Surface Processes at the Nanoscale:how crystals meet the outside world

1) Motivation: let's start with snowflakes

2) Microscopy and diffraction techniques

3) Nucleation, growth and nanofabrication

4) Specific systems: what do we want to know?

Pd/MgO(001); Cu/Cu(111); Ge/Si(001), etc

5) 2D Modeling: recent work in progress

6) Nanostructures : disciplines and technology

John A. VenablesPhysics Department, Arizona State University

and London Centre for Nanotechnology, UCL

Scientific and Technological Motivation

• We understand binding in bulk crystals: what is special and different at the surface?

• We understand thermodynamic equilibrium: but useful structures are grown kinetically...

• It's the science behind the chip business: epitaxial growth of heterostructures, lasers

• And catalytic reactions at small particles: only chemical firms don't share their secrets

• Plus energy, health, environment and art: enjoy nanotubes, tetrapods, snowflakes, etc.

Let's start with Snowflakes

http://www.its.caltech.edu/~atomic/snowcrystals/faqs/faqs.htm

Photos: Patricia Rasmussen, Website: Ken Libbrecht

Pioneers of photo-microscopy

• Wilson Bentley (1865-1931) was a farmer near Jericho, Vermont, who during his lifetime captured some 5000 snow crystal images. More than 2000 were published in 1931 in his famous book, Snow Crystals 

• Ukichiro Nakaya (1900-1962) was the first person to perform a true systematic study of snow crystals. Trained as a nuclear physicist, Nakaya was appointed to a professorship in Hokkaido, the North Island of Japan, in 1932, where there were no facilities for nuclear research.  Undaunted, Nakaya turned his attention to snow crystals, which were locally very abundant.  Book (1954), Snow Crystals: Natural and Artificial. 

The morphology diagramthe role of supersaturation

Driving force = kTln(S);

Supersaturation S = (p/pe)Early MC calculations, 1979John Weeks, George Gilmer

Facets, dendrites, pattern formation

• Pt(111) monolayers by high-T STM:

• top row 0.15 ML

• bottom row 1 ML

• Thomas Michely & Joachim Krug, book Islands, Mounds and Atoms, 2004

1.5 mm 310 nm field of view

• Ice Photos: Ken Libbrecht, book Snowflake, Winter's secret beauty, 2003

Early TEM pictures: Au/NaCl(001)

Donohoe and John L. Robins (1972) Journal of Crystal Growth

Field ion microscopy: diffusion of adatoms

Gert Ehrlich group UIUC, 1988-1997; Gary Kellogg 1994-1997

W(211) substrate23.06 kcal/mol = 1 eV/atom

Scanned Probe Microscopy

UHV STM: Pt/Pt(111) T = 424 K: ML

Helium atom scattering at different T (K)Thomas Michely, George Comsa group (1990-1995)

0.35 3.0

12 90

621 K

424 K

275 K

High resolution TEM: CoSi2/Si(111)

left: a) plan view TEM

b) platelets, c) wires:

lattice resolution cross

-section David Smith

(100)

(-111)

(111)

(-511)

[100

]

(100

)300 nm300 nm300 nm

above: platelets & nanowires by AFM Anouk Rougee

Zhian He, David J. Smith, Peter A. Bennett PRL 93 (2004) 256102

Growth modes

Island Layer + Island LayerVolmer-Weber Stranski-Krastanov Frank-VdM

Atomic-level processes

Variables: R (or F), T, time sequences (t)

Parameters: Ea, Ed, Eb, mobility, defects…

Competitive capture

dn1/dt = R – n1/;

an

c…

Venables (1987) Phys. Rev. B

Nucleation density predictions• Matlab Programs

(R, T-1 and cluster size, j)

• Input Energies

• Simultaneous output: Densities and critical cluster size, i.

McDaniels et al. (2001) PRL; Venables et al. (2003) Proc. Roy. Soc.

