two-dimensional electron gas (2deg) in a magnetic field
TRANSCRIPT
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Two-Dimensional Electron Gas (2DEG) in a Magnetic Field
In classical physics, an electron orbits around the magnetic field at
a well-defined radius
r
=
p /e B
with angular frequency ω = eB/m .
In quantum physics the energy E
=
ħω
is quantized into discrete
levels
En
(Landau levels). And a classical orbit becomes a probability distri-
bution
(the absolute square of the electron wave function).
Classical picture
(Quantum picture on Slide 6)
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Landau Levels in Two Dimensions
The continuous 2D density of states contracts into discrete levels.
As the B-field increases, the level spacing increases, and each level sweeps up a larger part of the continuum.
ContinuumThe spacing between Landau levels is the same as for a harmonic oscillator, including the zero point energy:
En =
(n+½) ·
ħωc
The corresponding angular frequency is the cyclotron frequency ωc
, which contains the effective mass m*
:
ωc =
e B/m*
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m*m*me
The magnetic moment
is related to the angular momentum of a rotating electron. There are two types of angular momentum, spin (left) and
orbital (center , right).
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Introduce Magnetic Interactions into the Schrödinger Equation
The electric potential
generates the electric field E : E = /r
The magnetic potential A produces the magnetic field B :
B = /r ×A
A enters the Schrödinger equation in the same way as .
Make the substitutions
Energy E
(+i ħ
/t +
e )
Momentum p
( i ħ
/r + e A)
in the classical equation for the kinetic energy E: E = p2 /2m
That leads to the Schrödinger equation in a magnetic field:
(iħ
/t + e)
= 1/2m
(
i ħ
/r + eA)2
Time-independent potentials produce energy levels En
: i ħ
/t
= En
(In a solid one has m = m*
= effective mass, e= V0
=
inner potential.)
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Landau’s Solution of the Schrödinger Equation in a Magnetic Field
A constant field Bz
is described by the vector potential A = (0,
Bz
x , 0) .B = /r A
Bz
= /x Ay /y Ax
Bx
= By= 0In two dimensions x,y
the Schrödinger equation takes the form:
En
= 1/2m*
[-iħ/r + eA]2
= 1/2m*
[-ħ2 2/x2
+ (-iħ
/y +
eBz
x)2
]
The trial wave function (x,y) = exp(i ky) (x)converts the y-derivative into a multiplication with i k .
After dividing by
exp(i ky) one obtains a one-dimensional Schrödinger equation for (x)
:En
= 1/2m*
[-ħ2 2/x2
+ (ħk + eBz
x)2
]
This becomes the Schrödinger equation of a harmonic oscillator
, if one
rewrites 1/2m*
(ħk + eBz
x)2
as ½ f (x-x0
)2 with the “force constant”
f .Then one can use the familiar energy levels En of the harmonic oscillator to obtain the Landau levels and their wave functions n
(
Lect.11
, p.
4
).
The same B-field can be created by other vector potentials, such asA =
½ (-Bz
y , Bz
x , 0) . This ambiguity is called gauge symmetry. It plays a fundamental role in our understanding of particle physics.
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Wave Functions of Electrons in Landau Levels
The radial wave functions are those of a harmonic oscillator, except that the count starts at n=1 instead of n=0.
n
=
1 n =
2 n =
3
n
=
1
n
=
2
n
=
3
Classical probability
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Magnetic Flux Quantization
The magnetic flux is quantized
in units of h/e
.
Thus, a magnetic field really can be viewed as composed of individual field lines, as shown in the previous slide. Each line carries one flux quantum h/e. The B-field is the flux density (flux quanta per area).
Flux quantization can be observed directly in superconductors, where the flux quantum is h/2e
because of electron pairs with charge 2e
:
Regular array of flux quanta crossing the surface of a superconductor (white dots). This STM image is taken with a very small applied voltage, less than the energy gap of the superconductor. Superconducting regions are dark, because electrons cannot tunnel inside a gap. The magnetic field of a flux quantum destroys superconductivity and allows tunneling, creating bright spots for the flux quanta.
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Integer vs. Fractional Quantum Hall Effect
Integer
Quantum Hall Effect
:n electrons
circle around
one
flux quantum
(more electrons than flux quanta).
Fractional
Quantum Hall Effect n
=
1/m :One
electron circles around
m
flux quanta (more flux quanta than electrons).
Each flux quantum gets a fraction of the electron.
n
=
2
n
=
1/3
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The normal Hall
effect gives the line. xy
Ey
/jx
Bz
The quantum Hall
effect gives steps. xy
=
h/e2
·
1/n
for n =
1,2,3,…
Hall Effect vs. Quantum Hall Effect
n=1
n=2
Ohmic
resistivity
xy
[h/e2]xx
xy
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Video: Landau Level Filling vs. Quantum Hall Effect
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Each of the plateaus has a very precise value of the Hall resistivity, which is determined purely by the fundamental constants h and e. For n=1 one obtains the value xy
=
h/e2
= 25.
8128…
k
. It can be measured so precisely that the quantum Hall effect has become the resistance standard.
The Quantum Hall Resistance Standard
n=1
n=2
xy
[h/e2]xx
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Resistivity = =
Resistance = = =V Voltage E · lI Current j · A
Resistivity vs. Resistance in 2D and 3D
Since resistivity does not contain a length in 2D, the quantum Hall effect becomes independent of the shape of the sample.
in 2D: Resistivity =
Resistance = in 3D: Resistivity = m
Resistance = samein
2D
E
Electric Fieldj Current Density
A
length
l
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Vanishing Ohmic
Resistivity
The Ohmic
resistivity
xx
nearly vanishes at the plateaus. (It looks like a superconductor, but the resistance is not exactly zero.) That helps making accurate measurements.
Ohmic
resistivity
xy
[h/e2]xx
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Edge States Carry the Current
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The Fractional Quantum Hall Effect
When the electron density is reduced or the B-field increased beyond the n
=
1 plateau
, additional plateaus appear at fractional values of n
, such as n
=
2/3, 3/5
.
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Quantized Conductance
Attach nanotubes
to a STM tip and dip them into a liquid metal electrode.
Conductance Quantum: G0
=
2 e2/h
1 /13 k( factor 2 for spin ,
)
Each
wave
function =
band =
“channel”
contributes G0 to the conductance.
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Quantum conductance: G =
G0
•TG0
=
2 e2/h per channel, T
1=
transmission at the contacts
Energy to switch one bit: E =
kBT • ln2
Time to switch one bit: t =
h / E
Energy to transport a bit: E =
kBT • f/c • d
at the rate f over a distance d
Limits of Electronics from Information Theory
Birnbaum
and Williams, Physics Today, Jan. 2000, p. 38. Landauer, Feynman Lectures on Computation .