simplicity of the adsorbent, make adsorbed helium on cesium an ideal model system for the physics of wetting in general. An introduction to the system ’liquid helium on cesium’ is given in Part I of this thesis.
However, there are a few unknown parameters complicating this system remarkably. One of them is the surface roughness of low temperature quench-condensed cesium films, which turned out to hugely influence the wetting properties of the system. For this purpose, scanning tunneling microscopy investigations of cesium films were conducted within the work for this thesis, leading to a deeper understanding of the growth of low temperature quench condensed cesium. These results are presented in Part II of this thesis.
Furthermore, up to now little is known on the ’non-wetting’ helium thin film existing below the wetting temperature. For its investigation, tunneling of photoemitted electrons was already used as a probe. However, the mechanism of electron tunneling and electron backscattering by gas atoms was not understood in detail1, leading to a broad spectrum of interpretation possibilities for the obtained photoelectron measurements. Therefore, a complete theoretical description of these phenomena is presented for the very first time in Part III. Experimental observations of both gas backscattering and electron tunneling obtained within this thesis are in very good accordance with the developed theory.
Constance, April 2007
M
ARTINZ
ECH1J.J. Thomson and L.B. Loeb investigated the backscattering of photoelectrons by gas atoms in 1928-32, but their work fell into oblivion leaving important questions unanswered.
1. Scanning tunneling microscopy 11
1.1. Theory of the tunneling microscope . . . 11
1.1.1. Tunneling from a many particle point of view as described by Bardeen, Tersoff and Hamann . . . 11
1.1.2. Tunneling between metals . . . 13
1.2. Components and operation . . . 14
1.2.1. Electroceramics . . . 14
1.3. Tunneling spectroscopy . . . 16
1.3.1. I(V) spectroscopy . . . 16
1.3.2. I(z) spectroscopy . . . . 18
2. Statistical analysis of STM images of rough surfaces 19 2.1. Thermodynamics of crystal growth . . . 19
2.2. Kinetics of film growth . . . 20
2.3. Shadow instability . . . 21
2.3.1. Reciprocal-space height correlation function . . . 22
2.3.2. Diffusion length . . . 22
2.4. Kardar-Parisi-Zhang equation . . . 22
2.4.1. Universality classes . . . 23
2.4.2. First simulations . . . 23
3. Experimental setup of the STM 25 3.1. Constructing the STM . . . 25
3.1.1. Coarse approach mechanism . . . 25
3.1.2. STM design . . . 27
3.1.3. Optical access . . . 28
3.1.4. Piezotube . . . 28
3.1.5. Slip-stick table . . . 29
3.1.6. STM tip . . . 30
3.1.7. Cesium source . . . 31
3.2. STM control electronics . . . 32
4.3. Calibration at low temperature . . . 46
4.3.1. N2 temperature . . . 47
4.3.2. Liquid He temperature . . . 48
4.4. Cesium evaporation . . . 48
5. Experimental results - STM 51 5.1. Grain size distribution . . . 52
5.1.1. Watershed-algorithm . . . 52
5.1.2. Gold grains at room temperature . . . 53
5.1.3. Gold and cesium grains at low temperatures . . . 54
5.2. Power spectral density function . . . 56
5.3. Spectroscopy . . . 62
III. Photoelectrons: Backscattering and Tunneling 65
6. Photoelectron transport through gases and films 69 6.1. Photoelectron transport through gases . . . 696.2. Thomson’s model . . . 70
6.3. Thermalisation of hot electrons . . . 73
6.4. Dependence of electron current on electric field . . . 75
6.5. Modelling of Bradbury’s results from 1932 . . . 77
7. Tunneling of electrons through liquid/solid films 81 7.1. Backscattering in presence of tunneling . . . 81
7.2. Calculation of pt . . . 82
8. Experimental setup - photoelectron measurements 87 8.1. Electric field distribution . . . 88
8.2. Fridge . . . 90
9. Experimental results - photoelectrons 91 9.1. H2 experiments . . . 91
9.2. He experiments . . . 95
9.3. Backscattering & Tunneling . . . 97
Publications 111
Acknowledgements 113
A. Calculations and references 115
A.1. Geometrical calculations of a colloidal monolayer . . . 115 A.2. Heat capacity of oxygen free copper . . . 115 A.3. Calculation of photon number . . . 115
B. Unusual behaviour of a helium droplet 117
B.1. Crude estimation . . . 118 B.2. Detailed calculation . . . 118
Bibliography 123
Index 133
atom. This is the reason why the density in metallic Cs is low; the spacing of the bcc lattice is 5.23 ˚A [FI88]. These 6s electrons extend far out from the atomic centres of the Cs surface and repel the 4He atoms which prevents them coming close to the Cs. This in turn means that the attractive van-der-Waals force, which varies as 1/r3 for a semi-infinite slab, is unusually small. So although the cohesive force between 4He atoms is small, it is larger than between a 4He atom and Cs.
The difference between liquid 3He and liquid 4He, at temperatures around 1K, is mainly due to their difference in mass: the superfluidity of4He does not play a direct role in wetting.
Both liquids have large zero point energies which makes their density low as the balance between the kinetic energy of the zero point motion and the attractive van-der-Waals force favours a large atomic separation. The smaller mass of 3He increases its kinetic energy as the atom’s wave-vector is determined by the nearest neighbour distance. This causes a lower liquid density (the molar volumes of liquid 3He and4He are 35 and 28 cc, respectively) and a lower cohesive energy (the normal boiling points for liquid 3He and 4He are 3.2 and 4.2K, respectively). Although the binding energy of 3He to Cs is less than for 4He, it is higher than the binding energy between 3He atoms, which is opposite to4He, therefore liquid 3He wets Cs [RPRT97] but 4He does not.
The Cs surface around a drop of liquid 4He is not completely free of 4He, there is always a thin film of 4He which is in equilibrium with the vapour. This thin film state (often referred to as the ’non-wetting’ thin film) is supposed to be a quasi 2D gas [SKW94a].
Flow measurements of 4He atoms across a Cs surface have shown that this gas has a very low density at T ! Tw, [SKW94b]. However, quartz microbalance measurements suggest that there could be a few monolayers of liquid 4He on Cs at T ∼ Tw [TR93a]. The thin film remains thin in a saturated vapour until T = Tw as expected for a first order wetting transition.
