ADVANCED TOPICS IN PHYSICS
ADVANCED TOPICS IN PHYSICS PHYS 600
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Fabrication of nanoscale structures August 9 2001 Now that we ve surveyed some of the physics ideas we ll need and some methods of seeing what s going on an nanometer scales it s time to discuss the ways in which nanoscale structures are made We can divide fabrication techniques into two broad categories top down and bottom up Since nanoscale top down techniques are mostly extensions of microfabrication methods we ll start by reviewing approaches to small scale assembly at the micron scale 1 Microfabrication photolithography The standard fabrication method used to make essentially all microelec tronic devices today is photolithography Actually the term photolithogra phy strictly refers to only the pattern de nition step in the process The silicon or other wafer on which devices are being created is referred to as the substrate 11 Pattern de nition Photolithography uses light activated chemistry to de ne the pattern that will be transferred to the substrate The keys to this are polymers called photoresz39sts Thousands of chemical engineers have labored thousands of hours to produce many varieties of resist each with subtly different proper ties The general idea is the following start with a clean substrate and place it on a vacuum chuck in a spinner Squirt a healthy puddle of resist onto the center of the substrate and spin it at high speeds typically a few thousand rpm for something like 30 sec The resist spreads out across the substrate forming a nearly uniform layer usually a few hundred nm thick usually resists are carefully characterized by viscosity so that the curve of layer thickness vs spin rate is known After spinning is complete the resist is baked at a carefully controlled temperature 120 C for a prescribed time 30 sec to drive off the carrier solvent leaving a hard layer of polymer The polymer chains have been carefully designed so that exposure to ultraviolet light produces chemical changes in the resist For positive tone77 resists exposure breaks down the polymer chains into shorter units that are soluble in some kind of developer Exposed resist will be dissolved and washed away in the development process leaving holes in the resist layer wherever exposure has occurred In negative tone77 resists UV light causes cross linking of the polymer chains and unexposed resist is washed away In this cass exposed areas are all that remain covered with resist after development Some resists can reverse their tone depending on baking temperatures and times and post exposure baking The desired pattern is originally de ned on a mask often especially in industry at a size substantially larger than the desired nal pattern The patterned mask is opaque to UV light in the shape of the pattern UV light is projected through this mask and reducing optics are used to project the image of the mask onto the resist layer This exposure step de nes the limits of conventional photolithography because of the diffraction limit of the projection optics Working at shorter wavelengths than the near UV is very dif cult because conventional glasses absorb rather than transmit at such scales After exposure and development a further step is frequently used called descumming The resist that gets washed away in the development process often leaves behind a monolayer of organic residue that can interfere with subsequent steps Brief exposure to ozone or an oxygen plasma can remove this contamination 12 Pattern transfer Now we have a substrate coated with a resist stencil The stencil can be used to protect parts of the substrate during an additive step like a metal deposition Alternately the stencil can allow some etching procedure to reach the substrate in well de ned locations in a subtractive step Additive steps These are most commonly deposition of insulating or metal layers The deposited material coats the entire wafer resist and all The resist layer is then lifted o by dissolving the polymer in a solvent like acetone leaving the deposited material only where holes had existed in the resist layer Deposition is usually through evaporation or sputtering ln evaporation physical vapor deposition PVD the wafer is placed in vacuum facing a heated piece of source material The source is heated either by a tungsten or molybdenum lament or via an electron beam until its temperature is high enough that it has a nonnegligible vapor pressure Metals are more commonly evaporated than insulators The material vapor hits the wafer surface and sticks ln sputtering plasma enhanced PVD PEPVD an argon plasma is used to knock atoms off the source material rather than simple heating In both cases the thickness of deposited material is monitored through a quartz oscillator thickness monitor The addition of material to the surface of a quartz resonator shifts it frequency by an amount that can be calibrated against thickness Note that oxygen contamination during deposition must be minimized for materials like copper and aluminum A good rule of thumb is that a partial pressure of oxygen of 10 6 torr corresponds to one monolayer of oxygen hitting the surface per second Subtractiue steps There are two types of subtractive steps wet and dry etching ln wet etching the wafer is exposed to a some wet chemistry such as a buffered HF solution to dissolve SiOg which can only reach the substrate through the holes in the resist ln dry etching a plasma or ion beam is used to chemically in reactive ion etching RIE or kinetically remove exposed material Again at the conclusion of the subtractive step the resist layer is removed with a solvent 13 Concluding not es Other steps are important in commercial device fabrication ion implan tation annealing damascene polishing but won t be discussed here The amazing thing about this whole business is that there are no less than 37 separate lithography steps each followed by an additive or subtractive step involved in the production of a wafer full of Pentium III chips Since the smallest feature sizes are approaching 01 pm layers must be aligned with a precision approaching that level uniformly across a 1277 diameter Si wafer These requirements get more and more stringent with each succeeding gen eration of chips as features shrink and complexity grows Further in just a few years the fab industry will run into the end of the road for 193 nm UV light Something new must happen to work at progressively smaller lengthscales 2 Topdown nanofabrication methods The procedure described above is top down and many of its elements are relevant at sub 100 nm scales We discuss a few examples of top down nanofabrication methods below 21 Extensions of photolithography Photolithography can be used to produce features at sub 100 nm scales but the dif culty is high Work is being done on using dielectric mirrors for re ectively focussing extreme UV EUV light through masks Further the wave nature of light can be used in certain clever ways to produce interference effects as small as 4 Some chemically ampli ed77 resists require two photon events to be exposed resulting in patterns that vary like the light intensity squared rather than simply the intensity Most nanoscale work doesn t currently use these techniques however 22 Electronbeam lithography The most common nanopatterning method out there these days is electron beam lithography EBL This is a direct analogy with photolithography where electron beam exposure alters the chemistry of the resist instead of light exposure The most common positive tone EBL resist is poly methylmethacrylate abbreviated PMMA Most university based EBL sys tems are based on modi ed SEMs in which computers are used to steer the electron beam over the surface of the sample Commercial EBL systems do exist and frequently resemble TEMS in their beam energies and electron optics Limitations EBL can routinely produce lines as narrow as 20 30 nm in PMMA Smaller features are possible in certain circumstances Limiting factors include feature broadening from secondary electron yields proximity exposure nite monomer size and dif culties in developing such small features The biggest problem with EBL on the large scale is speed In pho tolithography a whole wafer can be exposed simultaneously while EBL requires the e beam to draw each feature one at a time It remains an ideal technique for producing small patterns on few mm sized substrates however that are most relevant in university based nanostructure work mm A meam ulmLim 1 Emmy xsamh mu Figure 2 A pattern drawn with SiOg in Si using a nanotube as a conducting AFM tip for local oxidation From Dai s group at Stanfor V 23 Scanned probe lithography SPL Another topidown fabrication approach is to use variations on the wonderful scanned probe microscopy techniques we talked about last time This has some of the same dif culties with speed as EBL but has produced some irnpress ve patternin V Plowing Some research has been done on using an AFM tip to lite erally plow a groove through either a Very thin resist layer or a special selfrassembled monolayer SAM on the substrate This can produce lines in these layers as narrow as 20730 nrn7 though layers this thin and trenches produced this way make lifto processing Very challenging Local Orzdatwn Another approach has been to use either a conductive AFM tip or an STM tip to do local electrochemistry on the substrate with out any resistv This has been used to reduce the conducting crossisection of existing thin metal layers through oxidation It has also been used to grow SiOg patterns on clean Si wafers as shown in Fig 239 mm 3 Puma mm bum chm MAIIan s m at Noammu Wm WM th Arm mum mam mam mm m Amum mm mm usually mums the up and m a mm m mam w u s m mm dwn mm m thesnbsuai as shmm 3 mmwmmwmumumammmumm s a Bottomrup approaches Wm M in mm a 3901de mama mamas m mam mm m MAME 2125 mm m butwmrn 39 mama mm whaemdwdswmwlhmscmlmdeAmn sym my WM swmmm m v39 J39YquotJ 739 fr 1 x A H J Jy I 1 A V x 39 w n Figure 5 The Delft group s work on deposition onto STM modi ed surfaces Co on Si A103GamAs GaAs J xar l 511 Donor layer Figure 6 Diagram of energy bands at a semiconductor heterojunction show ing the triangular potential well