Applied Physics - hu-berlin.depeople.physik.hu-berlin.de/~bfmaier/data/fpraktikum/F4l_publish.pdf · half-sphere analysator (type SES-2002) with an angle resolution of 0:2 and an

FACULTY FOR MATHEMATICS AND NATURAL SCIENCE IDEPARTMENT OF PHYSICS

Applied Physics:Advanced Laboratory

4th experiment:ARPES

date of experiment 17.05.2011due to 25.05.2011advisor Mohamed Moustafa

experimentalists Lucas HacklBenjamin Maier

Humboldt University of BerlinFaculty for Mathematics and Natural Science I – Department of PhysicsApplied Physics: Advanced Laboratory | Lucas Hackl & Benjamin Maier

Contents

1 Introduction 31.1 The photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Three-step model of photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Further explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Experimental setup 4

3 Maesurement 53.1 Intensity of the monochromator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Fermi-edge and resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Work function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Conclusion 7

Used measurement devices

angle resolved photoelectron spectroscopy (ARPES) at the HU beamline (BESSY II)

Abstract

The angle resolved photelectron spectroscopy (ARPES) is a modern method to analyze the band structure ofsolids not only dependent on the energy but on the momentum angle as well. Therefore, it provides a goodopportunity to learn about the inner structure of solids, especially on their surface (because the interactionprocesses happen mostly there). In the first step we looked at the intensity of the incoming radiation andthe quality of our monochromator. In the second step we found the Fermi-edge by working with the energydependent response function. In the third step we were able to calculate the work function with gives us aquantity to describe the potential difference between the solid and the outer vacuum. Finally, we calculated thereciprocal lattice vectors.

4th experiment: ARPES 2


1 Introduction

ARPES stands for “Angle Resolved PhotoemissionSpectroscopy” and is a widely used experimentalmethod to analyze the electronical band structure ofmaterials. To be more accurate it enables the exper-imentalists to find the density of single-particle elec-tronic excitations within the reciprocal space.

1.1 The photoelectric effect

The photoelectric effect is the most likely interactionbetween photons and electrons for low energy photons(at higher energy compton scattering and pair produc-tion become more important). To explain this processAlbert Einstein suggested the equation

Ekin = hν−Φ

where the incoming photon has the energy Eγ = hνand the material dependent constant Φ represents thework function which is the potential difference be-tween Fermi energy (energy within the solid state) andthe outer vacuum.Fundamental new in this interpretation was the depen-dency on the photon frequence: The transfer is not af-fected by the intensity but by the frequence because inquantum mechanics we interpret the quantity Eγ as thecertain energy carried by a single photon.

1.2 Three-step model of photoemission

The three-step model is an often used descriptionwhich offers a basic understanding of the processes ofphotoemission. It consists of the following three parts:

1. Absorption of the incoming photon and excita-tion of an electronTo describe the process of absorbing the incom-ing electron we can use Fermi’s Golden Rulewhich follows from quantum theory. The proba-bility of transition from an initial state i to a finalstate f is given by

Γi→ f =2π~∣∣〈ψi | HI | ψ f 〉∣∣2 δ(E f −Ei−~ω)

At this we can understand the interaction be-tween photon and electron by absorbing the for-mer one with energy Eγ = ~ω. This interactioncan be described by an disturbance or interac-tion Hamiltonian

HI =e

2mc~A ·~p

which provides the opportunity to calculate thecorrect probability. Important to notice is thedelta function which implies the equality of pho-ton energy Eγ and excitation energy ∆E = E f −Ei – this guarantees energy conservation.

2. Transport to the surfaceThe photons may enter the solid state up to adepth of several 100Å depending on the energyof the incoming electrons. The mean free pathof the electrons is energy dependent as well andcan be seen in Fig. 1 – we can especially learnthat this free path is independent of the used ma-terial.

3. Emission to the vacuumIn the last step the excited state leads to an ef-fective free stadium of the electron inside thesolid state. Therefore we can write the energy-momentum-relation as

E f =~2

2m~k2f

By seperating the wavevector~k f in a parallel anda normal part we are able to describe them in-dependently. Neglecting photon scattering theparallel component will be invariant by surfacetransmission – we find:

~k|| =

√2m~2

Ekin sinθ

On the other hand the perpendicular compo-nent will not be invariant because of the changewithin the external potential (there is no innerpotential outside the solid state). This leads to

~k⊥ =

√2m~

(Ekin + |V0|)

In general the electron will not always be free and wehave to replace the electron mass m by an effectivemass m∗ (especially in the dispersion relation whichleads to former expressions). It is given by the follow-ing second derivation:

1m∗

=1~2

d2Edk2



Figure 1: energy dependence of mean free path

1.3 Further explanations

Angle resolved photoelectron spectroscopy is a mod-ern procedure to analyse the band structure of solidstates (mainly the surface). It enables us to findthe density of states as a function not only of theenergy but of the angle (resp. the direction of themomentum vector) as well. The illustration of theenergy-momentum relation is measured in one high-symmetric direction in the first Brillouin zone.