In contrast, a3He film on Cs is expected to increase in density continuously as the ambient vapour pressure is increased towards its saturated value at all temperatures. Following these facts, it is interesting to see how thin films of 3He and 4He on Cs actually behave.
State of the art
Very Challenging reasons to investigate the ’helium-cesium’ system are both its exceptional chemical purity and the simplicity of the adsorbent. These circumstances allow the inves- tigation of phenomena which are hardly accessible to systems using classical fluids. For
temperature quench-condensation and showed therefore similar optical surface properties.
This lead to the conclusion that a roughness on a microscopical scale is the origin of the observed wetting properties [DR02a].
In contrast, the very first measurements of Θ were performed on cesium substrates pre- pared by means of condensation from the gaseous phase at room temperature [KSW95a].
The measured contact angle was larger than theoretically expected [CCZ98] and showed a large temperature dependence. This led to the interpretation that ripplons (quantised surface waves) may be excited at the interface liquid helium-cesium [KW98]. Excitations of this kind have indeed been theoretically predicted [CKT98], but the calculated contribution to the surface free energy was lower than experimentally observed [KW98]. Furthermore, an irregularity in Θ(T) was found at roughly 1.5 K, which later on was interpreted as a phase transition from a 2D gas to a 2D liquid state [KSW95a] of the ’non-wetting’ helium thin film. Further results from measurements using the photoelectron tunneling technique were interpreted as growth of a 2D liquid close to coexistence [IKL03].
The contact line dynamics of liquid 4He films on cesium substrates with roughness of dif- ferent correlation lengths in the sub-micron scale was investigated by Prevostet al. [PRG99].
The wetting temperature Tw was found to be equal for all substrates; however, large dis- crepancies between Θ in respect to its hysteresis and temperature dependence were proven [PPRG00]. The temperature dependence was explained by the existence of different inter- face excitations, being dependent on the surface roughness of the substrate. Furthermore, different experiments showed that the roughness of a spreading contact line decreased with increasing optical surface quality [RGGR98]. However, the hysteresis in the moving contact line was larger than theoretically predicted and additionally independent on the used sub- strates. These observations were explained by pinning-effects on a microscopic scale. The dynamics of a 4He contact line on a lithographically created rough cesium surface was also investigated [PRG02]. By measuring the roughness of the contact line, strong pinning on the defects of the substrates was observed. Further measurements on the friction of helium droplets on an inclined cesium surface showed that the sticking of the droplets was very high; the influence of a sticking helium tail was mentioned as a possible explanation. In this context, the surface roughness should play a very important role [BTJ04].
Besides the system of ’liquid helium on cesium’, where the influence of surface roughness dominates most of the experiments on both static and dynamic wetting properties, there are lots of different systems for which roughness plays an important role, too. This is
fast moving contact line, whereas a pinning transition is expected to take place due to a roughness transition within the contact line [GR03]. The degree of surface roughness seemed to be an important parameter; however, experiments to prove this are lacking so far.
• A strong dependence of the contact angle hysteresis as function of defect density was found when investigating wetting properties of structured surfaces [RCBT03]. For a small defect concentration, the hysteresis increased as a linear function of the concen- tration as defects seemed to pin the contact line individually. For high defect densities, collective pinning phenomena came into play and the hysteresis decreased again.
In conclusion, all wetting experiments – independent on the details of the system – showed, that the surface morphology of the wetted substrate has a huge influence on wetting prop- erties and -phenomena. Qualitatively, this relationship sounds plausible; however, a quanti- tative link between surface morphology and wetting properties is still oustanding.
Therefore, a low temperature scanning tunneling microscope has been constructed by the author during this thesis. It is the aim of the following chapters to report on the STM experiments on cesium surfaces prepared at low temperature, see Chapter 4. Moreover, an interpretation of the obtained data with respect to different growth mechanisms will be given in Chapter 5.
Scanning Tunneling Microscopy
The scanning tunneling microscope (STM) has become a unique tool for the analysis of electrical conducting surfaces with a resolution well below the interatomic distance since its invention in 1981 at Zurich’s IBM research laboratories, Switzerland, by G. Binnig and H.
Rohrer [BRGW82], [BRGW83]. Even though tunneling spectroscopy by means of metal- insulator-metal sandwich structures has been investigated quite intensively already 20 years earlier [Gia60], this technique evidently had two inherent limitations: (1) once the tunnel junction has been made, access to the electrode surface for further treatment is lost, and (2) the information is averaged over an area limited in its size by lithographic techniques, which was about 100 nm at this time.
It was Binnig and Rohrer who realised that only vacuum as a tunneling barrier provides both access to the tunnel diodes by any time and, by appropriately shaping one of the electrodes, a spatial resolution far beyond that of sandwich structures can be reached. Binnig and Rohrer received the Noble prize for their pioneering research in 1986, only 5 years after they had published their first communication on tunneling microscopy.
State of the art
Today, the tunneling microscope has led to a whole family of scanning probe techniques, like AFM (atomic force microscopy), MFM (magnetic force microscopy) and SNOM (scanning near field optical microscopy). As the initial use was restricted to conductive surfaces, it is now open to a wide field of research, including magnetic domain analysis [RA02], DNA-imaging [LS93], and even imaging of viruses as the human immunodeficiency virus (HIV) [KV03]. Very recently, two remarkable experiments have been conducted at IBM Almaden, USA, which are worth noting: the first one is the assembly and probing of spin chains of finite size [BH06], where a chain of Mn-Atoms on a Cu(100) substrate has been created and investigated; this is an important step in the creation, understanding, and manipulation of low-dimensional spin systems. The second one is the detection of a single
V(r) =
− 0 −
0 , −d/2≤r≤+d/2
−V0 , +d/2< r ,
(1.1) one can solve the time-independent Schroedinger equation
$−!2
2m +V(r)
%
ψ(r) =Eψ(r). (1.2)
If suitable continuity conditions are applied (such as continuity of ψ(r) and ψ!(r) at the edges of the potential well), and if further on the potential barrier is assumed to be broad such that &
2m(V0−E)/!d >> 1, one receives the following expression for the tunneling probability through the potential barrier:
|T|2 = 16E(V0−E)
V02 e−2d!√2mV0−E. (1.3)
If one likes to describe a more general potential, the Wentzel-Kramers-Brillouin (WKB) [Wen26][Kra26] approach is suitable, in which the potential barrier is divided in square potentials of different height. For each potential n, having a width dn, the transmission coefficient|Tn|can be calculated. If the potential barrier is divided intoN steps, one obtains
|TN|2 = 'N n=1
e−2dn! √2mVn−E
|T|2 = lim
N→∞|TN|
= e−2!R−∞+∞√
2m(V(r)−Edr. (1.4)
Although Eq. 1.4 describes the tunneling of a single electron through a potential barrier quite accurately, it lacks the description from a many particle point of view.