Nucleation on point and line defects

(a) Point defects (vacancies) (b) Line defects (steps)

Extension to Defect Nucleation (parameters nt, Et)

  dn1/dt = R –n1/ n1(t), single terrace adatoms

dn1t/dt = 1tDn1nte - n1tdexp(-(Et+Ed)/kT) n1t(t)

.... empty traps trapped adatomsdnj/dt = Uj-1 - Uj = 0 nj(t), via local equilibrium

dnj’t/dt nj’t(t), not necessarily same i, i’

....dnx/dt = dnj/dt = Ui - ... nx(t),

(j > i +1) terrace cluster densitydnxt/dt = dnj’t/dt = Ui’t - ... nxt(t),

(j’ > i’ +1) trapped cluster density

Point defects and Nanofabrication?

For Fe/CaF2(111): Heim et al. 1996, JAP; Venables 1999

Specific systems: what do we want to know?

• Metals on metals (Pt, Ag, Au...): adatom energies, catalytic properties, templates for alloys, devices

• Semiconductors (Si, Ge, GaAs...): reconstructions, energies, device understanding and applications

• Metals on insulators (Au/NaCl, Fe/CaF2, Pd/MgO...): energies, role of defects, metal catalytic properties

• Metals on and in semiconductors (Ag/Si, Ti, Dy/Si...) energies, subsurface growth, nanowires, magnetism

Experiment - kinetic model - quantum calculation

A particular case: Pd/MgO (001)

G. Haas et al. 2000 PRB; Venables and Harding 2000 JCG

Defect nucleation, i = 3 at high T

Pd/MgO (001): parameter sensitivity

Venables and Harding 2000 J. Crystal Growth 211, 27-33

TrappingPair Binding

Rate equations & KMC with DFT parameters

Rate equations Venables, Giordano & Harding, J. Phys. C.M. 2006

KMC: L. Xu, G. Henkelman, 2005-07, G. Barcaro et al, 2005

Pd2

Pd4

Pd3

FS+ center

Conclusions #11) Nucleation & growth models have been developed

where "experimental" energies for adsorption, diffusion, binding & trapping can be extracted.

2) Small 2D and 3D clusters are mobile on the surface, can even be liquid; competing configurations

3) Many theoretical methods are now available to see if such energy values are reasonable. The cases of Pd and Ag/MgO(001) have been investigated in detail, but results have been controversial. Are we now OK?

4) The Chemists seem to be winning! Embedded clusters, spin polarized calculations seem to be needed to get good values, especially for Pd, which has competing singlet and triplet ground states.

Capture numbers: 1D radial rate-diffusion equations

dn1(r,t)/dt = G(r,t) –n1(r,t)/r,t +[D(r)n1(r,t)]

G(r,t), generation rate n1(r, t), adatom profile

dnx(r,t)/dt = dnj(r,t)/dt = Ui(r,t) – ... nx(r, t) nx(r, t) stable cluster density profile

Deals with deposition (G~F) and annealing (G~0), plus also potential energy landscapes, V(r), via Nernst- Einstein equation (t-dependence implicit),

j(r) = –D(r)n1(r) – [n1(r)D*(r)]V(r) radial current capture number

Diffusion and attachment limits

a) B=2exp(-EB)

b) BV=2exp(-V0)

Diffusion solution, at r = rk+ r0

D = 2Xk0.(K1(Xk0)/ K0(Xk0)),

with Xk0 = (rk+ r0)/D11/2

Attachment (barrier) solution:B = 2(rk+ r0)exp(-EB)

= B(rk+ r0) or BV(rk+ r0)

They combine inversely ask

-1 = B-1 + D

-1 Venables and Brune PRB 66 (2002) 195404

Delayed onset of nucleation

Reduced capture numbers: longer transient regime (nx) Venables and Brune 2002

Repulsive adsorbate interactions: Cu/Cu(111)

Knorr et al. PRB 65 (2002) 115420; Venables & Brune (2002) PRB

Annealing, low T (16.5K),Cu/Cu(111) Rate equations, full lines as f (rd); KMC, squares with error bars.