that results in a 2DEG at the interface 31 Crystal growth As we discussed previously interfaces between materials of different band gaps can result in charge transfer and band bending This can be controlled by crystal growers in certain systems so called lll V compound semiconduc tors in particular The crystal growth technique of choice in these systems is molecular beam epitaxy in which a UHV chamber is combined with substrate heating and exceedingly clean source materials in a very fancy evaporator With correct growth conditions in the right material system it is possible to grow crystals literally one atomic layer at a time One particular common structure is shown in Fig 6 Because of band bending at the GaAsAlongaoqu interface any electrons injected into the conduction band of the GaAs there sit in a triangular potential well In this system Si acts as a very shallow donor As the crystal is grown a dilute layer of Si atoms is placed a few hundred A on the AlGaAs side of the interface Some large fraction of the donated electrons the rest go to occupy surface states go to the interface where they become trapped in the triangular potential well and form a 2DEG The electron system is truly two dimensional less than a few nm in z extent and at such a low density that only one kz mode is occupied This structure is a modulation doped heterostructure and forms the backbone of an entire sub eld of condensed matter physics research and GaAs semiconductor electronics They also form the basis for much of ECE565 Quantum Semiconductor Devices Extensions of this technique can create truly 1d structures as well One can take one of these 2DEG structures cleave it inside the growth cham ber rotate it by 90 degrees and ouergrow onto the cleaved surface This cleaved edge overgrowth method has been used to produce a number of 1d quantum wires and has the advantage of employing atomically at and uniform boundaries unlike variations of lithographic approaches 32 Chemistry Synthetic chemistry is almost by de nition the science of producing struc tures on the sub 10 nm scale molecules The complication is that for most applications chemists are interested in producing very large numbers N 1022 of such molecules Physicists conversely would often like to ex amine much smaller quantities of such systems usually with some degree of isolation from one another Here we will just highlight a few particular systems that are made through chemical techniques and actively researched Nanocrystals Under certain circumstances chemical reactions can be used to produce large numbers of nanocrystals typically a few nm in size Such crystals particularly semiconducting ones can exhibit optical and electronic properties signi cantly different from the bulk due to quantum con nement Conducting polymers These long chain molecules such as polyacetylene can exhibit metallic conduction at room temperature In crystalline form the molecules align and are bonded to one another relatively weakly through the Van der Waals interaction forming a highly anisotropic system Fullerenes This family of carbon compounds has proven to be a remark ably fruitful source of research as you all are aware Buckyballs can be as small as a few A in diameter and can be produced in large quantities in a relatively pure form Carbon nanotubes can be single or multi walled can be metallic semiconducting or insulating depending on their structure and can have diameters as small as a few Awith lengths approaching the micron scale DNA DNA is not a very simple family of molecules but through the labors of biochemists and molecular biologists it is one of the most con trollable An enormous number of tools exist that allow one to buy DNA strands with base pair sequences literally made to order It has become a well studied molecule in physical nanosciences in part because of the ability of scientists to manipulate it A large number of researchers are looking to chemical synthesis methods and chemically synthesized materials as eventual replacements for the pro cesses and active elements in the microelectronics industry In principle the 10 ability to make trillions of complicated virtually awless sub 10 nm struc tures in parallel makes the idea of molecular electronics quite compelling As we shall see later however going from concept to practicality is extremely nontrivial 33 Selfassembly Closely related to the chemical synthesis techniques mentioned above is the idea of self assembly Under certain circumstances it is possible to take advantage of the same kinds of energetic and statistical forces that cause crystalline order in solids and to have the spontaneous formation of arrays of highly ordered nanostructures Here we list three examples Self assembled quantum dots Above we said that in MBE under the right growth conditions it is possible to assemble large single crystals one atomic layer at a time Other growth regimes are possible however particularly when two materials do not share exactly the same crystalline structure An example of this is the GaAslnAs system lnAs has a lattice constant somewhat smaller than GaAs resulting in a large surface energy cost if lnAs is forced to grow epitaxially on GaAs As a result if a small amount of lnAs is deposited on a GaAs surface it will spontaneously arrange itself into lnAs islands 1 3 nm in diameter and lt 1 nm high Further these islands can interact with each other elastically through strains in the GaAs and tend to arrange themselves into regular arrays These self assembled quantum dots can have remarkable optical and electrical properties and such systems are topics of current research Langmuir Blodgett lms Certain molecules have hydrophilic and hy drophobic portions For free energy reasons similar to those above when placed into water the molecules spontaneously arrange themselves into an ordered monolayer on the water s surface By carefully withdrawing a sub strate from such a volume of water the monolayer can be transferred intact onto the substrate Self assembled monolayers Other molecules can do the same sort of trick on different surfaces For example a family of relatively big molecules 1 2 nm called porphyrins will spontaneously form an ordered monolayer on Si if the substrate temperature is increased enough to enhance their surface mobility On Au surfaces alkylthiol molecules also tend to form ordered self assembled monolayers 4 Concluding remarks We ve now seen a number of methods for creeating structures on the sub 100 nm scale Now that we ve given an overview of the basics of charac terizing and fabricating nanoscale objects7 we will turn to the physics of such systems We ll use papers from the literature to examine a number of physical effects that arise in these systems A Far Too Brief Review of Solid State Physics August 97 2001 These rst lectures will be an attempt to summarize some of the key ideas in solid state physics These ideas form the intellectual foundation upon which most nanoscale physics has been based By summarize I mean just that It s extraordinarily dif cult to do a complete and modern treatment of these ideas even in a full two semester sequence therefore7 we ll be sacri cing depth and careful derivations as a trade off for breadth and developing a physical intuition We will supplement this introduction with additional discussions of par ticular subjects as they arise in our look through the literature At the end of these notes is a list of some references for those who want a more in depth treatment of these subjects It s pretty amazing that we can understand anything at all about the properties of condensed matter Consider a cubic centimeter of copper7 for example It contains roughly 1023 ion cores and over 1024 electrons7 all of which interact a prion39 not necessarily weakly through the long range Coulomb interaction However7 because of the power of statistical physics7 we can actually understand a tremendous amount about that copper s elec trical7 thermal7 and optical properties In fact7 because of some lucky breaks we can even get remarkably far by making some idealizations that at rst glance might seem almost unphysical In the rest of this section7 we ll start with a simple model system nonin teracting electrons at zero temperature in an in nitely high potential well Gradually we will relax our ideal conditions and approach a more realistic description of solids Along the way7 we ll hit on some important concepts and get a better idea of why condensed matter physics is tractable at all 1 Ideal Fermi Gas By starting with noninteracting electrons we re able to pick a model Hamil tonian for the single particle problem solve it and then pretend that the many particle solution is simply related to that solution Even without Coulomb interactions we need to remember that electrons are fermions Rigorously the many particle state would then be a totally antisymmetrized product of singleparticle states For our purposes however we can get the essential physics out by just thinking about lling up each singleparticle spatial state with one spin upT and one spin down L electron lntro quantum mechanics tells us that an eigenfunction with momentum p for a free particle is a plane wave with a wavevector k p h Consider an in nitely tall 1d potential well with a at bottom of length L the standard intro quantum mechanics problem the generalization to 2d and 3d is sim ple and we ll get the results below The eigenfunctions of the well have to be built out of planewaves and the boundary conditions are that the wave function iJm has to vanish at the edge of the well and outside the in nite potential step means we re allowed to relax the usual condition that 11 z has to be continuous The wavefunctions that satisfy these conditions are we sin nx 1 where n gt 0 is an integer So the allowed values of k are quantized due to the boundary conditions and the states are spaced in k by 7rL ln 3 dimensions we have states described by k km kg k1 where km 7rLnm etc Now each spatial state in k each takes up a volume of 7rL3 As usual the energy of an electron in such a state is z 2 2 2 Ekmkykz 2 An important feature here is that the larger the box the closer the spacing in energy of single particle states and vice versa