1.1.1. Tunneling from a many particle point of view as described by Bardeen, Tersoff and Hamann
When Giaever [Gia60] and later on Nicol, Shapiro, and Smith [NSS60] did their research on tunneling through sandwich structures in 1960, it was already clear that the tunneling
obtained:
Itun = 2πe
! (
µ,ν
f(Eµ)[1−f(Eν +eV)]|M|2µνδ(Eµ−Eν), (1.6) where f(E) is the Fermi function, V the applied voltage, and Mµν the tunneling matrix element between states ψµ of the probe and ψν of the surface. Eµ is the energy of state ψµ in the absence of tunneling. Note that Eq. 1.6 and all of its subsequent simplifications describes elastic tunneling only, i.e. pure vacuum tunneling. This is expressed as Dirac’s function ’δ’ in Eq. 1.6.
For low temperatures, i.e. at and below room temperature, and for small bias voltages, Eq. 1.6 can be simplified to
Itun = 2π
! e2V (
µ,ν
|Mµν|2δ(Eν−EF)δ(Eµ−EF). (1.7) It is now worthwhile to consider the simplest case, where the tip is replaced by a point probe, which represents the ideal of a nonintrusive measurement with a maximum possible resolution. If the tip wave function is arbitrarily localised, the matrix elementMµν is simply proportional to the amplitude of ψν at the position r&0 of the probe, and therefore Eq. 1.7 reduces to
Itun ∝(
ν
|ψν(r&0)|2δ(Eν −EF). (1.8) The quantity at the right side of the equation is the local density of states (LDOS), i.e.
the tunneling current is proportional to the LDOS of the surface being under investigation.
Furthermore, this means that a tunneling microscopy image depicts the contour of the constant LDOS of the surface.
In order to make use of Eq. 1.7, it is crucial to calculateMµν. As Bardeen showed [Bar61], the tunneling matrix element can be expressed by
Mµν = !2 2m
)
dS&·*
ψµ"∇ψν −ψν∇ψµ"+
, (1.9)
where the integral is over any surface lying entirely within the vacuum region separating the two electrodes. It was Tersoffand Hamann [TH85] who first tried to calculate Mµν in 1985.
They used Bloch waves in their approach to describe the surface wave function ψν; the tip
tip, the density of states of the tip Dt, the radius of curvature of the tip R, the inverse decay length for wave function into vacuum κ = (2mΦ)1/2!−1, and the applied voltage V. Furthermore, it reveals again (see Eq. 1.8) that the tunneling current is proportional to the LDOS of the sample and, since |ψν(&r0)|2 ∝e−2κ(R+d) that the tunneling current behaves as Itun ∝e−2κd.
For the sake of simplicity, one very often finds an approximation of Eq. 1.10, which is valid for tunneling through a planar barrier at small voltages V << Φ[Sim63]:
Itun = e
!
2 κ
4π2dV e−2κd. (1.11)
Among others, Binnig and Rohrer referred to Eq. 1.11 extensively.
1.1.2. Tunneling between metals
Figure 1.1.: Energy diagram for a tunneling junction having a trapezoidal barrier. a) Tip and sample are separated, i.e in non-tunneling distance. The tip and the sample have the workfunction ΦT and ΦS, respectively. b) The Fermi levels of both metals equilibrate after being put into tunneling contact. A contact voltageUC
arises. c) After a bias voltage V has been applied, a tunneling current flows from tip to sample. The highest current is located at the Fermi level, the lowest one at (EF −V)·e.
If both tip and sample are metals, they create a metal-insulator-metal structure. If no bias voltage V is applied, the fermi levels of both metals equilibrate and no tunneling current arises, independent of the barrier distance d. If V > 0, the fermi levels of both metals are separated by E = eV, and a current emerges being directed from occupied states to
where Φ is the average of both workfunctions ΦT and ΦS given in [eV]. As one can see from Eq. 1.11, a change in tunneling distance by about 1 ˚A leads to a variation in tunneling current by about one order of magnitude. This exponential decay was first observed by Young et al.[YWS71] in 1971 but it took another decade until Binnig and Rohrer realised that a surface structure can be mapped on an atomic scale if the current is kept constant by means of an electric control circuit.
1.2. Components and operation
The principle of operation of the STM is straightforward. Simplified it can be described in scanning a metal tip (one electrode of the tunnel junction) over the surface to be investigated (second electrode), see Fig. 1.2. The metal tip is fixed onto a tube-like piezodrive, which is usually made of PZT/LZT (Lead Zirconate Titanate).
The control unit ’CU’ applies a Voltage Vz to the piezotube such that Itun remains con- stant when scanning the tip over the sample surface. At a constant LDOS and a constant workfunctionΦ,Vz(x, y) yields the topography of the surfaces z(x, y,) directly provided the LDOS of the sample is constant. If for example a monoatomic step an a clean Au(100) surface is scanned, there will be a smearing outδ in lateral direction due to the finite radius R of the tip which is simply proportional to√
R [BRGW82]. For a constant tunnel current, local changes in Φ, e.g. due to adsorbates, are compensated by a corresponding change in z(x, y). Thus, a lower workfunction would mimic a topography change in +z direction, i.e.
the tunneling tip has to elevate a small pace. However, workfunction induced changes in z(x, y) can be clearly distinguished from topography induced changes when the tunneling distanced is modulated when scanning at a frequency higher than the cut-off frequency of the control unit. The modulation signal dItun/dd ∝Φ directly gives the workfunction in a simple situation (e.g. a smooth surface). In general, however, the modulation in distance δd equals δzcosφ, where δz is the length modulation of the piezotube in z-direction and φ the angle between the z-direction and grad(z). Separation is then, although involved, still possible since both Vz and modulation signal contain topography and workfunction in a different way [BR83].