Cu/Cu(111): STM, 0.0014 ML, preferred spacing

Interpolation scheme for annealing: i = 1

dn1/d(D1t) = -21n12 -xn1nx, dnx/d(D1t) = 1n1

2,

with k = init ft + kd(1-ft), init = Bft;

ft = K0(Xd)/K0(Xk0); Xd = (rk+r0+rd)/(D1)1/2

with time-dependent rd = (0.5D1t)1/2BV/2.

Full lines: Attachment limit

Dashed lines:Diffusion limit

Previous slides:Interpolation

Extrapolation to higher temperatures

Compare KMC-STM: 10 < V0 < 14 meV; Venables & Brune 2002

REs: integrate to 2 or 20 min. anneal with given V0.

KMC: hexagonal lattice simulations (1000 x 1155) sites with EB = V0.

Conclusions #2: time-dependent capture numbers

1) Explicit t-dependence involves the transient regime and a finite number of adatoms. Barriers or repulsive potential fields reduce capture numbers, lengthen transients and involve more adatoms.

2) Barrier capture numbers and diffusion capture numbers add inversely. An interpolation scheme is needed to describe t-dependence in the transient.

3) Large critical nucleus size lengthens transient. Annealing a low T deposit with potential fields is a very sensitive test of t-dependent capture numbers, as small capture numbers result in little annealing.

Extension to Ge/Si(001)stress-limited capture numbers

• Low dimer formation energy (Ef2 ~ 0.35 eV) gives large i,

even though condensation is complete • Stress grows with island size, x decreases

• Lengthened transient regime results, > 1 ML, source of very mobile ad-dimers (Ed2 ~ 1 eV) for rapid growth

eventually of dislocated islands• Interdiffusion, and diffusion away from high stress regions

around islands, reduces stress at higher T and lower F (e.g. at 600, not 450 oC for F ~1-3 ML/min.)

Chaparro et al. JAP 2000, Venables et al. Roy. Soc. A361 (2003) 311

Sizes and shapes in Ge/Si(001)

TEM, AFM: Chaparro, Zhang, Drucker, Smith J. Appl. Phys (2000)

Size distributions and alloying

T = 450 °C

1.5 x109

1 x109

0

5 x108

3.2 x109

1.6 x109

0

4.8 x109

0 40 12080 160

X 2

(d)

(b)T = 600 °C

5 ML6.5 ML8.0 ML9.5 ML11.0 ML12.5 ML

Diameter (nm)

Num

ber

of is

land

s /

cm2

/ 2.

5 nm

bin Strain relief via

1) interdiffusion 2) change of shape

Hut-dome transitionsreversible via alloying athigh T > 500 oC

S. Chaparro, Jeff Drucker et al. PRL 1999, JAP 2000

Ge/Si(001) STM Movies: watching paint dry at 450 OC

Mike McKay, John Venables and Jeff Drucker, 2007-08

gas-source MBE from Ge2H6

Ge = 5.0ML, 0.1 ML / minT = 450 °C, 26 min/frame62 hrs total elapsed timefirst frame after 33min annealField of view 600nm x 600 nm

Ge = 5.6ML, 0.2 ML / minT = 500 °C, 7 min/frame14 hrs total elapsed timefirst frame after 160min annealField of view 400nm x 400 nm

Ge/Si(001) hut clusters:Annealing at T = 450 oC

1

4

2

103

56

7

89

1,255

30 nm

1

4

2

103

56

7

89

33

1

4

2

103

56

7

89

2,503

1

4

2

103

56

7

89

3,751

100

150

200

250

300

350

400

450

500

10

15

20

25

30

0 1000 2000 3000 4000

9

Anneal Time (min)

Width

Volume

Length

100

150

200

250

300

350

400

450

500

10

15

20

25

30

0 1000 2000 3000 4000

8

Anneal Time (min)

Width

Volume

Length

Most islands static, smallest island grows (8).