2 Experimental setup

As source of our photons we used the electronsynchroton radiation of the BESSY II in Berlin-Adlershof, with our experimental setup connected tothe beamline of Humboldt-University. The scatteringhappens in the mainchamber as shown in fig. 2. Theposition and angle of our target can be adjusted bythe manipulator. Inside the mainchamber there is ahalf-sphere analysator (type SES-2002) with an angleresolution of 0.2 and an energy resolution of 2meV.The target is injected at the load lock and can be trans-ferred thorugh the UFO to the main chamber. Thewhole experiment takes place in ultra high vaccum(with pressure p < 1 ·10−7 bar).A more detailed explanation is given in the handout

[1].

Figure 2: experimental set-up



3 Maesurement

3.1 Intensity of the monochromator

The intensity characteristics of our monochromatorwas measured through the current on the mirror M3within a certain energy range as can be seen in fig. 3.

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7measurement

photon energy E [eV]

mirro

r curr

ent I [n

A]

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6


mirro

r curr

ent I [n

A]

smoothed

Figure 3: monochromator intensity depending on the photon en-ergy

In the next step we want to show theregion around the maximum of this dis-tribution which can be seen in fig. 4:

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

0.61

17 18 19 20 21 22 23 24

mirro

r cu

rre

nt

I [n

A]


smoothed measurement

Gaussian fit

Figure 4: monochromator intensity depending on the photon en-ergy (zoom of maximum) with Gaussian fit (χ2/d.o.F. = 6.8×10−6)

By fitting the maximum of the intensity with a Gaus-sian we gain the result:

EImax = (20.36±0.02)eV

3.2 Fermi-edge and resolution

After evaporating the samples of TeTi2 and Gold wemeasure the intensity of the emitted electrons at theΓ-point depending on their kinetic energy in the wide

range between E = 0 and E = 20eV. After this we canfind the approximate area of the Fermi edge. Thereforewe are able to measure this certain region in higher res-olution to improve our raw material for the fits. Thewhole material of this is accessible in the appendix.The measured raw material has to be corrected accord-ing to two different methods: On one hand we usedthe Savitzki-Golay algorithm to smooth the measuredcurve (which is important for a later step when we haveto derive numerically), on the other hand we correctedthe smoothed data I′ by subtracting the Shirley back-ground S:

Ii = I′i − I0∑k>i I′k∑k>0 I′k︸︷︷︸

S

This background is caused by inelastic scattering whenelectrons of higher energy scatter and lose energy –hence we measure too many events at lower energiesdependent on the state density of electrons with higherenergy. Therefore we subtract an integrated intensityfor higher energy (because we work with discretizeddata points the integral becomes an ordinary sum). Thescaling factor I0 is the intensity at maximum energybelow the valence band. Because we were not able tomeasure this intensity during the measurement in thearea of the Fermi edge and the effects of inelastic scat-tering are rather neglectable, we used the algorithm tocorrect the data in another way. By choosing a certainintensity I0 for every measurement, we were able toreach a constant intensity for E < EF and an intensityaround zero at E > EF.To calculate the location of the Fermi edge we usedthe derivative of the intensity function with respect tothe energy. In the region of the Fermi edge the Fermidistribution is similar to an error function – becauseof this the first derivative is alike a Gaussian whichshould be easy to fit. Then we can easily find the max-imum (which is equal to the mean value of the distri-bution) and this is the position of the Fermi-edge. Inthe next step we are able to determine the temperature-dependent broadening of the Fermi-edge. Because wehave already determined the Fermi-edge we can usethis to fit the smoothed data with the Fermi distribu-tion.The broadening s at T = 0 is composed by three ef-fects:

s =√

(∆EA)2 +(∆EBL)2 +(∆T )2

The three aspects are the following:

• Analysator uncertaintyThis quantity is a constant of our detector and inour case it is ∆EA ≈ 10meV.