1.2.1. Electroceramics
As mentioned in Section 1.2, a lead-zirconate-titanate ceramic (PbZrxTi1−xO3, PZT) is most frequently used to provide scanning and z-motion. This material is a ceramic perovskite featuring a marked piezoelectric effect and an extremely large dielectric constant at the
Figure 1.2.: Schematic drawing of an STM setup. The most important components dis- played: the piezo tube providing scanning motion, the tunneling tip, the sub- strate, the operational amplifier, and the feedback loop keeping the current constant. The step at A) will be smeared out at a lengthδ due to the finite ra- dius of the tunneling tipR, whereδ∝√
R[BRGW82]. At B), the workfunction of the surface changes suddenly due to an impurity which is compensated by a change in the tunneling distanced, thus mimicking a different surface structure.
’Op. Amp.’ is the operational amplifier, converting the tunneling current in a potential and ’CU’ is the control unit.
morphotropic phase boundary (transition from rhomboedric to tetragonal lattice structure) nearx= 0.52. These properties make the PZT-based compounds one of the most prominent and useful electroceramics. Figures 1.3 and 1.4 show the different lattice structures of PZT, which is an ideal cubic perovskit lattice above the Curie temperature (at x = 0.5, TC ≈ 650 K), and a tetragonal distorted phase below TC (for x > 0.52). This polyedric deformation leads to a charge shifting within the unit cell which is responsible for PZT’s ferroelectric properties. If the material is brought into an electric field it alters its shape due to the inverse piezoelectric effect. The correlation between deformation Si in i−direction and applied electric field Ez is given by
Si =sEiz·Tz+diz·Ez, (1.13)
where sEiz is the elastic modulus at constant electric field, diz is the piezoelectric strain coefficient, and Tz is the mechanical stress. For a piezo ceramic tube as shown in Fig. 1.2 operating on the transversal piezo effect (deformation is normal to the applied electric field),
1. Scanning tunneling microscopy
12 2.1 Die Perowskitstruktur
Die Polyederverzerrungen haben zur Folge, daß Ladungsschwerpunkte innerhalb der Elemen- tarzelle gegeneinander verschoben sind, es kommt zur Ausbildung einer spontanen Polarisa- tion. Diese spontane Polarisation, die durch äußere Faktoren, wie z.B. elektrische Felder
Abbildung 6: Perowskitstruktur in der ideal kubischen Form. Beispiel: Hochtemperaturphase von PZT (schwarz: Ti4+/Zr4+ Ionen, hellgrau: O2--Ionen, dunkelgrau: Pb2+-Ionen)
Abbildung 7: PZT in der tetragonal verzerrten ferroelektrischen Phase. Die zentralen Ti/Zr-Kat- ionen und die Sauerstoffionen sind gegenüber dem Pb2+-Untergitter verschoben. (schwarz: Ti4+/
Zr4+ Ionen, hellgrau: O2- Ionen, dunkelgrau: Pb2+ Ionen)
a b c
Powd erC el l 2 . 0
a b c
Po w d erCell 2 .0
Figure 1.3.: PZT in its ideal cubic perovskit structure as it appears above the Curie tempera- ture (paraelectric). In black: Ti4+/Zr4+ Ions, light-gray: O2− Ions, dark-grey: Pb2+ Ions [Lan03].
Die Polyederverzerrungen haben zur Folge, daß Ladungsschwerpunkte innerhalb der Elemen- tarzelle gegeneinander verschoben sind, es kommt zur Ausbildung einer spontanen Polarisa- tion. Diese spontane Polarisation, die durch äußere Faktoren, wie z.B. elektrische Felder
Abbildung 6: Perowskitstruktur in der ideal kubischen Form. Beispiel: Hochtemperaturphase von PZT (schwarz: Ti4+/Zr4+ Ionen, hellgrau: O2--Ionen, dunkelgrau: Pb2+-Ionen)
Abbildung 7: PZT in der tetragonal verzerrten ferroelektrischen Phase. Die zentralen Ti/Zr-Kat- ionen und die Sauerstoffionen sind gegenüber dem Pb2+-Untergitter verschoben. (schwarz: Ti4+/
Zr4+ Ionen, hellgrau: O2- Ionen, dunkelgrau: Pb2+ Ionen)
a b c
Powd erC el l 2 . 0
a b c
Po w d erCell 2 .0
Figure 1.4.: PZT in its tetragonal distorted phase as it ap- pears below the Curie temperature (ferroelec- tric). The central Ti/Zr Cations are shifted with respect to the Pb2+ sub- lattice [Lan03].
the lateral scanning range ∆x can be estimated as follows [Ins05]:
∆x≈ 2√
2d31L2V
π·ID·d , (1.14)
where d31 is the strain coefficient (displacement is normal to polarisation direction), V the operational voltage,L the length, ID the inner diameter, and d the wall thickness. For the change in length, we can use the following approximation [Ins05]
∆L≈d31
U ·L
d . (1.15)
1.3. Tunneling spectroscopy
Even though the STM visualises the nanoscale world in a way which is essential for the understanding of processes taking place at that scale, there is much more information in the tunneling current than just the surface geometry. If a structured barrier is considered with a non-uniform distribution of states, the tunneling current can also reflect this energetic variation. This information can be accessed by means of scanning tunneling spectroscopy (STS). Most commonly, I(V) and I(z) spectroscopy are the two different STS techniques used.
1.3.1. I(V) spectroscopy
I(V) spectroscopy is performed by keeping the tip/sample separation constant while ramp- ing the bias voltage V. Thus, the surface electronic structure can be measured although
Figure 1.5.: The tunneling current is influenced by the surface density of states, DOS(E).
Given that the tip has a constant DOS, its electronic structure will only con- tribute in a constant background to the STS spectroscopy, which enables direct investigations of the substrate’s DOS(E) usingI(V)-STS techniques.
