Scanning Tunneling Microscopy
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A scanning tunneling microscope (STM) is a type of
microscope A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisibl ...
used for imaging
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s at the
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ic level. Its development in 1981 earned its inventors,
Gerd Binnig Gerd Binnig (; born 20 July 1947) is a German physicist. He is most famous for having won the Nobel Prize in Physics jointly with Heinrich Rohrer in 1986 for the invention of the scanning tunneling microscope. Early life and education Binnig wa ...
and
Heinrich Rohrer Heinrich Rohrer (6 June 1933 – 16 May 2013) was a Swiss physicist who shared half of the 1986 Nobel Prize in Physics with Gerd Binnig for the design of the scanning tunneling microscope (STM). The other half of the Prize was awarded to Ernst R ...
, then at IBM Zürich, the
Nobel Prize in Physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 1986. STM senses the surface by using an extremely sharp
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tip that can distinguish features smaller than 0.1  nm with a 0.01 nm (10 pm) depth resolution. This means that individual atoms can routinely be imaged and manipulated. Most microscopes are built for use in
ultra-high vacuum Ultra-high vacuum (UHV) is the vacuum regime characterised by pressures lower than about . UHV conditions are created by pumping the gas out of a UHV chamber. At these low pressures the mean free path of a gas molecule is greater than approximately ...
at temperatures approaching zero kelvin, but variants exist for studies in air, water and other environments, and for temperatures over 1000 °C. STM is based on the concept of
quantum tunneling In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
. When the tip is brought very near to the surface to be examined, a
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
voltage applied between the two allows
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s to tunnel through the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
separating them. The resulting ''tunneling
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
'' is a function of the tip position, applied voltage, and the
local density of states In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states ...
(LDOS) of the sample. Information is acquired by monitoring the current as the tip scans across the surface, and is usually displayed in image form. A refinement of the technique known as
scanning tunneling spectroscopy Scanning tunneling spectroscopy (STS), an extension of scanning tunneling microscopy (STM), is used to provide information about the density of electrons in a sample as a function of their energy. In scanning tunneling microscopy, a metal tip is ...
consists of keeping the tip in a constant position above the surface, varying the bias voltage and recording the resultant change in current. Using this technique the local density of the electronic states can be reconstructed. This is sometimes performed in high magnetic fields and in presence of impurities to infer the properties and interactions of electrons in the studied material. Scanning tunneling microscopy can be a challenging technique, as it requires extremely clean and stable surfaces, sharp tips, excellent
vibration isolation Vibration isolation is the process of isolating an object, such as a piece of equipment, from the source of vibrations. Vibration is undesirable in many domains, primarily engineered systems and habitable spaces, and methods have been developed to p ...
, and sophisticated electronics. Nonetheless, many hobbyists build their own microscopes.