Suppose we dump N of our ideal electrons into the box This system is called an ideal Fermi gas Now we ask lling the single particle states from the bottom up ie in the ground state what is the energy of the highest occupied single particle state We can count states in k space and for values of k that are large compared to the spacing we can do this counting using an integral rather than a sum Calling the highest occupied k value the Fermi waueuector hp and knowing that each spatial state can hold two electrons we can write N in terms of kp as L 3 kF 1 N 2 x lt7 74wk2dk 7r 0 8 1 3 3 kFL 3 If we de ne ngd E NV and the Fermi energy as 2 EF E E 4 2m7 we can manipulate Eq 3 to nd the density of states at the Fermi level 1 2mE 32 ME 2 w lt5 dn V3dEEF E digilEEp 1 2m 32 12 EF39 6 The density of states V3dE as de ned above is the number of single particle states available per unit volume per unit energy This is a very important quantity because as we will see later in Sect 24 the rates of many processes are proportional to lntuitively 1E represents the number of available states into which an electron can scatter from some initial state Looking at Eqs 66 we see that for 3d increasing the density of electrons increases both E1 and lgd Let s review We ve dumped N particles into a box with in nitely high walls and let them ll up the lowest possible states with the Pauli restriction that each spatial state can only hold two electrons of opposite spin We then gured out the energy of the highest occupied single particle state EF and through 1EF and the sample size we can say how far away in energy the nearest unoccupied spatial state is from EF Thinking semiclassically for a moment we can ask what is the speed of the electron in that highest occupied state The momentum of that state is called the Fermi momentum pp kp and so we can nd a speed by hkp up 7 m h 3 2 13 mgd M 7 m dim ndEp ldEF UF 3 mil2 milZEN W 2 E2 2W2 h F m 2 l mEp h 27rn2dl2 r E2 H m 1 2 ZmEp 12 r m 12 1 mid 7r 7L2 7r 7 El2 2m Table 1 Properties of ideal Fermi gases in various dimensionalities So7 the higher the electron density the faster the electrons are moving As we ll see later7 this semiclassical picture of electron motion can often be a useful way of thinking about conduction We can redo this analysis for 2d and 1d7 where ngd E NA and am E NL7 respectively The results are summarized in Table 1 Two remarks notice that V2dEp is independent of ngd further7 notice that Illd actually decreases with increasing am This latter property is just a restatement of something you already knew the states of an in nite 1d well get farther and farther apart in energy the higher you go The results in Table 1 are surprisingly more general than you might expect at rst One can redo the entire analysis starting with Eq 1 and use periodic boundary conditions iJz L 1ampz 0 0z1 m L iJ z When this is done carefully7 the results in Table 1 are reproduced exactly From now on7 we will generally talk about the periodic boundary condi tion case7 which allows both positive and negative values of hm7 hy7 and hz7 and has the spacing between states in h be 27rL In the 3d case7 this means the ground state of the ideal Fermi gas can be represented as a sphere of radius hp in h space7 with each state satisfying h lt hp being occupied by a spin up and a spin down electron States corresponding to h gt hp are un occupied The boundary between occupied and unoccupied states is called the Fermi surface Notice that the total momentum of the ideal Fermi gas is essentially zero the Fermi sphere diskline in 2d1d is centered on zero momentum If an Fermi sphere zero eld Fermi sphere a er eld Figure 1 Looking down the z aXis of the 3d Fermi sphere before and after the application of an electric eld in the m direction Because there were allowed k states available the Fermi sphere was able to shift its center to a nonzero value of km electric eld is applied in say the m direction because there are available states the Fermi sphere will shift as in Fig l The fact that the sphere re mains a sphere and the picture represents the case of an equilibrium current in the z direction is discussed in various solid state books One other point Extending the above discussion lets us introduce the familiar idea of a distribution function fT E the probability that a par ticular state with energy E is occupied by an electron in equilibrium at a particular temperature T For our electrons in a box system at absolute zero the ground state is as we ve been discussing with lled singleparticle states up to the Fermi energy and empty states above that Labeling spin up and spin down occupied states as distinct mathematically f07 E EF 7 E 8 where 9 is the Heaviside step function x 0 xlt0 1zgt0 9 Note that f is normalized so that A fTEzEdE N 10 Occupation probability Energy per Boltzmann const Ek x 104 K Figure 2 The Fermi distribution at various temperatures At nite temperature the situation is more complicated Some states with E lt E are empty and some states above E are occupied because thermal energy is available In general one really has to do the statisti cal mechanics problem of maximizing the entropy by distributing N indis tinguishable electrons among the available states at xed T This is dis cussed in detail in many stat mech books and corresponds to minimizing the Helmholtz free energy The answer for fermions is the Fermi distribution function 1 ME exprE a MTkBT 1 11 where p is the chemical potential The chemical potential takes on the value which satis es the constraint of Eq 10 At T 0 you can see that pT 0 EF See Fig 2 A physical interpretation of p is the average change in the free energy of a system caused by adding one more particle For a thermodynamic spin on u start by thinking about why temperature is a useful idea Consider two systems 1 and 2 These systems are in thermodynamic equilibrium if when they re allowed to exchange energy the entropy of the combined system is already maximized That is i 831 832 63 lt6E1gt6E1lt6E2gt6E2 K3721 3 2l6E1 0 12 implying l W 8E1 8E2 We de ne the two sides of this equation to be 1kBT1 and 1kBT2 and see that in thermodynamic equilibrium T1 T2 With further analysis one can show that when two systems of the same temperature are brought into contact on average there is no net ow of energy between the systems We can run through the same sort of analysis only instead of allowing the two systems to exchange energy such that the total energy is conserved we allow them to exchange energy and particles so that total energy and particle number are conserved Solving the analogy of Eq 12 we nd that equilibrium between the two systems implies both T1 T2 and p1 p2 Again with further analysis one can see that when two systems at the same T and u are brought into contact on average there is no net ow of energy or particles between the systems So in this section we ve learned a number of things 0 One useful model for electrons in solids is an ideal Fermi gas Starting from simple particle in a box considerations we can calculate proper ties of the ground state of this system We nd a Fermi sea with full single particle states up to some highest occupied level whose energy is EF We also calculate the spacing of states near this Fermi energy and the semiclassical speed of electrons in this highest state We introduce the idea of a distribution function for calculating nite temperature properties of the electron gas Finally we see the chemical potential which determines whether par ticles ow between two systems when they re brought into contact 2 How ideal are real Fermi gases Obviously we wouldn t spend time examining the ideal Fermi gas if it wasn t a useful tool It turns out that the concepts from the previous section generally persist even when complications like actual atomic structure and electron electron interactions are introduced 21 Band theory Clearly our in nite square well model of the potential seen by the electrons is an oversimpli cation When the underlying lattice structure of crystalline solids is actually included the electronic structure is a bit more complicated and is typically well described by band theory Start by thinking about two hydrogen atoms very far from one another Each atom has an occupied 13 orbital and a number of unoccupied higher orbitals p detc If the atoms are moved suf ciently close a spacing com parable to their radii a more useful set of energy eigenstates can be formed by looking at hybrid orbitals These are the a and 0 bonding and antibond ing orbitals When a perturbation theory calculation is done accounting for the Coulomb interaction between the electrons the result is that instead of two singleelectron states 13 of identical energy we get two states 047 that differ in energy Now think about combinations of many electron atoms lt s reasonable to think about an ion core containing the nucleus and electrons that are rmly stuck in localized states around that nucleus and ualence electrons which are more loosely bound to the ion core and can in principal overlap with their neighbors For small numbers of atoms one can consider the dif ferent types of bonding that can occur Which actually takes place depends on the details of the atoms in question 0 Van der Waals bonding This doesn t involve any signi cant change to the electronic wavefunctions atom A and atom B remain intact and interact via uctuating electric dipole forces Only included on this list for completeness lonic bonding For Coulomb energy reasons atom A donates a valence electron that is accepted by atom B Atom A is positively charged and its remaining electrons are tightly bound to the ion core atom B is negatively charged and all the electrons are tightly bound Atoms A and B stick to each other by electrostatics but the wavefunction overlap between their electrons is minimal Covalent bonding As in the hydrogen case described above it becomes more useful to describe the valence electrons in terms of molecular or bitals where to some degree the valence electrons are delocalized over more than one ion core In large molecules there tends to be clustering of energy levels with intervening gaps in energy containing no allowed states There is a highest occupied molecular orbital