the interpretation of the STS data is complicated due to two usually unknown reasons: (1) the DOS of the tip and (2) the tunneling probability TT(E, V). The first problem is most frequently approached by comparing different STS-spectra at identical surface locations and by ensuring that all results are reproducible with different tips. Thus, if STS-spectra are measured at different surface locations, the tip electrode structure will at the minimum contribute a constant background. The second problem can be overcome by using an ap- proximate expression for the tunneling current [FS87]:
Itun ∝ ) eV
0
ρ(E)TT(E, V)dE , (1.16)
where TT(E, V)∝e−2κd as shown in Eq. 1.11. After differentiation, one obtains dItun
dV ∝eρ(eV)TT(eV, eV) +e ) eV
0
ρ(E) d
d(eV){TT(E, eV)} . (1.17) From Eq. 1.16 and Eq. 1.17, one obtains the expression
(dItun)/(dV)
Itun/U = ρ(eV) +,eV 0
ρ(E) TT(eV,eV)
d
d(eV){TT(E, eV)}
1 eV
,eV 0
-ρ(E)TTT(E,eV)
T(eV,eV)
.dE . (1.18)
As observed from Eq. 1.18, a normalisation of the differential conductance (dItun/dV) to the total conductance provides a measure for the surface DOS, which is the first term of the nominator in Eq. 1.18. The denominator in Eq. 1.18 can be considered to be a normalisation function of the surface DOS, whereas the second term in the nominator arises due to the fact that the tails of the electron wave functions are effected by the value of the electric field inside the metal-insulator-metal junction. Figure 1.5 shows a ’typical’ DOS(E) as it can be measured by means of I(V)-STS techniques.
nature of the surface being under investigation. This will turn out to be an important parameter for the work presented here. It is worth to note that instead of measuring I(z) at specific locations of the surface, the z-voltage of the piezo tube can be modulated while scanning. Thus, the local Φ can be mapped in parallel to a topographic scan. For details see Section 1.2.
analysing, but no information is obtained on how the surface has grown and why it does look like this. For this purpose, a detailed analysis is required on the growth mechanisms of vapour deposited films which are governed by both thermodynamics and kinetics.
2.1. Thermodynamics of crystal growth
It is generally accepted that there are three different modes of crystal growth, namely Frank- van der Merwe, Stranski-Krastanov, and Volmer-Weber growth, which can be classified according to their surface and interfacial energies σF, σS, and σFS, where ’F’ and ’S’ denote
’film’ and ’solid’.
In the Volmer-Weber or island mode (Fig. 2.2 c), small clusters are nucleated directly on the substrate growing into islands when further atoms are adsorbed. This happens when the nucleated atoms or molecules are more strongly bound to each other than to the substrate, i.e. ∆σ = σF +σFS −σS > 0. This mode is displayed by many systems of metals growing on insulators, including graphite and other layer compounds such as mica [DPM93] [EGN96] [BJD87]. The Frank-van der Merwe or layer mode (Fig. 2.2 a) exhibits
Figure 2.1.: Schematic presentation of the three different growth modes: (a) Frank-van der Merwe, (b) Stranski-Krastanow, and (c) Volmer-Weber.
opposite characteristics. Because of the atoms being more strongly bound to the substrate than to each other (∆σ ≤ 0), the first condensed atoms form a closed monolayer, which becomes covered afterwards with a somewhat less tightly bound second layer. Providing the decrease in binding energy is monotonic towards the bulk, the layering mode is obtained.
2.2. Kinetics of film growth
Besides the well understood thermodynamics, kinetics plays an equally important role in film formation, as it controls island nucleation and island growth. Its individual atomic processes being responsible for adsorption and crystal growth are presented in Fig. 2.2. As
Figure 2.2.: Schematic sketch representing characteristic processes and corresponding ener- gies taking place in nucleation and growth of adsorbates on surfaces [VSH84].
See text for details.
it turns out, many processes take place on the substrate, namely being re-evaporation or re-solution, nucleation of 2D or 3D clusters, capturing by existing clusters, dissolution into the substrate, and capturing at ’special’ sites such as steps [VSH84]. All of these processes are governed by characteristic times, which depend on single atom concentration and/or coverage. If these processes are thermally activated, their characteristic times are controlled by activation energies and frequency factors in turn. As an example, re-evaporation can be characterised by a characteristic timeτre which can be expressed as
τre ∝ν−1ekB1TSEa, (2.1)
whereTS is the temperature of the substrate,ν a characteristic approach frequency (1011− 1013s−1), andEa the binding energy.
In a complete description of the growth of a thin film, rate equations for all different pro- cesses taking place have to be taken into account. For real surfaces, this can be extraordinary complicated, as imperfections like distribution of dislocations and point defects can largely
section. For a more detailed discussion of thin film growth kinetics, the reader is referred to the review paper of Venables et al. [VSH84].
2.3. Shadow instability
Once the continuity of a film is attained and a thick films starts growing, processes like step adsorption and interdiffusion loose importance for determining the film shape. At this state of film formation, the process of growth induces a pronounced surface roughness when protruding parts of the surface mask the lower-lying regions during growth. This roughening mechanism is called shadowing instability [KBR89] and competes with various smoothing mechanisms, of which the most prominent one is, as for thin films, surface diffusion as discussed by Herring [Her50].
If we now consider the easiest model incorporating the basic physics of vapour deposition, the film height h(x, t) in real space can be described according to the shadow growth theory using the following differential equation [KBR89] [OAS+99]:
∂h(x, t)
∂t =−D∂4h
∂x4 +Jθ(x,{h(x, t)}) +η(x, t). (2.2) In this equation,J is the flux of arriving atoms,θ(x,{h(x, t)}) the exposure angle, andη(x, t) is the combined effect of shot noise and thermal noise of the substrate. The first term on the right-hand side is the divergence of the surface diffusion current withDproportional to the surface diffusion constant, or more preciselyD=DSσΩ2ρ/kBT, where DS is the surface diffusion constant, Ωthe atomic volume, ρ the number of surface atoms per area, andσ the surface energy per area. Furthermore, it is assumed that J is caused by the gradient of the local chemical potential. In principle, Eq. 2.2 could also be a model for sputter cleaning of rough surfaces by choosing a negative J.
The shadowing itself is hidden in θ(x,{h(x, t)}) as it presents the local flux being depen- dent on surface roughness of nearby regions, which is the key point of the shadowing growth equation. Thus, if one starts with a small periodically modulated surface with roughness h(x,0) =h0kcos(kx), then the exposure angle is θ ∝ kh0kcos(kx) neglecting higher harmon- ics. As a result, the perturbation grows exponentially hk(t) =h0keωkt with a rate constant
ωk ∝Dk4. (2.3)
Thus, short-distance details of the surface erode quickly, whereas the amplitude of long- wavelength mode grows. The mode with the highest growth rate has a wave vector k = (αJ/4D)1/3 and a growth rate ω = 34αJ(αJ/4D)1/3, where α is a constant being approxi- mately 0.5 [KBR89].
term D|q|4 for large values of q; in fact, as q→ ∞, Eq. 2.4 can be simplified to
qlim→∞ <|h(q, t)|2 > ∝ J
D|q|4. (2.5)
Thus, we should observe a slope of −4 for a large q if we plot the reciprocal-space height correlation function < |h(q, t)|2 > on a log-log scale for our data obtained by means of tunneling microscopy – providing surface diffusion and shadowing are the most important processes in the evolution of the film morphology. The calculation of the reciprocal-space height correlation function of STM images is described in section 5.2. However, in order to compare our results with the literature, we calculated the power spectral density which can be derived from the reciprocal-space correlation function as
S(q, t) = |h(q, t)|2 Nf
, (2.6)
whereNf is the total number of pixels per image [OAS+99].