Procedure

The tip is brought close to the sample by a coarse positioning mechanism that is usually monitored visually. At close range, fine control of the tip position with respect to the sample surface is achieved by
piezoelectric Piezoelectricity (, ) is the electric charge that accumulates in certain solid materials—such as crystals, certain ceramics, and biological matter such as bone, DNA, and various proteins—in response to applied Stress (mechanics), mechanical s ...
scanner tubes whose length can be altered by a control voltage. A bias
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
is applied between the sample and the tip, and the scanner is gradually elongated until the tip starts receiving the tunneling current. The tip–sample separation ''w'' is then kept somewhere in the 4–7 Å (0.4–0.7 nm) range, slightly above the height where the tip would experience repulsive interaction (''w''<3Å), but still in the region where attractive interaction exists (3<''w''<10Å). The tunneling current, being in the sub-
nanoampere The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to ele ...
range, is amplified as close to the scanner as possible. Once tunneling is established, the sample bias and tip position with respect to the sample are varied according to the requirements of the experiment. As the tip is moved across the surface in a discrete x–y matrix, the changes in surface height and population of the electronic states cause changes in the tunneling current. Digital images of the surface are formed in one of the two ways: in the ''constant height mode'' changes of the tunneling current are mapped directly, while in the ''constant current mode'' the voltage that controls the height (''z'') of the tip is recorded while the tunneling current is kept at a predetermined level. In constant current mode, feedback electronics adjust the height by a voltage to the piezoelectric height control mechanism. If at some point the tunneling current is below the set level, the tip is moved towards the sample, and vice versa. This mode is relatively slow as the electronics need to check the tunneling current and adjust the height in a feedback loop at each measured point of the surface. When the surface is atomically flat, the voltage applied to the z-scanner will mainly reflect variations in local charge density. But when an atomic step is encountered, or when the surface is buckled due to
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, the height of the scanner will also have to change because of the overall topography. The image formed of the z-scanner voltages that were needed to keep the tunneling current constant as the tip scanned the surface will thus contain both topographical and electron density data. In some cases it may not be clear whether height changes came as a result of one or the other. In constant height mode, the z-scanner voltage is kept constant as the scanner swings back and forth across the surface and the tunneling current, exponentially dependent on the distance, is mapped. This mode of operation is faster, but on rough surfaces, where there may be large adsorbed molecules present, or ridges and groves, the tip will be in danger of crashing. The
raster scan A raster scan, or raster scanning, is the rectangular pattern of image capture and reconstruction in television. By analogy, the term is used for raster graphics, the pattern of image storage and transmission used in most computer bitmap image s ...
of the tip is anything from a 128×128 to a 1024×1024 (or more) matrix, and for each point of the raster a single value is obtained. The images produced by STM are therefore
grayscale In digital photography, computer-generated imagery, and colorimetry, a grayscale image is one in which the value of each pixel is a single sample representing only an ''amount'' of light; that is, it carries only intensity information. Graysca ...
, and color is only added in post-processing in order to visually emphasize important features. In addition to scanning across the sample, information on the electronic structure at a given location in the sample can be obtained by sweeping the bias voltage (along with a small AC modulation to directly measure the derivative) and measuring current change at a specific location. This type of measurement is called
scanning tunneling spectroscopy Scanning tunneling spectroscopy (STS), an extension of scanning tunneling microscopy (STM), is used to provide information about the density of electrons in a sample as a function of their energy. In scanning tunneling microscopy, a metal tip is ...
(STS) and typically results in a plot of the local
density of states In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
as a function of the electrons' energy within the sample. The advantage of STM over other measurements of the density of states lies in its ability to make extremely local measurements. This is how, for example, the density of states at an
impurity In chemistry and materials science, impurities are chemical substances inside a confined amount of liquid, gas, or solid, which differ from the chemical composition of the material or compound. Firstly, a pure chemical should appear thermodynam ...
site can be compared to the density of states around the impurity and elsewhere on the surface.


Instrumentation

The main components of a scanning tunneling microscope are the scanning tip, piezoelectrically controlled height (z axis) and lateral (x and y axes) scanner, and coarse sample-to-tip approach mechanism. The microscope is controlled by dedicated electronics and a computer. The system is supported on a vibration isolation system. The tip is often made of
tungsten Tungsten, or wolfram, is a chemical element with the symbol W and atomic number 74. Tungsten is a rare metal found naturally on Earth almost exclusively as compounds with other elements. It was identified as a new element in 1781 and first isolat ...
or platinum-iridium wire, though
gold Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile met ...
is also used. Tungsten tips are usually made by electrochemical etching, and platinum-iridium tips by mechanical shearing. The
resolution Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual mak ...
of an image is limited by the
radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
of the scanning tip. Sometimes, image artefacts occur if the tip has more than one apex at the end; most frequently ''double-tip imaging'' is observed, a situation in which two apices contribute equally to the tunneling. While several processes for obtaining sharp, usable tips are known, the ultimate test of quality of the tip is only possible when it is tunneling in the vacuum. Every so often the tips can be conditioned by applying high voltages when they are already in the tunneling range, or by making them pick up an atom or a molecule from the surface. In most modern designs the scanner is a hollow tube of a radially-polarized piezoelectric with metallized surfaces. The outer surface is divided into four long quadrants to serve as x and y motion electrodes with deflection voltages of two polarities applied on the opposing sides. The tube material is a
lead zirconate titanate Lead zirconate titanate is an inorganic compound with the chemical formula (0≤''x''≤1), commonly abbreviated as PZT. Also called lead zirconium titanate, it is a ceramic perovskite material that shows a marked piezoelectric effect, meaning t ...
ceramic with a piezo constant of some 5 nanometers per volt. The tip is mounted at the center of the tube. Because of some crosstalk between the electrodes and inherent nonlinearities, the motion is
calibrated In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy. Such a standard could be another measurement device of known a ...
and voltages needed for independent x, y and z motion applied according to calibration tables. Due to the extreme sensitivity of the tunneling current to the separation of the electrodes, proper vibration isolation or a rigid STM body is imperative for obtaining usable results. In the first STM by Binnig and Rohrer,
magnetic levitation Magnetic levitation (maglev) or magnetic suspension is a method by which an object is suspended with no support other than magnetic fields. Magnetic force is used to counteract the effects of the gravitational force and any other forces. The ...
was used to keep the STM free from vibrations; now mechanical spring or
gas spring A gas spring is a type of spring that, unlike a typical mechanical spring that relies on elastic deformation, uses compressed gas contained within an enclosed cylinder sealed by a sliding piston to pneumatically store potential energy and withs ...
systems are often employed. Additionally, mechanisms for vibration damping using
eddy currents Eddy currents (also called Foucault's currents) are loops of electrical current induced within conductors by a changing magnetic field in the conductor according to Faraday's law of induction or by the relative motion of a conductor in a magn ...
are sometimes implemented. Microscopes designed for long scans in scanning tunneling spectroscopy need extreme stability and are built in
anechoic chamber An anechoic chamber (''an-echoic'' meaning "non-reflective") is a room designed to stop reflections of either sound or electromagnetic waves. They are also often isolated from energy entering from their surroundings. This combination means ...
s—dedicated concrete rooms with acoustic and electromagnetic isolation that are themselves floated on vibration isolation devices inside the laboratory. Maintaining the tip position with respect to the sample, scanning the sample and acquiring the data is computer controlled. Dedicated software for scanning probe microscopies is used for
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
as well as performing quantitative measurements. Some scanning tunneling microscopes are capable of recording images at high frame rates. Videos made of such images can show surface
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
or track adsorption and reactions on the surface. In video-rate microscopes, frame rates of 80 Hz have been achieved with fully working feedback that adjusts the height of the tip.