HOMO and a lowest unoccupied molecular orbital LUMO 0 Metallic bonding Like covalent bonding only more extreme the delo calized molecular orbitals extend over many atomic spacings Now let s really get serious and consider very large numbers of atoms arranged in a periodic array as in a crystal This arrangement has lots of interesting consequences Typically one thinks of the valence electrons as seeing a periodic potential due to the ion cores and to worry about bulk properties for now we ll ignore the edges of our crystal by imposing periodic boundary conditions When solving the Schrodinger equation for this situation the eigenfunctions are plane waves like our old free Fermi gas case multiplied by a function that s periodic with the same period as the lattice 1kr ukrexpikr ukr ukrrn 14 These wavefunctions are called Bloch waves and like the free Fermi gas wavefunctions are labeled with a wavevector k See Fig 3 The really grungy work is two fold nding out what the the function ukr looks like for a particular arrangement of particular ion cores and g uring out what the corresponding allowed energy eigenvalues are In practice this is done by a combination of approximations and numerical techniques It turns out that while getting the details of the energy spectrum right is extremely challenging there are certain general features that persist First not all values of the energy are allowed There are bands of energy for which Bloch wave solutions exist and between them are band gaps for which no Bloch wave solutions with real k are found Plotting energy vs k in the 1d general case typically looks like Fig 4 The details of the allowed energy bands and forbidden band gaps are set by the interaction of the electrons with the lattice potential In fact looking closely at Fig 4 we see that the gaps really open up77 for Bloch waves whose wavevectors are close to harmonics of the lattice potential The Coulomb interactions between the electrons only matter here in the indirect sense that the electrons screen the ion cores and self consistently contribute to the periodic potential Now we consider dropping in electrons and ask what the highest occupied singleparticle states are as we did in the free Fermi gas case The situation here isn t too different though the properties of the ground state will end up depending dramatically on the band structure Notice too that the Fermi sea is no longer necessarily spherical since the lattice potential felt by the electrons is not necessarily isotropic prtype Iallicepenodm lunchun uk m wavemncuon cos 1kx m L V7 m r 777 Rea pan of IV Bloch wave uk 0 cos kx a Dws39ance x Figure 3 The components of a Bloch wave7 and the resulting wavefunction From Ibach and Luth Allowed band For d Allowed ban d Fov TI 7 50 k l Brillouin zone Figure 4 Allowed energy vs wavevector in general 1d periodic potential from Ibach and Luth Figure 5 shows two possibilities In the rst the number of electrons is just enough to exactly ll the valence band Because there is an energy gap to the nearest allowed empty states this system is a band insulaton if an electric eld is applied the Fermi surface can t shift around as in Fig 1 because there aren t available states Therefore the system can t develop net current in response to the applied eld A good example of a band insulator is diamond which h as a gap of around 10 eV The second case represents a metal and because of the available states near the Fermi level it can support a current in the same way as the ideal Fermi gas in Fig 1 Other possibilities exist as shown in Fig 6 One can imagine a band insulator where the gap is quite small so small that at room temperature a detectable number of carriers can be promoted from the valence band into the conduction band Such a system is called an intrinsic semiconductor A good example is Si which has a gap of around 11 eV and a carrier density at room temperature of around 15 x 10 cm Further it is also possible to dope semiconductors by introducing ime purities into the lattice A donor such as phosphorus in silicon can add an electron to the conduction band At zero temperature this electron is boundto the P donor but the binding is weak enough to be broken at higher E E comma E E mm insulator metal Figure 5 Filling of allowed states in two different systems On the left7 the electrons just ll all the states in the valence loand7 so that the next unoccupied state is separated by a band gap this is an insulator On the right7 the electrons spill over into the conduction loand7 leaving the system metallic Q l conduction band m intrinsic nite T Iadoped nim T pdnped ail T Figure 6 More possibilities On the left is a band insulator7 as before Next is an intrinsic semiconductor7 followed by two doped semiconductors temperatures leading to usual electronic conduction This is the third case shown in Fig 6 Similarly an acceptor such as boron in silicon can grab an electron out of the valence band leaving behind a positively charged hole This hole acts like a carrier with charge 5 at zero temperature it is weakly bound to the B acceptor but at higher temperatures it can be freed leading to hole conduction One more exotic possibility not shown is semimetallz39c behavior as in bismuth Because of the funny shape of its Fermi surface parts of Bi s valence band can have holes at the same time that parts of Bi s conduction band can have electrons While the existence of Bloch waves means it is possible to label each eigenstate with a wavevector k that doesn t necessarily mean that the en ergy of that state depends quadratically on k as in Eq The approxi mation that Ek N k2 is called the e ectwe mass approximation As you might expect the effective mass is de ned by k k k3 2m 2m Ek h lt 15 where we re explicitly showing that the effective mass m isn t necessarily isotropic One nal point there is one more energy scale in the electronic structure of real materials that we explicitly ignored in our ideal Fermi gas model The work function Q is de ned as the energy difference between the vacuum a free electron outside the sample and the Fermi energy EV6VC 7 EF Our toy in nite square well model arti cially sets Q oo Unsurprisingly Q also depends strongly on the details of the material s structure and can vary from as low as 24 eV in Li to over 10 eV in band insulators The important ideas to take away from this subsection are c Bonding in small systems is crucially affected by electronic binding energies and Coulomb interactions Large systems typically have energy level distributions well described by bands and gaps Eigenstates in systems with periodicity are Bloch waves that can be labeled by a wavevector k 0 Whether a system is candnctinu in lila rinu or semiconductinu de pends critically on the details of its band structure including the num ber of available carriers Systems can exhibit either electronic or hole conduction depending on structure and the presence of impurities 13 Figure 7 Surface states on copper imaged by Crommie and Eigler at IBM With a scanning tunneling microscope o The energy needed to actually remove an electron from a material to the vacuum also re ects the structure of that material 22 Structural issues The previous section deliberatly neglected a number of what I ll call struc tural issues h These include nonidealities of structure such as boundaries impurities and other kinds of structural disorderi Further we ve treated the ion cores as providing a static backgroundd potential When in fact they can have important dynamics associated With t emi Let s deal With structural defects rsti We ll treat some speci c effects of these defects later on First let s ask What are the general consequences of not having an in nite perfect crystal lattice Blush waves in nite in extent are no lungex exact eigenstates 0f the system That s not necessarily a big deal lntuitively an isolated defect in the middle of a large crystal isn t going to profoundly alter the nature of the entire electronic structure In fact the idea that there are localized electronic states around the ion cores and can be delocalized extended states Which span many atomic spacings is still true even Without any lattice at all this happens in amorphous and liquid metals Special states can exist at free surfaces Unsurprisingly these are called surface states The most famous experimental demonstration of this is shown in Fig 7 Suppose the surface is the z 7 y plane Because of the binding energy of electrons in the material states exist which are pinned to the surface having small 2 extent and wavelike or localized depending in surface disorder character in z and y Surface states can have dramatic implications when a samples are very small so that the number of surface states is comparable to the number of bulk states and b the total number of carriers is very small as in some semiconductors so that an appreciable fraction of the carriers can end up in surface states rather than bulk states Interfaces between di erent materials can also produce dramatic ef fects The boundary between a material and vacuum is just the lim iting case of the interface between two materials with different work functions When joining two dissimilar materials together there are two conditions one has to keep in mind a The vacuum for materials A and B is the same though ltIgtA 7 113 in general That means that prior to contact the Fermi levels of A and B are usually different b Two systems that can exchange particles are only in equilibrium once their chemical potentials are equal That implies that when con tact is made between A and B charge will ow between the two to equalize their Fermi levels The effect of a and b is that near interfaces space charge layers can develop which bend the bands to equalize the Fermi levels across the junction The details of these space charge layers eg how thick are they and what is the charge density pro le depend on the avail ability of carriers is there doping and the dielectric functions of the two materials In general one has to solve Poisson s equation self consistently while considering the details of the band structure of the two materials A term related to all this is a Schottky barrier It is possible to have band parameters of two