Conclusions #3: Long term annealing with barriers

1) Long term meta-stability in the Ge/Si(001) system at intermediate T = 450 oC, ripening at 500 oC, over long times, several days.

2) Some hut clusters to grow via growth of the short side, but other sides do not grow. Individual facets nucleate and grow: volume proportional to length; nucleation barrier smallest on the shorter sides.

3) Large ad-dimer mobility and some coarsening on and in the wetting layer. Finally dislocated dome clusters grow, and coarsening accelerates, with much mass transfer over large distances (many m).

Extension to general 2D potential

dc(r)/dt = G–c(r)/ +Dc(r))+ c(r)D*) V(r))]

dc(r)/dt = G–c(r)/ +D2c(r))+A.c(r))+ B. V(r))

1st 3 terms, linear diffusion, sources, sinksA. = (D+ D*V(r)). dot product operatorB. = (c(r)D*+ c(r)D*. dot product

Starting Simplifications: 1) low concentration D = D*; 2) no distributed sinks = 0; 3) annealing G = 0.

2D Rate-diffusion simulations

J.A. Venables, J. DeGraffenreid, D. Kay & P. Yang, PRB 2006

R. Grima, DeGraffenreid, Venables, PRB 76, 233405 2007

Frame from 2D Movies

Connect to MatLab filesmovie #1; isometricmovie #2;plan viewmovie #3; capture numberwith/ without repulsive fields

Mean-field equations from microscopic dynamics

From Shu, Liu, Gong et al:

For Ge/Si(001): 1 = 1.75 eV; at lattice sites

21 = .75 eV fast diffusion direction

1 2 1

1 1

( , ) exp( ( ) ( , ))

( ) ( )ˆ ˆ( , ) ( , )

D x y D x y

x y D x y x yx y

V

Strain dependent Diffusion D and Drift velocity Vas deduced by Grima, DeGraffenreid, Venables 2007

Ge/Si(001) concentration profiles

R. Grima, J. DeGraffenreid and J.A. Venables, 2007, PRB

2= 1= 1.75 eV

2 1= 0.75 eV

2- 1= 1.50 eV

Conclusions• Three approaches to diffusion in potential fields (Ovesson

2002, Venables & Brune 2002, Grima & Newman 2004) "same for constant D"; but this is not generally the case. V&B thermodynamics correct, G&N advection-diffusion

• Capture numbers are much reduced due to island potential fields; (rectangular) updateable potentials for "strain".

• Grima & Newman's MED algorithm has been solved for 2D problems; Sum rules are exactly satisfied, including general potential fields. Nanowire systems studied with Ge/Si(001) model parameters (Venables et al., 2006).

• MED with "potentials due to strain" are studied (Grima, DeGraffenreid, Venables 2007). Explicit microscopic expressions for D & drift velocity V obtained; D changes are more important than drift for the Ge/Si(001) model.

ReferencesReview of capture numbers, etc

C. Ratsch & J.A. Venables: JVSTA 21 S96 (2003)

Anisotropic substrates, Restricted corner diffusion

Y. Li, M.C. Bartelt, J.W. Evans et al.: PRB 56 12539 (1997)

P. Yang & J.A. Venables: MRS 859E JJ3.2 (2005)

Numerical methods, Capture numbers in potential fields

S. Ovesson: PRL 88 116102 (2002)

J.A. Venables & H. Brune: PRB 66 195404 (2002)

R. Grima & T.J. Newman: PRE 70 036703 (2004)

J.A. Venables, J. Degraffenreid et al.: PRB 74 075412 (2006)

R. Grima, DeGraffenreid, Venables, PRB 76 233405 (2007)

Nanostructures : disciplines and technology

• Interdisciplinary environment: Physicists, Chemists, Materials Scientists, Engineers. Interchangeable jobs: what does each discipline bring to the table?

• Electrochemistry, solution chemistry, single molecules: more knobs to turn, but fewer in-situ analysis tools? Do all Inventions lead to Innovation? If not, why not?