Principle of operation

Quantum tunneling of electrons is a functioning concept of STM that arises from
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. Classically, a particle hitting an impenetrable barrier will not pass through. If the barrier is described by a potential acting along ''z''-direction in which an electron of mass ''m''e acquires the potential energy ''U''(''z''), the electron's trajectory will be deterministic, and such that the sum ''E'' of its kinetic and potential energies is at all times conserved, :E=\frac+U(z) The electron will have a defined, non-zero momentum ''p'' only in regions where the initial energy ''E'' is greater than ''U''(''z''). In quantum physics, however, particles with a very small mass, such as the electron, have discernible wavelike characteristics and are allowed to ''leak'' into classically forbidden regions. This is referred to as tunneling.


Rectangular barrier model

The simplest model of tunneling between the sample and the tip of a scanning tunneling microscope is that of a
rectangular potential barrier In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. ...
. An electron of energy ''E'' is incident upon an energy barrier of height ''U'', in the region of space of width ''w.'' An electron's behavior in the presence of a potential ''U''(''z''), assuming one-dimensional case, is described by
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
s \psi(z) that satisfy Schrödinger’s equation, :- \frac \frac + U(z)\,\psi(z) = E\,\psi(z) Here, ''ħ'' is the reduced Planck’s constant, ''z'' is the position, and '' me'' is the mass of an electron. In the zero-potential regions on two sides of the barrier, the wave function takes on the following form :\psi_L(z) = e^+r\,e^, for ''z''<0 :\psi_R(z) = t\,e^, for ''z''>''w'' Here, k=\tfrac\sqrt. Inside the barrier, where ''E'' < ''U'', the wave function is a superposition of two terms, each decaying from one side of the barrier :\psi_B (z) = \xi e^+\zeta e^, for 0 \kappa = \tfrac\sqrt. The coefficients ''r'' and ''t'' provide measure of how much of the incident electron's wave is reflected or transmitted through the barrier. Namely, of the whole impinging particle current j_i=\tfrac only j_t=, t, ^2\, j_i will be transmitted, as can be seen from the
probability current In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is th ...
expression :j_t=-i\tfrac \left\ which evaluates to j_t=\tfrac, t, ^2. The transmission coefficient is obtained from the continuity condition on the three parts of the wave function and their derivatives at ''z''=0 and ''z''=''w'' (detailed derivation is in the article
Rectangular potential barrier In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. ...
). This gives , t, ^2= +\tfrac \sinh^2\kappa w where \varepsilon=E/U. The expression can be further simplified, as follows: In STM experiments, typical barrier height is of the order of the material's surface
work function In solid-state physics, the work function (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" m ...
''W,'' which for most metals has a value between 4 and 6 eV. The
work function In solid-state physics, the work function (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" m ...
is the minimum energy needed to bring an electron from an occupied level, the highest of which is the
Fermi level The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''µ'' or ''E''F for brevity. The Fermi level does not include the work required to remove ...
(for metals at ''T''=0 kelvin), to
vacuum level In physics, the vacuum level refers to the energy of a free stationary electron that is outside of any material (it is in a perfect vacuum). It may be taken as infinitely far away from a solid, or, defined to be near a surface. Its definition and m ...
. The electrons can tunnel between two metals only from occupied states on one side into the unoccupied states of the other side of the barrier. Without bias, Fermi energies are flush and there is no tunneling. Bias shifts electron energies in one of the electrodes higher, and those electrons that have no match at the same energy on the other side will tunnel. In experiments, bias voltages of a fraction of 1 V are used, so \kappa is of the order of 10 to 12 nm−1, while ''w'' is a few tenths of a nanometer. The barrier is strongly attenuating. The expression for the transmission probability reduces to , t, ^2=16\,\varepsilon(1-\varepsilon)\,e^. The tunneling current from a single level is therefore :j_t = \left tfrac\right2 \, \tfrac\,e^ where both wave vectors depend on the level's energy ''E''; k=\tfrac\sqrt and \kappa = \tfrac\sqrt. Tunneling current is exponentially dependent on the separation of the sample and the tip, and typically reduces by an order of magnitude when the separation is increased by 1 Å (0.1 nm). Because of this, even when tunneling occurs from a non-ideally sharp tip, the dominant contribution to the current is from its most protruding atom or orbital.