materials be such that the space charge layer which forms upon their contact can act like a substantial potential barrier to electronic transport from one material to the other For example pure Au on GaAs forms such a barrier These barriers have very nonlinear I 7 V characteristics in contrast to Ohmic contacts between materials an example would be In on GaAs A great deal of 15 semiconductor lore exists about what combinations of materials form Schottky barriers and what combinations form Ohmic contacts We also need to worry about the dynamics of the ion cores rather than necessarily treating them as a static background of positive charge The quantized vibrations of the lattice are known as phonons We won t go into a detailed treatment of phonons but rather will highlight some important terminology and properties A unit cellis the smallest grouping of atoms in a crystal that exhibits all the symmetries of the crystal and when replicated periodically reproduces the positions of all the atoms The vibrational modes of the lattice fall into two main categories dis tinguished by their dispersion curves wq where q is the wavenumber of the wave When q O we re talking about a motion such that the displace ments of the atoms in each unit cell is identical to those in any other unit cell Acoustic branches have w0 0 There are three acoustic branches two transverse and one longitudinal Optical branches have w0 7 O and there typically three optical branches too The term optical is historic in origin though optical modes of a particular q are typically of higher energy than acoustic modes with the same wavevector For our purposes it s usually going to be suf cient to think of phonons as representing a bath of excitations that can interact with the electrons ln tuitively the coupling between electrons and phonons comes about because the distortions of the lattice due to the phonons show up as slight variations in the lattice potential through which the electrons move Electron phonon scattering can be an important process in nanoscale systems as we shall see The Debye model of phonons does a nice job at describing the low energy behavior of acoustic modes which tend to dominate below room tempera ture The idea is to assume a simple dispersion relation w ULTq where v is either the longitudinal or transverse sound speed This relation is assumed to hold up to some high frequency short wavelength cutoff tap the Debye frequency set by the requirement that the total number of acoustic modes 3rNuC where r is the number of atoms per unit cell and NuC is the num ber of unit cells in the solid Without going into details here the main result of Debye theory for us is that at low temperatures T lt TD E wDkB the heat capacity of 3d phonons varies as T3 One more dynamical issue for extremely small structures like clusters of tens of atoms guring out the equilibrium positions of the atoms requires a self consistent calculation that also includes electronic structure That 16 is the scales of electronic Coulomb contributions and ionic displacement energies become comparable Electronic transitions can alter the equilibrium conformations of the atoms in such systems The important points of this subsection are 0 Special states can exist at surfaces and in small or low carrier density systems these states can be very important lnterfaces between different materials can be very complicated involv ing issues of charge transfer band bending and the possible formation of potential barriers to transport 0 The ion cores can have dynamical degrees of freedom which couple to the electrons and the energy content of those modes can be strongly temperature dependent o In very small systems it may be necessary to self consistently account for both the electronic and structural degrees of freedom because of strong couplings between the two 23 Interactions We have only been treating Coulomb interactions between the electrons indirectly so far Why have we been able to get away with this As we shall see one key is the fact that our electronic Bloch waves act so much like a cold T lt TF E EpkB Fermi gas Another relevant piece of the physics is the screening of point charges that can take place when the electrons are free to move eg particularly in metals Let s look at that second piece rst working in 3d for now Suppose there s some isolated perturbation 6Ur to the background electrical po tential seen by the electrons For small perturbations we can think of this as causing a change in the local electron density that we can nd using the density of states 6717 V3dEFlel6Ur 16 Now we can use Poisson s equation factor in the change in the potential caused by the response of the electron gas V26Ur 26717 V3dEFlel6Ur 17 In 3d the solution to this is of the form 6Ur N 1rexp7r where E 1rTF de ning the Thomas Fermi screening length 62 712 rTF ltV3dEFgt 18 60 If we plug in our results from Eqs 66 for the free Fermi gas in 3d we nd 716 TTF 05 lt1 19 00 where a0 is the Bohr radius Plugging in n N 1023 cm s typical for a metal like Cu or Ag we nd rTF N 1 A So the typical lengthscale for screening in a bulk metal can be of atomic dimensions which explains why the ion cores are so effectively screened Looking at Eq 18 it should be clear that screening is strongly affected by the density of states at the Fermi level This means screening in band insulators is extremely poor since 1 N 0 in the gap Further because of the changes in 1 with dimensionality screening in 2d and 1d systems is nontrivial Now let s think about electron electron scattering again We have to conserve energy of course and we have to conserve momentum though because ofthe presence of the lattice in certain circumstances we can dump momentum there The real crystal momentum77 conservation that must be obeyed is k1k2k3k4G where G is a reciprocal lattice vector For zero temperature there just aren t any open states the electrons can scatter into that satisfy the conservation conditions At nite temperature where the Fermi distribution smears out the Fermi surface slightly some scattering can now occur but a screening reduces the scattering cross section from the bare value and b the Pauli principle reduces it further by a factor of kBTEF2 This is why ignor ing electron electron scattering in equilibrium behavior of solids isn t a bad approximation We will come back to this subject soon though because sometimes this weak scattering process is the only game in town and can have profound implications for quantum effects The full treatment of electron electron interactions and their consequences is called Landau Fermi Liquid Theory We won t get into this in any signif icant detail except to state some of the main ideas In LFLT we consider starting from a noninteracting Fermi gas and adiabatically turn on electron electron interactions Landau and Luttinger argued that the ground state of 18 the noninteracting system smoothly evolves into the ground state of the in teracting system and that the excitations above the ground state also evolve smoothly The excitations of the noninteracting Fermi gas were electron above EF holebelow EF so called particlehole excitations In the inter acting system the excitations are quasiparticles and quasiholes Rather than a lone electron above the Fermi surface a LFLT quasiparticle is such an elec tron plus the correlated rearrangement of all the other electrons that then takes place due to interactions This correlated rearrangement of the other electrons is called dressing An electron injected from the vacuum into a Fermi liquid is then dressed to form a quasiparticle The effect ofthese correlations is to renormalize certain quantities like the effective mass and the compressibility The correlations can be quanti ed by a small number of Fermi liquid parameters that can be evaluated by experiment Other than these relatively mild corrections quasiparticles usually act very much like electrons Note that this theory can break down under various circumstances In particular LFLT fails if a new candidate ground state can exist that has lower energy than the Fermi liquid ground state and is separated from the LFLT ground state by a broken symmetry The symmetry breaking means that the smooth evolution mentioned above isn t possible The classic example of this is the transition to superconductivity In 1d systems with strong interactions Fermi liquid theory can also break down The proposed ground state in such circumstances is called a Luttinger liquid We will hopefully get a chance to touch on this later The main points here are 0 Screening depends strongly on 1Ep and so on the dimensionality and structure of materials 0 Electron electron scattering is pretty small in many circumstances be cause of screening and the Pauli principle 0 LFLT accounts for electron electron interactions by dressing the bare excitations 24 Transitions and rates Often we re interested in calculating the rate of some scattering process that takes a system from an initial state takes it to a nal state Also often we re not really interested in the details of the nal state rather we want to consider all available nal states that satisfy energy conservation The 19 result from rst order timedependent perturbation theory is often called Fermi s Golden Rule If the potential associated with the scattering is Vs the rate is approximately 5 2 1lt lvslgt126ltAEt lt21 where the 6 function only strictly a 6 function as the time after the collision gt 00 takes care of energy conservation If we want to account for all nal states that satisfy the energy condition the total rate is given by the integral over energy of the right hand side of Eq 21 times the density of states for nal states The important point here is as we mentioned above the density of states plays a crucial role in establishing transition and relaxation rates One good illustration of this in nanoscale physics is the effect of dimensionality on electron phonon scattering in for example metallic single walled carbon nanotubes Because of the peculiarities of their band structure these objects are predicted to have onedimensional densities of states There are only two allowed momenta forward and back and this reduction of the Fermi surface to these two