• What will we really learn from biology? Is nano-bio-anything the wave of the future, or is it just the latest bubble, and already past its prime? Stick to basics...

• A great field for "emergent phenomena": simple rules lead to complex results (P.W. Anderson, 1972)

Nanotechnology, modeling & education

Interest in crystal growth, atomistic models and collaborative experiments, as illustrated in this talk

Interest in graduate education: web-based, and web-enhanced courses and resources, book

See http://venables.asu.edu/index.html for current projects, reference list, links to courses, resources

New Professional Science Masters (PSM) in Nanoscience degree at ASU, now in second year http://phy.asu.edu/graduate/psm/overview

A flurry of theoretical activityExperiment seems to give for Pd/ MgO(001)

Et > 1.5, Ea = 1.2, Eb = 1.2 and Ed < 0.3 eV, and much lower values for Ag/MgO(001)

Several groups try to calculate these values

J.A. Venables and J.H. Harding (2000)

D. Fuks, E. Kotomin et al. (2002-03)

A. Bogicevic and D.R. Jennison (2002)

L. Giordano... G. Pacchioni (2003-06)

L. Xu, G. Henkelman, C.T. Campbell (2005-07)

Ionic crystal + semi-classical metalsExperiment seems to give for Pd/ MgO(001)

Et > 1.5, Ea = 1.2, Eb = 1.2 and Ed < 0.3 eV, and much lower values for Ag/MgO(001)

J.A. Venables & J.H. Harding (2000) J. Cryst. Growth:

discussion Et > 1.5, neutral F-centre, Eb free Pd2 dimer

Pd Ea Ed Ag Ea Ed

Mon 0.85 0.2 Mon 0.66 0.1

Dim 1.47 0.3 Dim 1.27 0.3

DFT-GGA and all that VASPExpt: Pd/ MgO(001) Et > 1.5, Ea = 1.2, Eb = 1.2

and Ed < 0.3 eV, Ea for Ag ~ 0.65 eV?

A. Bogicevic and D.R. Jennison (2002) Surface Sci.

Calculation Ads-Ea F trap-Et Bind-Eb Dim+t-E2t

Pd 1.34 2.72 0.03 0.09

Ag 0.53 1.27 1.81 1.86

Pt 2.67 3.83 0.72 0.14

Au 0.90 2.22 2.15 2.21

Cluster chemistry: DFT-GGA + VASP

First emphasis on the F-centre charge state: neutral F centre (2e in vacancy) binds Pd (Et = 1.55 eV), not Ag;

F+ centre (e in vacancy) binds Pd (Et = 0.77), Ag 0.99 eV

F++ centre (no e) captures an e from both Pd and Ag to give F+ centre + Pd+ or Ag+ Ferrari & Pacchioni (1996)

"Oxygen vacancy: the invisible agent on oxide surfaces" mini-review, on MgO, SiO2 and TiO2 Pacchioni (2003)

Recent cluster details: Giordano... & Pacchioni (2003-05)

Expt: Pd/ MgO(001) Et > 1.5, Ea = 1.2, Eb = 1.2 and Ed < 0.3 eV, and Ea for Ag/MgO(001) ~ 0.65 eV?

Wait a moment, that can't be right...Expt: Pd/ MgO(001) Et > 1.5, Ea = 1.2, Eb = 1.2 and Ed <

0.3 eV, and Ea for Ag ~ 0.65 eV?

Bogicevic & Jennison (2002)Pair-binding on the surface Eb > or << free space dimer E2?

