A review of perturbation theory

Most quantum mechanics problems are not solvable in closed form with analytical techniques. To extend our repertoire beyond just particle-in-a-box, a number of approximation techniques have been developed. A large class of these fall under the heading of “perturbation theory”, in which we consider our system to obey Hamiltonian H that may be written as

H = H0 + λH1 + λ2H2 + … , (A.1)

where H0 is an exactly solvable Hamiltonian, λ is a small parameter, and the other terms may therefore be taken as small corrections.

These notes are a quick review of how to deal with systems that obey such Hamiltonians. We'll be using Dirac notation and the Schrödinger formalism, in which the states are time-dependent. We begin with time-independent perturbation theory, and will then move on to consider time-dependent problems. For now, we'll only deal with single-particle problems, too.

Time-independent

We're going to use the time-independent Schrödinger equation, and look for the energy eigenvalues and eigenstates of H. First, suppose that our unperturbed problem,

H0|ψ0〉 = E0|ψ0〉 (A.2)

may be solved exactly, giving an energy eigenvalue spectrum E0j with corresponding eigenstates |ψ0j〉. Remember, because H0 is a Hermitian operator, its eigenvalues are real, and its eigenvectors |ψ0j〉 form a complete set.

The strategy we're going to take is straightforward. We're going to assume that our perturbative corrections to the Hamiltonian, λH1 + λ2H2 + … , lead to corresponding perturbative corrections to the eigenvalues and eigenstates. That is,

Ej = E0j + λE1j + λ2E2j + … ,

|ψj〉 = |ψ0j〉 + λ|ψ1j〉 + λ2|ψ2j〉 + ….

In our discussion of the electronic properties of bulk and nanoscale materials, we have so far concentrated on the charge carried by the electrons. However, the spin degree of freedom is also tremendously important, from the standpoints of fundamental science and technological impact. The collective response of electronic spins (in concert with their orbital motion) gives rise to the magnetic properties of materials. Magnetism's technological impact cannot be overstated, from the first primitive compasses thousands of years ago to advanced magnetic data storage and magnetic imaging techniques today.

In this chapter we introduce magnetism and magnetic materials, beginning with bulk systems. After discussing how to characterize magnetic response down to very small scales, we look at the effects of surfaces and interfaces, including confinement down to the nanoscale. After reviewing the history and evolution of magnetic data storage and magneto electronic devices, we consider nanoscale magnetism and its potential impact.

Definitions and units

One unfortunate and unavoidable challenge of discussing magnetism in materials is the unwieldy nature of magnetic units. This is largely the result of the historical development of our understanding of electricity, magnetism, and their relationship.

Let's begin by introducing the different types of vector field relevant to magnetism. The truly fundamental field is B, the magnetic induction or magnetic flux density (though, truth be told, most physicists would call this the magnetic field, as was done in the previous chapter). This is the field that shows up in the Lorentz force law and determines the force on a moving charge. It is also the field that determines magnetic resonance frequencies, and is related to the vector potential by ∇ × A = B. The SI unit of B is the Tesla. The Earth's magnetic field is about 3 × 10− 5 T over most of the planet.

One of the greatest advantages of the AFM, compared to other imaging techniques, is the possibility of modifying the morphology of a surface while scanning it. This possibility is well exemplified by the number of nanolithography and nanomanip-ulation experiments reported in the literature. However, to use these techniques for practical applications, the motion of the probing tip must be controlled in order to pattern the surfaces in a desired way or to rearrange the manipulated objects in a desired configuration. In the case of nanomanipulation, assembling nanoparticles in a well-defined arrangement is usually a difficult and time-consuming task. Fric- tion and adhesion forces between particles and substrates indeed play a major role in the motion on the nanoscale and, even if one works in a controlled environment, the size of the nanoparticles is usually comparable to that of the tip apex, which makes any attempt of controlling the manipulation process quite challenging.

Contact mode manipulation

One of the first examples of AFM manipulation was reported by Lüthi et al. [199], who succeded in moving compact C60 islands on a NaCl(001) surface by pushing them with the probing tip (Fig. 19.1). From the area of the islands determined by the AFM topographies and the values of the kinetic friction force recorded while sliding, a shear stress between C60 and NaCl of the order of 0.1 MPa was estimated. A larger shear stress, caused by the static friction and more difficult to quantify, accompanied the onset of motion. In another experiment, Sheehan and Lieber recognized the importance of the misfit angle in the manipulation of MoO3 islands on an MoS2 surface [307]. In this case, the islands could only move along low index directions of the substrate. Another experiment on Sb islands manipulated on the same substrate is presented in Section 19.3. Note that, instead of pushing a nanoisland, the tip can also be positioned on top of it.