points greatly suppresses the electron phonon scattering rate compared to that in a bulk metal As a result despite large phonon populations at room temperature ballistic transport of carriers is possible in these systems over micron distances In contrast the scattering time due to phonons in bulk metals is often three orders of magnitude shorter We won t go into detail calculating rates here any that we need later we ll do at the time Often this simple perturbation theory picture gives im portant physical insights More sophisticated treatments include diagram matic expansions equivalent to higher order perturbation calculations One other thing to bear in mind about transition rates If multiple independent processes exist and their individual scattering events are un correlated and well separated in time it is safe to add rates 1 1 1 ii 22 Ttot 7391 7392 If a single effective rate can be written down this implies an exponential relaxation in time as we know from elementary differential equations This is not always the case and we ll see one major example later of a nonexpo nential relaxation 20 3 Transport One extremely powerful technique for probing the underlying physics in nanoscale systems is electrical transport the manner in which charge ows or doesn t into through and out of the systems Besides the simple slap on ohmmeter leads dc resistance measurement one can consider transport and noise time dependent uctuations of transport with multiple lead con gurations as a function of temperature magnetic eld electric eld measuring frequency etc We ll begin with some general considerations about transport and progress on to different classical and quantum treatments of this problem 31 Classical transport Let s start with a collection of 71 classical noninteracting electrons per unit volume whizzing around in random directions with some typical speed 120 Suppose each electron typically undergoes a collision every time interval 739 that completely scrambles the direction of its momentum and that an elec tric eld E acts on the electrons Without the electric eld the average velocity of all the electrons is zero because of the randomness of their di rections In the steady state with the eld the average electron velocity is ieET m Vdft 23 The current density is simplyj 77175Vdft and we can nd the conductiv ity a and the mobility u not to be confused with the chemical potential 71527 039 m 5739 E 7 24 M m This is a very simpli ed picture but it introduces the idea of a charac teristic time scale and the notion of a mobility Notice that longer times between momentum randomizing collisions and lower masses mean higher mobilities and conductivities Now think about including statistical mechanics here One can de ne a classical distribution function by saying that probability of nding a par ticle at time t with within dr of r with a momentum within dp of p is ft rpdrdp We know from the Liouville theorem that as a function of time the distribution function has to obey a kind of continuity equation 21 that expresses conservation of particles and momentum This is called the Boltzmann equation and looks like this gvvrfFfoltgtmu 25 Here we ve used the idea that forces F p and that V r This is all explicitly classical since it assumes that we can specify both r and p The right hand side is the contribution to changes in 1 due to collisions like the ones mentioned above In the really simple case we started with one effectively replaces that term with f 7 f0T where f0 is the equilibrium distribution function at some temperature For more complicated collision processes naturally one could replace 739 with some function of r p t A good treatment of this is in the appendices to Kittel s solid state book as well as many other places If you assume that 739 is a constant you can do the equilibrium problem with a classical distribution function and nd that this treatment gives you Fick s law for diffusion That is to rst order a density gradient produces a particle current 1 jn 713W 7 ltu2gtm 26 in 3d where we ve de ned the particle diffusion constant D 32 Semiclassical transport We can do better than the above by incorporating what we know about band theory and statistical mechanics Note that to preserve our use of the Boltzmann equation for quantum systems in general we need to think hard about the fact that rp don t commute For now we ll assume that quantum phase information is lost between collisions We ll see later that preservation of that phase info leads to distinct consequences in transport We also need to assume that the disorder that produces the scattering isn t too severe one reasonable criterion is that kF UFT gtgt 1 27 This says that the electron travels many wavelengths between scattering events so our Bloch wave picture of the electronic states makes sense Remembering to substitute k for p one can solve the Boltzmann equa tion using the Fermi distribution function and again nd the diffusion con stant 1 D gugr 28 22 Here 739 is the relaxation time at the Fermi energy In general we can relate the measured conductivity to microscopic pa rameters using the Einstein relation 0 EZVD 29 where 1 is again our density of states at the Fermi level In principal one can measure 1 by either a tunneling experiment or through the electronic heat capacity If 1 is known D can be inferred from the measured conductivity 33 Quantum transport A fully quantum mechanical treatment of transport is quite involved though some nanoscale systems can be well understood through models involving simple scattering of incident electronic waves off various barriers This sim ple scattering picture is called the Landauer B ttiker formulation and we will return to it later Another approach to quantum transport is to use Greens functions often involving perturbative expansions the shorthand for which are Feynman diagrams The small parameter often used in such expansions is kp fl as hinted at in Eq 27 We ll leave our discussion of quantum corrections to conductivity until we re examining the relevant papers 34 Measurement considerations and symmetries Just a few words about transport measurements These can be nicely divided into two terminal and mum terminal con gurations In the world of theory one measures conductance by specifying the potential difference across the sample and measuring the current that ows through the sample with a perfect ammeter In practice it is often much easier to measure resistance That is a current is speci ed through the sample and an idea volt meter is used to measure the resulting potential difference In two terminal measurements the voltage is measured using the same leads that supply the current as in Fig 8a This is usually easier to do than more complicated con gurations especially if the device under examination is extremely small A practical problem can result though because usually the sample is relatively far away from the current source and volt meter ie at the bottom of a cryogenic setup The voltage being measured could include funny junction voltages at joints between the leads and the sample as well as thermal EMFs due to temperature gradients along the 23 leads etc Some of these complications can be avoided by actually doing the measurement at some low but nonzero frequency Contact and lead resistances however remain and can be signi cant Four terminal resistance measurements can eliminate this problem though care has to be taken in interpreting the data in certain systems The idea is shown in Fig 8b The device under test has four or more leads two of which are used to source and sink current and two of which are used to measure a potential difference Ideally the voltage probes are at the same chemical potential as their contact points on the sample and those con tact points are very small compared to the sample size The rst condition implies that no net current ows between the voltage probes and the sam ple avoiding the problem of measuring contact and lead resistances The second condition is necessary to avoid having the voltage probes shorting out77 signi cant parts of the sample and seriously altering the equipotential lines from their ideal current leads only con guration A third condition that the voltage probes be weakly coupled77 to the sample ensures that carriers are not likely to go back and forth between the voltage probes and the sample This is particularly important when the carriers in the sample are very different than those in the probes such as in charge density wave systems or Luttinger liquids In such cases the nature of the excitations in the sample near the voltage probes could be strongly changed if carriers could be exchanged freely While more complicated to arrange especially in nanoscale samples multiprobe measurements can provide more information than two probe schemes An example in 2d samples it is possible to use the additional probes to measure the Hall voltage induced in the presence of an external magnetic eld as well as to measure the longitudinal along the current resistance The classical Hall voltage balancing the Lorentz force by a Hall eld transverse to the longitudinal current is proportional to B and can be used to nd the sign of the charge carriers as well as their density In general with multiple probes it is possible to measure with all sorts of combinations of voltage and current leads When the equations of mo tion are examined certain symmetry relations Onsager relations must be preserved For two terminal measurements we nd that the measured con ductance G must obey CB C7B In the four probe case it is possible to show the argument treats all four probes equivalently that Rmnkl7B7 where RkLmn means current in leads kl voltage measured between leads mn 24 sample I IV R0gtgtRs IVR0 V Figure 8 Schematics of 2 and 4 probe resistance measurements 25 4 Concluding comments That s enough of an overview for now of basic solid state physics results Hopefully we re all on the same page and this has gotten you thinking about some of the issues that will be relevant in the papers we ll be reading The next two meetings we ll turn to the problems of characterizing and fabricating nanoscale systems This should familiarize you with the state of the art in this eld again