Pd: Eb = 0.03, E2 = 1.06 ±0.16 eV; Ag: Eb = 1.81 eV , E2 =

1.65 ±0.06 eV; E2 from Gringerich (1984-85)

Charge redistribution, and hence Et in F-centre too large? Pd: Et = 2.72 eV > Hf (PdO) = 0.9 eV; Ag: Et = 1.27 eV > Hf (Ag2O) = 0.34 eV; Hf from Reuter & Scheffler (2004)

Fuks, Kotomin et al. (2002-03)HF+Correl, Ea, Ed too small?: Ag: Ea ~ 0.20, Ed ~ 0.05 eV

Embedded DFT cluster + classical shell

Expt: Pd/ MgO(001) Et > 1.5, Ea = 1.2, Eb = 1.2 and Ed < 0.3 eV? Calculated Ed = 0.34, not 0.86 eV (B&J)

Many details, several XC functionals explored, etc; can only give a flavor here. Giordano..& Pacchioni (2004-05)

Terrace Step F F DiVac

Pd Ea 1.36 1.85 3.99 2.70 3.00

trap Et 0.49 2.63 1.20 1.64

Pd2 Eb 0.50 0.66 0.57 0.91 1.71

trap E2t 1.14 1.34 cluster 1.49

Nanoclusters: Pd2, Pd3 and Pd4 ...Expt: Pd/ MgO(001) Et > 1.5, Ea = 1.2, Eb = 1.2 and Ed < 0.3 eV? Indications of i=3 and desorption at high T

Extension of same Pd2 approaches to Pd3 and Pd4:

Pd2 has minimum binding (0.57 eV) at F-center, Pd3 by a further 0.75 eV, and Pd4 by a another 1.38 eV (all relative to Pdn-1 on the defect and Pd1 on the terrace)

Spin polarized cluster configuration (spin singlets d10 on surface, versus triplet d9s1 in gas phase, E ~ 0.19 eV)

Giordano..& Pacchioni (2005)

Is i=3 likely at high T at defects? Looks good: obvious next question in context of "believing" these energies...

Main Recent ReferencesA. Bogicevic &D.R. Jennison Surface Sci. 515 (2002) L481-6

A.M. Ferrari & G. Pacchioni J. Chem. Phys. 100 (1996) 9032-7

D. Fuks, E.A. Kotomin et al: Surface Sci. 499 (2002) 24-40

L. Giordano... & G. Pacchioni Phys. Rev. Lett. 92 (2004) 096105; Chem. Phys. 309 (2005) 41-7; Surface Sci. 575 (2005) 197-209

G. Haas et al. Phys. Rev. B. 61 (2000) 11105-8

G. Pacchioni Chem. Phys. Chem. 4 (2003) 1041-7 (mini-review)

C. Ratsch & J.A. Venables J. Vac. Sci. Tech. A 21 (2003) S96-109

K. Reuter & M. Scheffler Appl. Phys. A 78 (2004) 793-8

J.A. Venables & J.H. Harding J. Crystal Growth 211 (2000) 27-33

J.A. Venables et al. Phil. Trans. Roy. Soc. A 361 (2003) 311-329

Shape transitions: S. Chaparro, Jeff Drucker et al. JAP 2000

Side Length (nm)

10 20 807060504030 90 120110100 150140130 160 170

10 20 807060504030 90 120110100 150140130 160 170

10

20

30

50

40

10

20

30

50

40<100> Section

<110> Section

{510} {510}

Steeper facets + {211} + {311} + {511}

Huts Domes Defective DomesBig Huts

{110} + {320} + {210} + {510}

{110} appears

T = 600 °C

{511} disappearsDislocations

Trenches Steeper facets appear

{510} disappears

Dislocations

Trenches

{320} + {210} + {510}

F

D

v

• Islands as continuum in the plane, but individual atomic layers

• Velocity of island boundaries ?

• How do islands nucleate ? Where ?