Similarly to standard AFM (Fig. 17.1), friction force microscopy (FFM) is based on the relative motion of a sharp tip on a solid surface. This motion is realized by a scanner formed by piezoelectric elements, which moves the surface perpendicularly to the tip with a certain time periodicity. The scanner can be also extended or retracted in order to vary the normal force FN between tip and surface. This force is responsible for the deflection of the microcantilever supporting the tip. If FN increases while scanning due to local variations of the surface height, the scanner is retracted by a feedback loop. If FN decreases, the surface is brought closer to the tip by extending the scanner. In such a way, the surface topography can be reconstructed line by line from the vertical deformation of the scanner. An accurate control of the imaging process is made possible by a light beam, which is reflected from the rear of the cantilever into a photodetector. When the bending of the cantilever changes, the light spot on the detector moves up or down. This causes a variation of the photocurrent corresponding to the value of FN to be controlled.

The scan motion is also accompanied by friction. A tangential force F with the opposite direction to the scan velocity v hinders the sliding motion. The force F causes the torsion of the cantilever, and can be recorded simultaneously with the topography if the photodetector can measure not only the normal deflection but also the torsion of the lever while scanning. In practice this is made possible by a four-quadrants photodetector, which converts the photocurrent corresponding to the lateral force into a voltage VL. Note that the friction also causes lateral bending of the cantilever, but this effect is modest if the thickness of the lever is much less than its width.

The first atomic friction measurements by Mate et al. [206] were actually based on the deflection of a tungsten wire.

The study of friction, wear and lubrication between two surfaces in relative motion is called tribology. This term is derived from the Greek verb ‘tribos’, which means ‘to rub’. On one hand tribology aims at a scientific foundation of these phenomena. On the other hand it aims at a better design, manufacture and maintenance of devices which are affected by these ‘annoyances’. Tribology has a very important economical outcome. According to one of the first reports on this issue, tribological problems accounted for 6% of the Gross Domestic Product in industrialized countries in the 1960s [160]. This percentage may have increased by now. Tri-bological problems are found in pinions, pulleys, rollers and continuous tracks, in pin joints and electric connectors, and may cause more failure than fracture, fatigue and plastic deformation. On the other hand, friction is highly desirable, or even essential, in power transmission systems like belt drives, automobile brakes and clutches. Friction can also reduce road slipperiness and increase rail adhesion. Before starting our rather theoretical description of tribology, it is important to recall the milestones that have marked the progress in this subject from the dawn of civilization.

Historical notes

More than 40 000 years ago a complex process such as the generation of frictional heat from the lighting of fire was already well known. Nowadays the same process is studied by a branch of tribology, which is known as ‘tribochemistry’ and is focusing, more generally, on friction-induced chemical reactions. The early use of surface lubricants to reduce friction is unambiguously proven by a famous painting from ancient Egypt, in which a ‘prototribologist’ supports the work of a few dozen slaves by pouring oil in front of the heavy sled that they are pulling (Fig. 1.1). More than four thousand years later Leonardo da Vinci (1452–1519) started a systematic investigation of tribology, as documented by his drawings (Fig. 1.2). Leonardo's intuition and perseverance resulted in the formulation of the first friction law, which states the proportionality between friction and normal force.

Friction permeates every aspect of our life. It accompanies us when we walk and our fingers when they slide on the display of a tablet. Friction produces very annoying results when a chalk is rubbed against a blackboard and may cause tremendous damage when it fails to hold two tectonic plates together and a powerful earthquake is suddenly generated. Friction can also be very useful, when a cat suddenly jumps in front of our car and the brake pedal avoids serious consequences; and even pleasant, when a talented violinist takes up a bow and starts playing his Stradivarius. In any case, friction is certainly not a boring subject, and writing a book about friction is definitely not an easy task.

In spite of an immense amount of experimental data, a general theory of sliding friction between two solid surfaces is still missing. The simple Amontons' law, stating that the friction is proportional to the normal force, has been found to work exceptionally well in a variety of situations. Based on this law, theoretical models with different degrees of complexity have been derived and successfully applied to reproduce real situations. Even if Amontons' law is universally accepted as empirical evidence rather than as a consequence of first principles, the attitude is rapidly changing and it is now possible to prove by analytical means that the friction between two rough elastic surfaces has to be almost proportional to the loading force. A different situation is encountered when studying the drag force accompanying the motion of a solid object in a viscous liquid. Here, the Navier–Stokes law works usually quite well, which made hydrodynamic lubrication an established subject a long time ago. Still, problems arise when the lubricants are confined and the friction can only be investigated, theoretically, using atomic-scale models.

In the past 25 years, significant progress has been achieved in the understanding of the basic principles of sliding friction.