2.3.2. Diffusion length
According to Oliva et al. [OAS+99], the surface diffusion length ld can be estimated from the double logarithmic power spectral density as
ld= 2π qd
, (2.7)
where qd is the value of q for which the −4 slope levels off into a somewhat smaller slope.
The parameterlddescribes the average range that a deposited atom/molecule is able to move before it looses energy to the surface due to phonon emission and the likelihood of finding an adhesion site increases drastically. This method is used to obtain the surface diffusion length for both CuCl [TSW91] and gold films [OAS+99] and we try to apply this technique to the cesium morphology presented within this work (see section 5.2) – if possible.
2.4. Kardar-Parisi-Zhang equation
Besides the shadow growth theory, many further continuum growth equations exist. One of the most studied equations is the one by Kardar-Parisi-Zhang [KPZ86], which turned out to
to the surface tension of the substrate-film interface. The second term is the lowest order nonlinear term that can appear in the interface growth equation. The function η(x, t) is the random noise associated with the deposition and incorporation of atoms on the surface.
Equation 2.8 was analysed by Edwards and Wilkinson [EW82] for λ= 0, and is therefore often referred to as the Edwards-Wilkinson-Equation. Later on, it was demonstrated that the non-linear term is not only necessary but even responsible for most unusual interface growth behaviours [KPZ86].
2.4.1. Universality classes
Different kinds of growth processes are represented by different growth equations, which are usually characterised by their universality class. A whole set of critical exponents is associated with the universality class: the roughness exponent α, the growth exponent β, and the dynamic exponent z ≡α/β, describing the scaling properties of a surface.
For the KPZ-equation 2.8, these critical exponents are α= 0.385, β= 0.24, and z = 1.58 [BRS+02]. It is very important to notice that the power spectral density function (PSD) of a surface (see Eq. 2.6) is correlated to the critical exponent α as for a infinite growth time
PSD(t→ ∞)∝q−2(1+α) ∝q−2.77 . (2.9)
For a finite growth time, this asymptotic behaviour can be observed only for q larger than some cut-off value qc ∝t−1/z [BRS+02].
2.4.2. First simulations
Very recently (2006), surface simulations using the KPZ-equation were performed by A.
Ballestad et al. [BRS+02] [BLST06] for the time evolvement of a GaAs surface grown by means of molecular beam epitaxy (MBE) techniques. It was shown that a single set of parameters successfully describes the growth on surfaces with different initial roughness for a given growth condition, demonstrating the applicability of the KPZ equation (see Fig.
2.3). Regardless of initial conditions, the surface tends toward an equilibrium roughness level determined by the interplay between smoothing rate and random noise in the system, as predicted by the KPZ equation for long growth times. Furthermore, it is demonstrated that an increase of the temperature leads to smoother surfaces, implying increased values for the smoothing terms in Eq. 2.8. This is in agreement with investigations of cesium growth at different temperatures as performed by A. Fubel [Fub]. It is worth noting that the PSD generated from the simulated surfaces is in very good agreement with the PSD from AFM images as displayed in Fig. 2.4.
equations generate inversion symmetric surfaces,1– 4whereas with our inclusion of the nonlinear terms both equations cause the etch pits on the starting surface to develop into mounds separated by V grooves, in agreement with the ex- perimental surface morphology. However, while the surfaces generated by the KPZ equation have similar roughness to the real surfaces at all length scales, the MBE equation generates less smoothing of the largest features than is seen in the experiment. Modifying the parameters of the MBE equation to enhance smoothing of the deepest pits causes the smaller mounds to be less prominent than those on the real surface.
Figure 5!a"shows the PSDs of a thermally desorbed sub- strate!sampleT0), a sample grown for 10 min!sampleT1), and a sample grown for 150 min!sampleT3), measured along the#11¯0$direction. Figure 5!b"shows the PSD from the same surfaces measured along the#110$direction. PSDs FIG. 2. 10!10%m2AFM images from samples grown under
nominally identical conditions, but for different times, on thermally desorbed substrates.!a"10 min growth!sampleT1).!b"37.5 min growth!sampleT2).!c"150 min growth!sampleT3). The scale bars are in nm, and the arrows point along the#11¯0$direction.
FIG. 3. 10!10%m2simulations generated using the conserva- tive form of the KPZ equation with growth times of!a"10 min and
!b"37.5 min. The scale bars are in nm, and the arrows point along
the#11¯0$direction. The simulations compare well with the real surfaces shown in Fig. 2.
205302-5
Figure 2.3.: 10×10µm2 AFM images (left) and simulations (right) [BRS+02] of a GaAs film.
The growth time for both a) images was 10 min, for both b) images 37.5 min.
All scale bars are in nm and the arrows point along the 110 direction.
calculated from AFM scans ranging in size from 1
!1 !m2 to 100!100 !m2 were combined to generate these figures. The PSD shrinks rapidly as a function of time during growth, until it reaches a saturated level for spatial frequencies greater than a crossover frequencyqc "indicated by the vertical dashed lines#. The crossover frequency de- creases monotonically as a function of growth time, going from around 36 !m"1 to around 9 !m"1 ($11¯0% direc- tion#, or from around 60 !m"1to around 17 !m"1($110%
direction# after 10 and 150 min growth, respectively. In the saturated region (q#qc) the PSD is well described by a power law, with slope close to"2, and a magnitude which is the same for all samples grown under similar conditions. The
quencies during growth as the average size and spacing of the mounds increases.
The solid lines on Fig. 5 show the PSD’s obtained from simulations using the conservative form of the KPZ equa- tion. Figure 6 shows the PSD of the same samples shown in Fig. 5, but in this case the solid lines are PSDs obtained from simulations using the MBE equation. The simulation times match the experimental data, and the parameters are the same as those used to generate the images shown in Figs. 3 and 4.