Sample Tempera- Slit Fermi energy EF [eV] Broadening s [eV] Expected χ2/d.o.F.ture T [K] (Gaussian fit) (Fermi fit) broadening [eV] (Gaussian fit) (Fermi fit)

Gold 100.0 546 17.3021±0.0057 0.1214±0.0029 0.020 7996 6123.0 746 17.2826±0.0036 0.0916±0.0040 0.024 64444 129

33.5 546 17.2871±0.0043 0.0886±0.0046 0.012 22962 8333.5 746 17.1762±0.0043 0.1022±0.0018 0.012 1301249 506

TiTe2 123.0 746 17.2941±0.0029 0.1047±0.0027 0.024 136924 243

Table 1: Data gained from measurements at Fermi energy (see figures in appendix)

• Resolution error of beamlineThis quantity is determined by the experimen-tal setting and the beamline and in our case it is∆EBL ≈ 2meV.

• Temperature-dependent broadeningAs we know the Fermi distribution depends onthe temperature because energy can be stored ina solid by exciting the electron states. Thereforesome electrons with an energy near the Fermi-edge get excited to a state above the Fermi-edge.This broadening can be quantized to ∆T = 2kBT

3.3 Work function

To measure the work function ΦA of Gold we Firstly,we measured the density of states in dependence onthe kinetic energy of the detected electrons. Therearetwo significant points within the spectrum which wecan use to estimate the work function:

1. Ekin = Emax = maxAt this pont the binding energy EB = 0 is mini-mal.

2. Ekin = Emin = 0At this pont the binding energy EB = max ismaximal.

Therefore, we find as difference L=Emax−Emin whichenables us to to calculate the work function:

Φ = hν−L

The following table shows the different work functionswe have determined:

3.4 Dispersion relation

Because of the noise within our measured dispersionrelation we were not able to analyze the band struc-ture and escpecially we could not calculate the effec-tive mass of the electrons in the solid state potential.This seems to be a problem of the experimental setupor more specifically a problem of its current propor-ties.

3.5 Reciprocal lattice

The direct lattice of Titaniumditellurid (TiTe2) corre-sponds to a hexagonal lattice structure, which can beseen in Fig. 5). Therefore the angle between two latticevectors of the basement is 120, while the third vectoris perpendicular to both.

a

bφ

Figure 5: illustration of hexagonal lattice structure

The lattice constants (absolute value of the lattice vec-tors) in the lattice are given as follows:

a = b = 3.778Å

c = 6.6.493Å

We choose the following directions in our 3-dimensional coordinate system:

~a =

100

~b =

−12√320

~c =

001

In the first step we calculate the volumina of the unitcell, which is given by the following well-known term:

V = a2 c sin(120) = a2 c

√3

2= 80.26Å3



After that we are now able to derive the reciprocallattice vectors by using the following formular (re-member, you can reach the reciprocal lattice by fouriertransformation) and cyclic rotation of its parameters:

~a′ = 2π~b′×~c′

V

By proceeding this calculation for each vector of thereciprocal lattice we get:

~a′ =2πa

11√3

0

~b′ =2πb

0− 2√3

0

~c′ =

2πc

001

Finally, the Γ-point is that point of the reciprocalspace where maximal symmetry appears. In termsof the Brillouin zone this is exactly its center wherewe can find rotation and translation symmetry simul-tane1ously.As mentioned in our introduction we can calculate thechange of~k⊥ by applying the following formular:

~k⊥ =

√2m~

(Ekin + |V0|)

In our case we can assume the given value of V0 =10eV which leads us to the following result for theperpendicular component of the original momentumvector (as an example we give here the value at theFermi-edge):

~k⊥ = 2.75 ·10−7 kgm/s

There it is important that the formular above is onlyvalid orthogonal emission. In fact, we have also to no-tice that the mass has to be replaced by the effectivemass which depends on the momentum vector.

4 Conclusion

The angle resolved photo electron spctroscopy wassuccessfully used to analyse the electronic band struc-ture of our targets). Firstly, we were able to find theintensity distribution of the incoming photons with itsmaximum. Furthermore, we determined the Fermi-edge and discussed its broadening dependent on the

temperature. Additionally, we calculated the workfunction for the used materials, whereas we were un-able to fit the dispersion relation correctly due to dom-inating noiose. Finally, we looked at the reciprocallattice of TiTe2 and were able to find the momentumchange by leaving the inner potential of the solid.

Literature

[1] Winkelaufgelöste Photoelektronspektroskopiemit Synchrotonstrahlung, Anleitung für dasFortgeschrittenen-Praktikum, Humboldt-Universität zu Berlin, Institut für Physik, AGElektronische Eigenschaften und Supraleitung



Appendix

The following plots show our approach to determinethe Fermi-edge. For each case (temperature T and slitwidth w) we produced three plots:

1. Preparation dataHere we plot the measured data and addition-ally a smoothed curve. For smoothing we useda Savitzky-Golay-Filter. Finally, we subtract theShirley underground.