to better prepare you for the papers to come After those talks we ll start to examine quantum effects in transport and then it s time to start doing seminars 5 References General solid state references H lbach and H Luth Solid State Physics an Introduction to Theory and Experiment Springer Verlag This is a good general solid state physics text with little experimental sections describing how some of this stuff is actually measured lts biggest aw is the number of typographical mistakes in the exercises N Ashcroft and ND Mermin Solid State Physics The classic graduate text Excellent and as readable as any physics book ever is Too bad that it ends in the mid 1970 s C Kittel Introduction to Solid State Physics Also a classic and also fairly good Like AampM the best parts were written 25 years ago and some of the newer bits feel very tacked on W Harrison Solid State Physics Dover Very dense written by a master of band structure calculations Has the added virtue of being quite inexpensive PM Chaikin and M Lubensky Condensed Matter Physics More recent and contains a very nice review of statistical mechanics Selection of topics geared much more toward soft condensed matter Nanoelectronics and mesoscopic physics 26 Y lmry Introduction to Mesoscopz39c Physics Oxford University Press Very good introduction to many issues relevant to nanoscale physics Occasionally so elegant as to be cryptic DK Ferry and SM Goodnick Transport in Nanostmctures Cam bridge University Press Also very good7 and quite comprehensive 27 Nanoscale characterization August 9 2001 The task of making structures on nanometer scales already dif cult is further complicated by the dif culty of seeing what s going on at these extreme dimensions Below we outline a number of nanoscale probes de scribing some of the pluses and minuses of each Generally speaking we leave transport measurements until later when we re discussing particular experimental results 1 Scattering methods Roughly a zillion different powerful scattering techniques exist for probing various material systems These techniques can be nanoscale in the sense of being sensitive to sub A variations in lattice spacing in a crystal such as x ray diffraction various electron diffraction methods etc However they tend to achieve this sort of resolution by averaging over many many atoms rather than providing a true local probe which is frequently what nanoscience types are interested in Therefore we re not going to discuss them here Instead we re going to concern ourselves primarily with mi croscopy techniques that offer truly local probes 2 Scanning electron microscopy SEM The old standby SEM operates by magnetically scanning a focused electron beam typically from 1 kV to 40 kV in energy across the surface of a sample in high vacuum The incident electrons upon interacting inelastically with the sample material produce secondary electrons Positioned close to the sample is a detector that picks up some of the secondary electron yield By measuring the detector response as the beam is rastered over the sample an image is constructed Electron beam sources are either thermal emission from heated W or LaB laments or eld emission Sample requirements The sample in SEM needs to be at least a little bit conducting and grounded Really insulating samples present an imag ing challenge because charging of the sample can de ect later beam scans resulting in a distorted image Often insulating samples are sputter coated with a very thin layer of metal to aid in imaging Further a clean sample surface is highly desirable since the local energy deposition of the electron beam can cause chemical changes in any surface layer of contaminants crud ding up the sample Sample materials with high Z such as Au produce high secondary yields and look bright in SEM micrographs while light elements are more dif cult to image Resolution limits SEM resolution is limited by beam spot size can be as small as 1 nm beam jitter and the spread of secondary electron yield There is a tradeoff that goes on between beam energy and resolution of ne surface features On the one hand high incident energies mean reduced transit times between emitter and detector and so reduced vulnerability to vibrational noise However higher energies mean that the secondary yield takes place deeper into the sample material making it dif cult to resolve surface detail Other limitations In most SEM chambers there is frequently a dilute gas of hydrocarbon junk drifting around often due to backstreaming from pumps or improperly degreased chamber construction components The electron beam can cause some kind of reaction which deposits carbon from this junk onto the sample On larger scale objects this isn t a big deal since the layer is usually thin 20 Afor a few minutes of viewing over a 05 umz area in one machine I ve used and carbon is fairly transparent because of its low Z However on real nanoscale objects this is clearly very serious and can limit SEM imaging to a oneshot procedure for noncritical samples Variations Two common variants on SEM are back scattered SEM which looks at back scattered electrons rather than secondaries and SEM with X ray detection Because both the quantity of back scattered electrons and the spectra of emitted X rays are strongly dependent on the Z of the sample these techniques are used for elemental analysis 3 Transmission electron microscopy TEM TEM is an extremely powerful technique that can give truly atomic resolu tion images As you might guess from the name it s similar to SEM but works by ring an energetic between 100 and 500 kV beam of electrons with a very small spot size throught the sample through some more op Figure 1 SEM images I took of some of my sample using a eld emission SEM at Bell Labs Figure 2 TEM images of in situ fabricated Au nanowires7 from Kondo and Takanayagi7 Science 2897 606 2000 tics7 and onto a detector Imaging techniques include bn39ght eld imaging7 Where Widely scattered electrons miss the detector7 and bright spots indi cate transmission and dark eld imaging7 Where the minimally scattered electrons are diverted7 and scatterers show up as bright spots One can also do electron diffraction and holography7 looking at interference effects be tween unscattered and scattered electrons All of these approaches require a bright7 coherent source of electrons Sample requirements To get decent transmittance7 bulk samples must be thinned down to a few hundred nm thickness7 usually by mechanical polishing followed by ion milling TEM sample prep is truly an art Some very impressive results have been seen recently in nanoscale images With in situ sample fabrication Resolution limits True atomic resolution images are possible With TEM7 though its a balancing act Thicker samples lead to more contrast while simultaneously leading to more multiple scatterings and reduced transmit tance O K edge intensity arb units ADF image Energy loss eV Figure 3 TEM and EELS spectra of thin gate oxides in FETsi From Muller et al Nature 399 758i Variations One very powerful variation is electron energy loss spectroscopy where a sensitive calorimeter looks at the energy of transmitted electronsi This allows one to look at the band structure of whatever s being imaged see Fig 3 Another technique is called Lorentz microscopy where one looks at the de ection change in electron diffraction pattern caused by the magnetic eld of a sample acting on the electron beami This allows high resolution examination of magnetic structure 4 Scanned Probe Microscopy SPM SPM is a general term for a class of microscopy techniques two of which scanning tunneling microscopy and atomic force microscopy you ve almost certainly heard of The general features of SPM are small samples usually mounted on some sort of piezoelectric scanneri A probe tip is then brought into extremely close proximity to the sample surface and some parameter is measured to be used in a feedback circuit that controls the relative 2 positions of the tip and sample The sample is then scanned laterally and usually the strength of the feedback signal or error signal or some other quantity is monitored as a function of zy position A computer runs all Figure 4 AFM image of a nanotube bridging deposited electrodes from Cees Dekker7s group at Delft this and can then be used to produce cool 3d rendered images ofthe sample 5 Atomic Force Microscopy AFM AFM is the workhorse upon which many SPM techniques are based The general idea is to use a micromachined cantilever with a very sharp point on it as the probe tip The de ections of the cantilever are usually sensed ope tically through a re ected laser beam or piezoresistively through changes in the resistance of a nanostructure on the cantilever AFM can be done with a number of modes and variations 51 Contact mode This is the original concept for AFM The tip acts like a stylus or record needle lt7s pressed against the sample with some small force on the order of nN and the 2 position of the sample is usually varied during the scan to keep the cantilever de ection and hence the force constant The map of 2 position vs zy gives the sample topography Resolution limit Typically a few nm The problem is the nite rai dius of curvature of the tip which means that the tipisurface interaction is formed by a large number of atoms Quotes of true atomic resolution for contactimode AFM are almost always hype Occasionally one can get the appearance of atomic resolution ie can see the lattice spacing if imag ing a periodic surface such as atomically smooth graphite However its very hard to spot isolated singleiatom defects which is a better de nition of atomic resolution Limitations and advantages This technique can be done in air and even in liquids with reasonable success It s very exible Limitations besides the resolution issue involve the tip surface interaction For very small and fragile structures you really don t want to drag the tip across the sample because of the risk of damage Further this tends to damage the tip over time Variations Probably the two cookest variations of contact AFM are lateral force microscopy LFM and nanolithography ln LFM the sideways twist of the cantilever is measured as the cantilever is dragged over the sample