As above, the PSD’s are generated from a combination of simulated surfaces of different sizes. In the saturated region q#qc the magnitude of the PSD is determined by both the coefficients in the growth equation and the strength of the noise included in the simulations. The values of the param- eters &n "KPZ equation# and &c "MBE equation# quoted above were determined by matching the simulated and mea- sured PSD’s in the saturated region at high spatial frequen- cies.
The KPZ simulations in Fig. 5 are in excellent agreement with the experimental data over the entire q range. For q
#qc the MBE equation with conservative noise predicts the
"Fig. 6#. However, unlike FIG. 4. 10!10 !m2 simulations generated using the MBE
equation with growth times of"a# 10 min and "b#37.5 min. The scale bars are in nm, and the arrows point along the$11¯0% direc- tion. The fourth order equation also produces a mounded surface, although the depth of the cusps between the mounds is greater than on the real surfaces.
FIG. 5. PSD after different growth times on thermally cleaned substrates, measured along "a# the $11¯0% direction and "b# the
$110% direction. The symbols represent the experimental data. The
solid lines, representing 10 and 150 min simulations generated with the KPZ equation, are in excellent agreement with the experimental data. The vertical dashed lines indicate the cutoff frequenciesqc at 10 and 150 min"right to left#.
A. BALLESTADet al. PHYSICAL REVIEW B65205302
Figure 2.4.: PSD after different growth times on thermally cleaned substrates measured along the 110 direction. Symbols represent experimental data, solid lines simulated data generated with the KPZ equation [BRS+02].
24
2. Ability to create pinning centres by means of either STM tip contact or by manipulating the Cs surface using a high tunneling current (see e.g. [Zec04])
3. Ability of measuring helium contact angles and their hysteresis
4. Investigation of wetting properties at such low temperatures, where the gaseous phase can be neglected
5. Possibility to move the STM tip to an arbitrary location on the whole substrate area (≈10×10 mm).
3.1. Constructing the STM
3.1.1. Coarse approach mechanism
Even though many publications exist reporting on low temperature STM (e.g. [MCP01], [PKH+00], [RNKF90a], [PSS+01]), it is still a demanding task to construct an STM in a prosperous manner. Including abilities to quench condense cesium, measure helium con- tact angle and trying to minimise vibrations (e.g. due to pumps and boiling cryogenic liq- uids) is even more challenging. One of the most crucial issues, however, is designing a low-temperature reliable coarse approach mechanism, moving the tip from a macroscopic distance between substrate and tip to a microscopic (tunneling) one. There are many differ- ent techniques published, ranging from slip-stick type ([Poh87], [RNKF90b], [WMCE98]), over an inch-worm-technique [PSS+01] to the Besocke-style approach ([Bes87], [GCK96]) and the ’walker’ coarse approach [GN01].
All of those principles are based somehow on friction forces between a piezo-drive and the part to be driven, but not all of them are based on inertia. For example, the walker type approach functions by applying a voltage step to six independent piezos, one by one, moving the piezos in the same direction. Following this, the voltage on each piezo is brought slowly down to zero at the same time for all piezos, resulting in a movement of the STM tip or whatever is meant to be moved (see Fig. 3.2).
In contrast to the walker principle, a slip stick approach mechanism is based on a delicate interplay between static and dynamic friction forces on the one hand and inertial forces of
Figure 3.1.: STM setup designed within the work of this PhD thesis showing all parts before being gold plated (upper left), the slip-stick table in which temperature sensor and heater are already integrated (lower left), and the finished setup as it was used in the experiments (right).
the sliding parts on the other hand. The system normally consists of two parts: (1) the slider which is typically made out of brass or bronze, and (2) the sapphire rods on which the slider moves. The sapphire rods are glued onto the piezo tube, providing both scanning motion and coarse approach. If a trapezoidal voltage is applied to the piezo, the slider rides on the sapphire rods as long as the inertial forces of the slider are well below the acceleration forces due to the expansion of the piezo. This state is reached by increasing the piezo voltage quite slowly until the piezo has reached its maximum elongation. At this point, the voltage is switched to zero very rapidly, causing the slider to stay at its position. Repeating those steps at a typical frequency of about 1 kHz leads to an approach speed of approximately 0.1 mm/sec. As it turned out, the system works best when applying a cycloidal shape rather than a trapezoidal [MKLM96].
This principle has the advantage of being extremely lightweight and thus being hardly effected by external vibrations (as ωres ∝&
D/m for which ωres is the resonance frequency, Da spring constant, andmthe mass of the slider), but it is extraordinary sensitive to changes in the shape of the slider (e.g. due to abrasion) and the sapphire rods (e.g. condensation of adsorbates). This sensitivity becomes obvious, if one has a closer look at the condition for the onset of sliding:
a >(µstatFspring/m)±g , (3.1)
where a is the acceleration, µstat the coefficient of static friction, Fspring the normal force between track and tip holder created by a spring, and m the mass of the slider. The
Figure 3.2.: Schematic sketch of an approach mechanism based on the ’walker’ principle. The outer shell is made from Macor, a beryllium-copper spring allows fine tuning of the friction forces between sapphire (grey) and the alumina (green). The driving force is generated by a piezotube (blue) cut into six pieces. The piezo, providing scanning motion for the STM itself, is glued into the walker piezo drive (not shown) [SCZC06].
parametergis the acceleration due to gravity. Since the mass of the slider is fixed, essentially three parameters govern the reliability of the coarse approach system: the spring-forceFspring, the static friction µstat, and the acceleration a. If the correct parameters for Fspring and a are obtained by changing the spring properties and testing different frequencies of the cycloids, the slider works well even when temperature is varied. Unfortunately,µstat changes drastically due to different facts, mainly abrasion of the slider and change of its surface properties during the cesium evaporation.
Although the slip-stick system has some disadvantages, they are levelled out by it’s re- markable insensibility to external vibrations. This fact is supported by discussions with other groups ([Deb], [DL], [Hei]) and led to the decision to use the slip-stick principle as coarse approach system for the STM.
3.1.2. STM design
The construction of the STM was performed using modern 3D-development tools1. Most of the parts were made out of oxygen free copper, a few others by Macor ceramics and stainless steel. To increase the systems impassivity towards mechanical resonances, the STM was suspended on three springs. As the total weight of the STM could be calculated with the construction software, the spring constants of the three springs could be selected in such a way that the resonance frequency of the total system was about 10 Hz. To achieve a good vacuum, all parts were gold-plated after fabrication.
