quantum mechanics course microsoftpowerpoint-harmonicoscillator-motioninamagneticfield

8/14/2019 Quantum mechanics course microsoftpowerpoint-harmonicoscillator-motioninamagneticfield

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Harmonic Oscillator:

Motion in a Magnetic Field

* The Schrdinger equation in a magnetic field

The vector potential

* Quantized electron motion in a magnetic field

Landau levels

* The Shubnikov-de Haas effect

Landau-level degeneracy & depopulation


2/16

An important example of harmonic motion is provided by electrons that moveunder the influence of the LORENTZ FORCE generated by an applied MAGNETICFIELD

* From CLASSICAL physics we know that this force causes the electron toundergo CIRCULAR motion in the plane PERPENDICULAR to the direction of themagnetic field

* To develop a QUANTUM-MECHANICAL description of this problem we needto know how to include the magnetic field into the Schrdinger equation

In this regard we recall that according to FARADAYS LAW a time-varying magnetic field gives rise to an associated ELECTRIC FIELD

The Schrdinger Equation in a Magnetic Field

)1.16(BvF e

)2.16(t

BE


3/16

To simplify Equation 16.2 we define a VECTOR POTENTIAL A associated with themagnetic field

* With this definition Equation 16.2 reduces to

* Now the EQUATION OF MOTION for the electron can be written as


)3.16(AB

)4.16(ttt

AEA

BE

)5.16()( AkkAk

Ep

eBt

et

et

o

1 2

1. MOMENTUM IN THE PRESENCE OF THE MAGNETICFIELD2. MOMENTUM PRIOR TO THE APPLICATION OF THEMAGNETIC FIELD


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Inspection of Equation 16.5 suggests that in the presence of a magnetic field weREPLACE the momentum operator in the Schrdinger equation by

* To incorporate this result into the Schrdinger equation we recall that the

first term on the LHS of its (B= 0) time-independent form represents theKINETIC ENERGY

* Since kinetic energy is related to momentum asp2/2mthis in turn suggests

that we define the MOMENTUM OPERATOR as


)6.16(App e

)7.16()()()(2

22

xExVxm

)8.16(

zyxii

NOTE THETHREE-DIMENSIONALFORM


5/16

With our definition of the momentum operator at ZERO magnetic field (Equation16.8) we now use Equation 16.6 to obtain the momentum operator in thePRESENCE of a magnetic field

* With this definition we may now REWRITE the Schrdinger equation in aform that may be used to describe the motion of electrons in a magnetic field

The first term in the brackets on the LHS of this equation is known asthe CANONICAL momentum and is the momentum in the absence of a magneticfield

The entire term in brackets is called the MECHANICAL or KINEMATICmomentum and corresponds to the KINETIC ENERGY of the electron


)9.16(

Ae

i

)10.16()()()(2

12

xExVxeim

A


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We now apply the results of the preceding analysis to describe the motion ofelectrons in a magnetic field

* We assume that this magnetic field is CONSTANTand points in the z-direction

* ONE possible choice of vector potential that satisfies this equation is knownas the LANDAU GAUGE and is given by (you can check this using Equation 16.3)

* With this choice of gauge the Schrdinger equation now becomes

Quantized Electron Motion in a Magnetic Field

)11.16(zB B

)12.16(yA Bx

)13.16(2

)(

2

22

2

Em

eBx

ym

Bxei

m


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Equation 16.12 reveals that the magnetic field produces TWO effects

* The first effect is a derivative that COUPLES the motion in the x- and y-directions as we would EXPECTfor a particle that undergoes CIRCULAR motion inthe xy-plane

* The second effect is that the magnetic field generates a PARABOLICMAGNETIC POTENTIAL of the form that we have studied for the harmonicoscillator!

* Now since the form of the vector potential we have chosen does NOT

depend on ythis suggests that we write the WAVEFUNCTION solutions for theelectrons as

)13.16(2

)(

2

22

2

Em

eBx

ym

Bxei

m

1 2

)14.16()(),(yikyexuyx

NOTE HOW THE y-COMPONENT CORRESPONDSTO AFREELY-MOVINGPARTICLE



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Substitution of our wavefunction into the Schrdinger equation

* This is just the Schrdinger equation for a one-dimensional HARMONICOSCILLATOR with the magnetic-field dependent CYCLOTRON FREQUENCY

* An important difference with our previous analysis however is that theCENTER of the parabolic potential is NOTlocated at x= 0 but rather at


)15.16()()(2

1

2

2

2

2

22

xEuxueB

kxm

xm

y

c

)16.16(m

eBc

)17.16(eB

kx

y

k


9/16

Solution of the Schrdinger equation yields a quantized set of energy levels knownas LANDAU LEVELS

* The wavefunctions are the usual HERMITE POLYNOMIALS and may bewritten as

where we have defined the MAGNETIC LENGTH lBas


)18.16(2

1cn nE

)19.16(2

)(exp

!2

1)(),(

2

2yik

B

kn

B

k

B

n

yik yy el

xxH

l

xx

lnexuyx

)20.16(eB

lB


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An important property of the quantized Landau levels is that they are highlyDEGENERATE

* By this we mean that each Landau level is able to hold a LARGE number ofelectrons

* To obtain an expression for this degeneracy we begin by assuming that thecircular motion of the electrons occurs in a plane with dimensions Lx Ly

By assuming PERIODIC boundary conditions along the y-direction (ECE352) we may write a quantization condition for the wavenumber ky

