
[2] C.P. Poole, Jr., Electron Spin Resonance Interscience Publishers, New
York 1967.
fields distribution E, H, J of the H102 (TE102) mode in
a parallelepipedic resonator
J
H
z
y
E
x
sample surface
coupling hole
resonator
Figure 8: The microwave resonator.
5
5.1
Running the experiments
Preparation
The following explains the main steps in the preparation of the measurement.
The microwave system is switched on. The frequency of the Gunn
oscillator is adjusted to the resonator frequency.
5.2 Measurements
17
One …lls the external Dewar with liquid nitrogen. When the resonator
is cooled at nitrogen temperature (77 K), one can transfer liquid helium.
Observe the change of the resonator frequency! What is the reason of
this change?
When the Gunn oscillator is adjusted exactly on the resonator frequency, one adjusts the variable short circuit in order to have a broad
maximum and in the middle a sharp resonance minimum. The short
circuit from the detector maximizes the detected voltage for a given
input power.
The magnetic …eld is changed and one choses a strong cyclotron resonance peak. With this signal one can now adjust the phase of the
lock-in ampli…er.
5.2
Measurements
The surface of the Bi sample is almost trigonal, i.e. the normal on the
sample’s surface is parallel to the trigonal axis. The goal of the measurement
is, on one side, to observe the symmetry of the crystal, represented by the
dependence on direction of the cyclotron mass; on other side, to determine
the cyclotron mass of the high symmetry direction.
1. The dependence on orientation. One measures the @R=@B spectrum of
the cyclotron peaks as a function of the angle of magnetic …eld (each
10 degree is enough). Before every measurement one must readjust the
frequency of the Gunn oscillators! One represents the values of the
fundamental resonance (n = 1) as a function of angle; in this way one
get a picture of the angular dependence of the mass.
2. Measurement of the cyclotron mass. From your diagram you can determine the high symmetry directions. Now, careful measure (i.e. with
longer integration time) the cyclotron spectra with the magnetic …eld
along these directions!
5.3
Data processing
In order to estimate the cyclotron mass with the highest precision, one represents the values Bn for the subharmonics versus 1=n (x axis) and set the
best …t straight line between points (B1 = 0; 1=1 = 0, inclusively). From
the slope of this line hnBn i one gets the cyclotron mass (see equation (4)).
18
5 RUNNING THE EXPERIMENTS
1. Estimate the mean square deviation! Compare this mass with that
given in the literature!
2. Compare the experimental angular dependence of the mass with the
prediction from task 3c and discuss the deviation from 1= cos variation
or from the expected three fold symmetry!
3. Estimate the scattering time ! Use for this (a) the full width half
maximum of the resonance and (b) the number of the resonances. What
is the reason of this scattering?
4. For …nally, describe how did you enjoy the experiment and what can
be done, in your opinion, in order to improve the experiment!
5.4
Measurement program
The measuring program is written in LabVIEW. The next following pictures
(Figs. 9,10,11) explain the use of the program.
19
5.4 Measurement program
1. File name (angle) is written here
e.g. w110. The program asks afterward, if
the file name MRw110.dat is accepted
(save) or not (stop the measurement)
normalized measurement
(not used)
2. press OK
instruments for command and measurement
x-axis DAC, y-axis ADC
Path of the measurement files
(preinstalled)
Figure 9: Start menu of the program.
5 RUNNING THE EXPERIMENTS
20
1. The temperature should be approx. 4 K up to 10 cm
and below 20 K up to 30 cm.
2. The Gunn oscillator should be adjusted
on the resonance frequency
3. press "Start the measurement"
range of B-field,
number of steps and
delay time (preinstalled)
Figure 10: Start the measurement when all were adjusted and checked.
stop the measurement and the program. Finally B=0.
Interrupt the measurement for adjustement. B=const.
Figure 11: Running the measurement.
21
5.4 Measurement program