1Autodesk Mechanical Desktop 2004
Figure 3.3.: Microscopical image of the slider used in the low temperature STM setup. Each part of the slider has got three legs in order to avoid tilting of the slider which turned out to work reasonable well. The STM tip is clamped in a small Cu capillary (outer diameter 1 mm), which is bonded onto the slider. The electrical connection from the STM tip to the shielded coaxial cable is made by a 40µm diameter copper wire (lower right).
3.1.3. Optical access
As the STM setup was not only intended for surface characterisation but also for the inves- tigation of wetting properties, optical access was provided by four indium sealed windows allowing investigations of adsorbed films using surface plasmon spectroscopy (SPS), surface plasmon microscopy (SPM), and photo electron tunneling (PET). In order to precisely de- termine the helium level inside the STM cell, a stainless steel capacitor was added as well.
Its resolution was well below .1 mm (depending on capacitance meter).
Furthermore, there were mirrors mounted inside the cell offering the possibility to directly monitor droplets and their behaviour on the cesium surface. For a good image quality, PCO’s pixelfly digital camera was used in combination with a long focal microscope objective2. The total resolution of the system was.81µm per pixel at a focal distance of 15 cm and maximum magnification.
3.1.4. Piezotube
The main part of the STM, the piezoscanner, was made from industry type PZT8 3, with the following dimensions for length × outer diameter × inner diameter: 31×6.35×5 mm.
Using Eq. 1.14 andd31= 0.95 as strain coefficient, the maximum scanning range turned out to be 7.31µm for±150 V scanning voltage atT = 300 K. Assuming a reduction to 1/3 of its room temperature value ford31at liquid helium temperature, the scanning range of 7.31µm could still be received by simply switching the high voltage amplifier to±450 V (see Section 3.2).
One of the main advantages of PZT8 is its high depoling field which is about 1.5 MV/m.
Since the wall thickness of the scanning tube is .675 mm, the maximum applicable voltage
2Karlheinz Hinze Optoengineering GmbH&Co, LVM1200
3provided by www.eblproducts.com as EBL#4
Figure 3.4.: Schematic drawing of the STM as it looks like in the 3D-development tool. The STM is suspended on three springs leading to a resonance frequency of about 10 Hz. Materials: brown – gold plated oxygen-free copper; red – stainless steel;
turquoise & white – macor, and dark blue – sapphire.
before depoling is 952 V. Thus, the high voltage amplifier could be switched to ±450 V even at room temperature, leading to a scan range of about 21.9µm (even in the worst case, i.e. z-Voltage =−450V and outer electrodes at 450 V, the electric field was well below the critical field). The piezoscanner was made out of two piezotubes, the inner one provided scanning motion and coarse approach whereas the outer one was used for passive temperature compensation only (see Fig. 3.6). In principle, the outer scanner could be used to increase the scan range or to perform the coarse approach in addition.
There were two sapphire rods4 bonded onto the piezotube on which the slider moved, see also Section 3.1.1.
3.1.5. Slip-stick table
As different locations on the cesium substrate were investigated, a slip stick table was used instead of a normal sample holder, see Fig. 3.4. The slip-stick table works similar to the coarse approach of the slider, but instead of moving in only one direction, it can move both in
4Saphirwerk Br¨ugg, CH-2555 Br¨ugg
Figure 3.5.: Different optical applications are included in the STM setup: the lower part shows the plasmon spectroscopy/microscopy setup, where ’B’ denotes bimorph,
’Li’ lenses, ’IF’ interference filter, ’PMT’ a photomultiplier tube, and ’LD’ a laser diode. The right part ’D’ shows the illumination for droplet imaging using the LFO (’long focal objective’) and the VC (’video camera’).
xandy-direction. The maximum displacement was about 12 mm for each axis, the maximum moving speed was≈3 mm/s at room temperature and slightly below 1 mm/s at liquid helium temperature. The substrate holder (or prism if plasmon spectroscopy measurements are performed) was mounted on top of the slip-stick table including a DT-470 silicon temperature diode and a resistivity heater allowing precise temperature control of the substrate (∆T ≈ 1 mK). The slip-stick table itself was made out of six 10×10×1 mm PZT8 shear piezos, each three of them providing movement in one direction. In order to minimise abrasion, the contact surface between the piezos and the table is made out of silicon nitride (Si3N4) on one side and from V2A stainless steel on the other side.
3.1.6. STM tip
The tunneling tip used in the experiments presented here was made from a .25 mm 90/10 Pt/Ir-wire. Even though etching has been tested both on tungsten [KKN98] and on Pt [LHG95] [UIT96], it has turned out in the experiments that cutting the Pt/Ir wire is the
Figure 3.6.: Schematic drawing of the piezoscanner, consisting of two PZT8-tubes. The inner one (orange) provides scanning motion, the outer one (dark grey) is for passive temperature compensation. As PZT8 and Macor nearly have the same thermal expansion coefficient (αT ≈ 7×10−6 11
K
2) from room to liquid helium temper- ature [PGO], the two piezotubes are bonded together using a low temperature epoxy and Macor ceramics as spacer (light grey).
easiest, most reliable way to produce clean STM tips. However, for structures with sharp edges and high slopes such as e.g. nanotriangles, the cut Pt/It tips lack a high aspect ratio when compared to an etched tungsten tip yielding to image artefacts.
18.5 µm 695 nm 250 µm 100 nm
a) b)
Figure 3.7.: Electro-chemically etched Pt/Ir tip (a) vs. a typically Pt/Ir cutted tip (b). Etch- ing Pt/Ir is much more difficult than etching tungsten, but it has the advantage of failing to form oxygen compounds. Although having a high aspect ratio, the STM tip shown here is useless for scanning tunneling microscopy purposes as it has got a tip radius of ≈ 240 nm. Controversially, the cut STM tip has a tip radius of only ≈30 nm.
3.1.7. Cesium source
A dispenser5was used as cesium source. It consists of a small stainless steel housing including cesiumchromate Cs2CrO4 and the getter ’St101’ (84% Zr, 16% Al). The stainless steel housing can be heated up by simply applying a current (typically 7.5 A), leading to a redox reaction
Cs2CrO4+ Zr→ZrCrO4+ 2Cs (3.2)
5SAES Getters, 50937 K¨oln