The Shubnikov-de Haas Effect

)18.16(2

1cn nE

)21.16(,2,1,0,2 jjL

ky

y

WHEN A PARTICLE ISCONFINEDIN A ONE-DIMENSIONAL BOXOF LENGTH Ly ITS ALLOWED WAVENUMBERS AREQUANTIZED

ACCORDING TO EQUATION 16.21


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Since the CENTER COORDINATE of the harmonic oscillator must lie somewherewithin the sample we may use Equation 16.21 to write the following condition

* According to Equation 16.22 EACH Landau level contains the same number ofstates at any given magnetic field

The number of states in each level PER UNIT AREA of the sample isjust given by

Since each state within the Landau level can hold TWO electrons withOPPOSITE spins the number of ELECTRONS that can be held in each Landau levelis


)22.16(00h

LeBLjxL

yx

kx

)23.16(h

eB

LL

j

yx

Max

)24.16(2

h

eBn


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Now let us consider what happens to a sample containing a FIXED number ofelectrons as we VARY the magnetic field

* Starting at some INITIAL magnetic field a specific number Nof Landaulevels will be occupied by electrons

(N 1) of these levels will be filled completely while the Nth will

typically be PARTIALLY filled with the remaining electrons that cannot beaccommodated in the lower levels


FILLING OFLANDAU LEVELSBY A FIXEDNUMBER OF ELECTRONSAT AN ARBITRARY MAGNETIC FIELD

EACH LANDAU LEVEL IS CAPABLE OF HOLDING THESAME

NUMBER OF ELECTRONS AND THESE LEVELS WILL BE FILLED INA MANNER THATMINIMIZESTHETOTAL ENERGYOF THE SYSTEM

BECAUSE OF THIS AT ANY MAGNETIC FILLED THE LOWEST(N-1)LANDAU LEVELS WILL BECOMPLETELYFILLED BY ELECTRONSACCOUNTING FOR (N-1)n ELECTRONS

THE REMAINING ELECTRONS WILL BE ACCOMMODATED IN THEPARTIALLY-FILLEDUPPERMOST LEVEL

EN-1

E3

E1

E2

EN


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If we now raise the magnetic field we increase the ENERGY SPACING of theLandau levels and also increase their DEGENERACY

* Since more states are available in each level electrons DROP from higherlevels to occupy empty states in the lower levels

* Consequently we eventually reach a point where the Nth Landau level isCOMPLETELY emptied of electrons and the number of occupied Landau levels is

now just N-1


EMPTYINGOF THE UPPERMOST OCCUPIEDLANDAULEVEL IN ANINCREASINGMAGNETIC FIELD

WITH INCREASING MAGNETIC FIELD THESPACINGOF THE LANDAU LEVELSINCREASESBUT THENUMBEROF ELECTRONS HELD BY EACH LEVELALSOINCREASES

AS ELECTRONSDROPTO FILL NEW STATES THATBECOME AVAILABLE WITH INCREASING FIELDTHEUPPERMOSTLANDAU LEVEL EVENTUALLYEMPTIES

THIS PROCESS IS REFERRED TO ASMAGNETICDEPOPULATIONOF LEVELS

EN-1

E3

E1

E2

ENEN-1

E3

E1

E2

EN

B B+ DB


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The MAGNETIC DEPOPULATION we have described CONTINUES withincreasing magnetic field until all electrons occupy only the LOWESTLandau levelat VERY HIGH magnetic fields

* To obtain an expression for the magnetic field values at which thedepopulations occur we consider a sample containing nselectrons PER UNIT AREA

* Since nis the number of electrons per unit area that occupy EACH Landau

level we require those values of the magnetic field for which the following ratio isan INTEGER

This relation shows that as expected the number of occupied Landaulevels DECREASES with increasing magnetic field It also shows that an increasingly LARGER magnetic field increment is

required to depopulate successively LOWER Landau levels


)25.16(,3,2,1,2

LLss NNeB

hn

n

n


15/16

The depopulation of Landau levels can actually be seen in the MAGNETO-RESISTANCE of semiconductors which OSCILLATES at low temperatures

* The period of the oscillations INCREASES with magnetic field as expectedfrom Equation 15.24 and the oscillations appear PERIODIC when plotted on anINVERSE-FIELD scale

The periodicity of these SHUBNIKOV-DE HAAS oscillations is often

used in experiment as a means to determine the electron CARRIER DENSITY


SHUBNIKOV-DE HAAS OSCILLATIONSMEASURED IN AGaAs/AlGaAs HETEROJUNCTION AT 4 K

THE NUMBERS ON THE FIGURE INDICATE THE NUMBEROFOCCUPIEDLANDAU LEVELS AT SPECIFIC VALUES OF

THE MAGNETIC FIELD

THESPLITTINGOF THE PEAK IN THE REGION OF AROUND2.5 T RESULTS AS THE MAGNETIC FIELD BEGINS TO LIFTTHESPIN DEGENERACYOF THE ELECTRONS

QUANTUM MECHANICS, D. K. FERRY, IOPP (2001)


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quantum Hall history

discovery: 1980

Nobel prize: 1985

K. v. Klitzing

H. Strmer R. LaughlinD. Tsui

discovery: 1982

Nobel prize: 1998

IQHE

FQHE

quantum mechanics course microsoftpowerpoint-harmonicoscillator-motioninamagneticfield

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