2. Derivation with Gaussian fitHere we use a numerical derivation to find theFermi-edge by using the mean value of an Gaus-sion fit.

3. Fermi distributionFinally, we make a fit with the Fermi distribu-tion with the Fermi-edge as we have calculatedin the last step. This is to find the broadening atthe Fermi-edge dependent on the temperature.

Gold

T = 33.5K and w = 546

16.4 16.6 16.8 17 17.2 17.4 17.6 17.8−50

0

50

100

150

200

250

300

350

400

450

Energy E [eV]C

ou

nts

N

measurement

smoothed and reduced

background

Figure 6: Preparation data

-1600

-1400

-1200

-1000

-800

-600

-400

-200

0

200

400

600

16.7 16.8 16.9 17 17.1 17.2 17.3 17.4 17.5 17.6

Au, T = 33.5K, Slit = 546Gaussian fit

Figure 7: Derivation with Gaussian fit

-50

0

50

100

150

200

250

300

16.6 16.7 16.8 16.9 17 17.1 17.2 17.3 17.4 17.5 17.6

Au, T = 33.5K, Slit = 546

Fermi distribution fit

Figure 8: Fermi distribution



T = 33.5K and w = 746

16 16.2 16.4 16.6 16.8 17 17.2 17.4 17.6 17.8 180

100

200

300

400

500

600

700

800

Energy E [eV]

Co

un

ts N

measurement


background


-2500

-2000

-1500

-1000

-500

0

500

1000

16 16.2 16.4 16.6 16.8 17 17.2 17.4 17.6 17.8

De

riva

tive

of

co

un

ts d

N/d

E [

1/e

V]

Energy E [eV]

Au, T = 33.5K, Slit = 746

Gaussian fit


0

50

100

150

200

250

300

350

400

450

500

16 16.2 16.4 16.6 16.8 17 17.2 17.4 17.6 17.8

Co

un

ts N

Energy E [eV]

Au, T = 33.5K, Slit = 746



T = 100K and w = 546

16.9 17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.80

50

100

150

200

250

Energy E [eV]

Co

un

ts N

measurement


background


-800

-700

-600

-500

-400

-300

-200

-100

0

17.1 17.15 17.2 17.25 17.3 17.35 17.4 17.45 17.5 17.55 17.6

De

riva

tive

of

co

un

ts d

N/d

E [

1/e

V]

Energy E [eV]

Au, T = 100K, Slit = 546

Gaussian fit


0

20

40

60

80

100

120

140

160

17.1 17.15 17.2 17.25 17.3 17.35 17.4 17.45 17.5 17.55 17.6

Co

un

ts N

Energy E [eV]

Au, T = 100K, Slit = 546





T = 123K and w = 746

16.9 17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8−100

0

100

200

300

400

500

600

700

Energy E [eV]

Co

un

ts N

measurement


background


-3000

-2500

-2000

-1500

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-500

0

500

17 17.1 17.2 17.3 17.4 17.5 17.6 17.7

De

riva

tive

of

co

un

ts d

N/d

E [

1/e

V]

Energy E [eV]

Au, T = 123K, Slit = 746

Gaussian fit


-50

0

50

100

150

200

250

300

350

400

450

17 17.1 17.2 17.3 17.4 17.5 17.6 17.7

Co

un

ts N

Energy E [eV]

Au, T = 123K, Slit = 746



TiTe2, T = 123K and w = 746

17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.90

100

200

300

400

500

600

700

800

900

1000

Energy E [eV]

Co

un

ts N

measurement


background


-6000

-5000

-4000

-3000

-2000

-1000

0

17.15 17.2 17.25 17.3 17.35 17.4 17.45 17.5 17.55 17.6 17.65

De

riva

tive

of

co

un

ts d

N/d

E [

1/e

V]

Energy E [eV]

TiTe2, T = 123K, Slit = 746

Gaussian fit


0

100

200

300

400

500

600

700

800

900

17.15 17.2 17.25 17.3 17.35 17.4 17.45 17.5 17.55 17.6 17.65

Co

un

ts N

Energy E [eV]

TiTe2, T = 123K, Slit = 746




IntroductionThe photoelectric effectThree-step model of photoemissionFurther explanations

Experimental setupMaesurementIntensity of the monochromatorFermi-edge and resolutionWork functionDispersion relationReciprocal lattice

Conclusion

Documents

Applied Physics - hu-berlin.depeople.physik.hu-berlin.de/~bfmaier/data/fpraktikum/F4l_publish.pdf · half-sphere analysator (type SES-2002) with an angle resolution of 0:2 and an