• Evolve island boundaries with the level set method

• Treat atoms as a mean field quantity, at least initially

Compare with

•Continuum Models (deterministic, lacks atomic detail)

•Atomistic KMC (stochastic, expensive)

Island Dynamics Model for Epitaxial Growth

Alternative approaches to modeling

1) Rate and diffusion equations

2) Kinetic Monte Carlo simulations

3) Level-set and related methods

plus

4) Correlation with ab-initio calculations

Issues: Length and time scales, multi-scale; Parameter sets, lumped parameters;

Christian Ratsch and John Venables, JVST A S96-109 (2003)

Level Set Function Surface Morphology

t

=0

=0

=0

=0=1

• Continuous level set function is resolved on a discrete numerical grid

• Method is continuous in plane, but has discrete height resolution

The Level Set Method: Schematic

• Governing Equation:

• Obtain velocity of island boundaries by solving diffusion equation:

• Boundary condition

0||

nvt

dt

dNDF

t22

0

=0

• Velocity:

• Nucleation Rate:

• Seeding position chosen stochastically (weighted with local value of 2)

2),( tDdt

dNx

)()( aveedge DDvn nn

Level Set Slides: Christian Ratsch, UCLA Applied Math Department

The Level Set Method: Formalism

KMCLevel SetData: Fe/Fe(001)

J.A. Stroscio and D.T. Pierce, Phys. Rev. B 49 8522 (1994)

Petersen, Ratsch, Caflisch, Zangwill, Phys. Rev. E 64, 061602 (2001)

Reversibility: Sharpening of the Island Size Distribution

Microscopy and Diffraction Techniques • Early TEM: Au/NaCl(001) island growth example• FIM: the first to "see atoms" and diffusion paths • Scanned probe microscopy: STM, AFM and MFM• High resolution TEM, EDAX, EELS, holography • UHV analytical SEM and STEM, AES, EELS• LEEM and PEEM, SPLEEM, etc • coupled with scattering and diffraction techniques:

LEED, THEED, RHEED, X-rays and neutrons, Helium atom scattering (HAS), RBS, ICISS....

Acronym heaven: techniques in red available at ASU

UHV SEM and STEM: AES & SAM• ASU development, mid 80's- present: John Cowley, Peter

Crozier, Jeff Drucker, Gary Hembree, Mohan Krishnamurthy, Jingyue Liu, Mike Scheinfein, John Spence, John Venables et al.

• UHV Applications to electronic materials and catalysis: see my web page at http://venables.asu.edu/research/index.html

• Cowley memorial volume: J. Electron Microscopy 54 (2005) 151

LEEM and PEEM: SPLEEM & XMCDPEEM• ASU and Trieste development, early 90's- present: Ernst Bauer,

Peter Bennett, Assia Pavlovska, Ig Tsong and co-workers

• UHV applications to surface morphology and reconstructions, electronic and magnetic materials and catalysis: see Bauer/ Pavlovska web page at http://physics2.asu.edu/homepages/bauer/

• Review article: Reports on Progress in Physics 57 (1994) 895

Rate Equations (experimental variables T, R,t)

  dn1/dt = R –n1/ n1(t), single adatoms

....dnj/dt = Uj-1 - Uj = 0 nj(t), via local equilibrium

....dnx/dt = dnj/dt = Ui - ... nx(t),

(j > i +1) stable cluster densityalso:dZ/dt = f(cluster shape) Z(t), surface coverage

  and dax/dt ax(t), d/dt (t), instantaneous

mean cluster size condensation coefficient

Differential equations versus Algebra

Using cluster shape, assumed or measured, express

nx(Z) (Z). f1(Rpexp(E/kT))

t(Z) (Z). f2(Rpexp(E/kT));

where p and E are functions of i, critical nucleus size

similarly f3 and f4, for ax(Z) and (Z), not much used.

Choice of 1) integrating differential equations, or

2) evaluating near the maximum of nx(Z).

Steady state conditions (dnx/dt, etc = 0) converts a set of ODE’s into a (nonlinear) algebraic solution. 

DNA/genes

Proteinscomplexes/

reactionnetworks

CellsCellular

aggregatesOrganisms Populations Ecosystems

Life on earth

910 510 110 310 710

Informationfeedback

Molecularbiology

Bio-chemistry

Cellbiology

Develop-mental biology/genetics

PhysiologyEcology andpopulationgenetics

Ecosystem biology

Evolutiontheory

(adaptation/speciation)

metres 710

Length scales in biology (Newman)