Changes in the twist show changes in the tip surface friction and are a local probe of the surfaces elastic properties Nanolithography involves using the tip to produce local modifactions of the surface We ll talk about this more suring our look at fabrication techniques 52 Tappingnoncontact mode NCAFM is an innovation over contact mode The cantilever is driven near its mechanical resonance frequency and moved near the sample The tip surface interaction changes the resonance frequency of the cantilever which affects the amplitude of the cantilever s motion The 2 position is adjusted to keep constant amplitude providing an image of surface topography ln tapping mode the tip is allowed to just touch the surface sampling the short range repulsive part of the interatomic potential Resolution limits Generally similar to contact mode AFM though through clever techniques one can improve this to really see atomic reso lution in some circumstances Variations Since NCAFM is sensitive to all forces acting between the tip and the sample people take advantage of this in the many variations to the technique Examples of this include 0 Magnetic Force Microscopy Here a magnetized tip is used to detect variations in sample magnetization over very small lengthscales Contact mode can the be used to nd topography Electrostatic Force Microscopy A conducting tip is used and a nite bias is applied between tip and sample One can measure differences in contact potential between tip and sample as well as mapping out potential variations along the sample This has been used for example to study the voltage drop along carbon nanotube devices 0 Chemical Force Microscopy One can chemically functionalize the AFM tip and see spatial variations across the sample in the chemi cal forces between tip and sample Extremely neat possibilities for biotechnology and nanochemistry 0 Magnetic Resonance Force Microscopy This one s really clever Try to get polarized nuclear spins in the sample precessing at the reso nance frequency of the cantilever and have a magnetized cantilever tip The idea is that when the tip passes over the precessing spins it will be driven on resonance which should be detectable It works and numbers suggest it should be possible to get enough sensitivity to see single spinsl 6 Scanning tunneling microscopy STM STM is an extremely useful tool for examining conducting samples A con ducting tip set at some bias voltage N1 V with respect to the sample is lowered very carefully to within a few A of the sample surface At this dis tance electrons can tunnel between the tip and sample and the tip sample distance is maintained to keep the tunneling current constant Let s assume we re tunneling carriers from the tip into the sample From our discussion of rates of quantum processes one can nd that the tunneling current looks like 00 00 W 2 0 dEmp 0 dEsamumaEmpwsmEmfEm17IEsam6EmpeEmeev lt1 where M is the tunneling matrix element Here we re taking the Fermi s Golden Rule expression for a transition rate and integrating over available initial and nal electronic states that satisfy an energy difference of eV The product of Fermi functions 1 says that carriers need to be going from an occupied state in the tip to an unoccupied state in the sample The rst point the tunneling matrix element M depends exponentially on the distance between the tip and sample the effective barrier thickness This is clear from simple WKB ideas of elementary quantum This extremely steep dependence of current on distance is what leads to true atomic res olution in STM measurements Even with a tip rounded on the nm scale tunneling through the single tip atom closest to the sample dominates A second point we re interested here in the tunneling from a normal metal into the sample that may not be a simple metallic system It is possible to use the STM as a local probe of electronic structure because of Figure 5 Crommie and Eigler s famous quantum corral77 of Fe atoms on the Cu 100 surface showing standing waves composed of electronic surface states the ability to vary the tipsample voltage Starting from Eq 1 do the integration over Em to eliminate the 6functioni One can then make various manipulations and eventually nd that the di emential conductance dIdV is proportional to the local density of states of the sample This allows the STM to perform tunneling spectroscopy as a function of position The tip is moved to a position Ly the feedback loop is brie y opened and the tipsample bias is swept over some range The feedback loop is closed the tip is repositioned and this is repeated to build up a map of the sample s electronic structure The ability to do this well requires an extremely stable positioning system for the STM tip vertical drift during the bias voltage sweep will obscure real effects because of the steep dependence of I on 2 So STM images are a convolution of the surface topography and the local electronic density of states This allows the imaging of electronic surface states in conducting materials as in Fig 5 One can also perform scans at different bias levels and map out the presence of different features in the electronic spectrum such as unoccupied states due to dangling bonds or the impact of impurities on some correlated electronic state Limitations The biggest limitation of STM is the need for relatively conducting sample surfaces This makes working with structures on some Flguxe s STM nnage of a cazbon nanetnhe 10m cees Dekkez s gxoup at Delft the nest commonly available snhstnates eg oxidized s wafezs vezy dxf cult Fuxthex fox enhanced stabxhty and Lo avmd suzface contamination ss es b x m ultxarhxgh wcuum U39HV and low tennpenatnnes 42 K These constzmnts nnahe STM Lough Lo pexfoxm on 1 p e man one xecent wuauon s spmrpolauzed STM wheze a magne t t t y y y be vexy sensitive to the mayheuc stucLuxe of the sample A second wzmtxon 15 when almtmn amuswn mtcmscapy BEEM If y y y t h h a mg suhstnate some of the tunnehng electzons mm the hp can balhstxcally penetxate Into the substzate By measuung this cuuent n s possible to get mfoxmauon about the scatteung pmesses at wozk m the Lhm sample lagex 7 Near eld scanning optical microscopy NSOM Usual optical mlcxoscopy 1s diffwcuon limited and Lhexefoxe 1s vezy dxf cult to the hp dwmetex Lamatataens The snna11en the hp dmmetez the mole dxf cult ht s to 10 get ef cient optical coupling between the sample and the ber One must trade off resolution against signal to noise Further reliable fabrication of tips signi cantly smaller than 100 nm is not really possible at this time 8 Combined SPM and transport It is also possible to combine SPM with transport measurements in nanos tructures There are two main approaches both sometimes called scanned gate microscopy In the rst one applies some bias voltage to a conducting AFM tip and measures the conductance of the device under test as a function of tip position When the tip is positioned over a portion of the device where no current is owing the current distribution in the device is relatively unperturbed Conversely when the tip can strongly affect nearby regions of high current density A map of the current paths in a device can be built up this way In the second approach one does feedback on the tip sample bias of a conducting AFM tip in noncontact mode to null out the potential difference between the tip and sample This allows mapping of the potential distri bution on the sample If current is owing through the device one can see where the voltage drops occur and how big they are 9 Nanomagnetic sensing In addition to MFM and MRFM mentioned above there are two primary techniques for measuring the magnetic response of nanoscale samples su perconducting quantum intereference devices SQUle and Hall magne tometers SQ UIDS We won t get into a detailed explanation of SQUle here The general idea utilizes the fact that the ux through a superconducting loop is quantized in units of the ux quantum 10 7125 where the 2 comes from the 25 charge of a Cooper pair The dc Josephson effect implies take a superconducing loop of area A with a couple of weak links77 in it and bias it at constant current near the critical current of the weak links To enforce ux quantization supercurrents will be set up circulating around the loop As a result the voltage measured across the loop will be oscillatory in the applied magnetic eld with a period of ltIgtltIgt0 Such superconducting loops can be exquisitely precise detectors of mag netic eld They can be used as nanoprobes by a fabricating a SQUID 11 pickup on the end of an AFM style cantilever or b placing a sample on a tiny SQUID directly so that the sample s magnetic behavior affects the ux through the loop Limitations Because of their superconducting nature SQUle require low temperatures to operate though high TC SQUle can mitigate this to some degree The other major problem is that SQUle can be too sensitive The fact that SQUle are sensitive to any magnetic eld applied to the sample means that shielding and clever experimental design are essential when using them Hall e ect A second approach to magnetic sensing is to use the classic Hall effect A small Hall bar is fabricated from two dimensional electron gas 2DEG usually in a GaAsAlGaAs heterostructure As with SQUle the bar can either be on the end of an AFM style cantilever or the bar can be xed and the sample placed directly on top of it Measuring the Hall voltage at xed current can determine the local magnetic eld Further it is sometimes possible to null out the effects of applied elds by comparing the Hall voltages from two nominally identical Hall bars one next to the sample and one nearby but isolated from the sample The main limitation of the Hall technique is that its sensitivity simply isn t as high as that of SQUle 10 Conclusions We have now looked at a considerable number of techniques for character izing the properties of nanoscale samples The list presented here is by no means exhaustive and new techniques are being developed all the time That is of course one reason this area is so exciting with the development of each new technique a wealth of new information is learned some of which may lead to other new methods
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