Tunneling between two conductors

As a result of the restriction that the tunneling from an occupied energy level on one side of the barrier requires an empty level of the same energy on the other side of the barrier, tunneling occurs mainly with electrons near the Fermi level. The tunneling current can be related to the density of available or filled states in the sample. The current due to an applied voltage ''V'' (assume tunneling occurs from the sample to the tip) depends on two factors: 1) the number of electrons between the Fermi level ''EF'' and ''EF−eV'' in the sample, and 2) the number among them which have corresponding free states to tunnel into on the other side of the barrier at the tip. The higher the density of available states in the tunneling region the greater the tunneling current. By convention, a positive ''V'' means that electrons in the tip tunnel into empty states in the sample; for a negative bias, electrons tunnel out of occupied states in the sample into the tip. For small biases and temperatures near absolute zero, the number of electrons in a given volume (the electron concentration) that are available for tunneling is the product of the density of the electronic states ''ρ''(''E''F) and the energy interval between the two Fermi levels, ''eV.'' Half of these electrons will be travelling away from the barrier. The other half will represent the
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
impinging on the barrier, which is given by the product of the electron concentration, charge, and velocity ''v'' (''I''i=''nev''), :I_i = \tfrace^2v\,\rho(E_F)\,V. The tunneling electric current will be a small fraction of the impinging current. The proportion is determined by the transmission probability ''T'', so :I_t = \tfrace^2v\,\rho(E_F)\,V\,T. In the simplest model of a rectangular potential barrier the transmission probability coefficient ''T'' equals , ''t'', 2.


Bardeen's formalism

A model that is based on more realistic wave functions for the two electrodes was devised by
John Bardeen John Bardeen (; May 23, 1908 – January 30, 1991) was an American physicist and engineer. He is the only person to be awarded the Nobel Prize in Physics twice: first in 1956 with William Shockley and Walter Brattain for the invention of the tran ...
in a study of the
metal-insulator-metal Metal-insulator-metal (MIM) diode is a type of nonlinear device very similar to a semiconductor diode that is capable of very fast operation. Depending on the geometry and the material used for fabrication, the operation mechanisms are governed eith ...
junction. His model takes two separate orthonormal sets of wave functions for the two electrodes and examines their time evolution as the systems are put close together. Bardeen's novel method, ingenious in itself, solves a time-dependent perturbative problem in which the perturbation emerges from the interaction of the two subsystems rather than an external potential of the standard Rayleigh–Schrödinger perturbation theory. Each of the wave functions for the electrons of the sample (S) and the tip (T) decay into the vacuum after hitting the surface potential barrier, roughly of the size of the surface work function. The wave functions are the solutions of two separate Schrödinger's equations for electrons in potentials ''U''S and ''U''T. When the time dependence of the states of known energies E^S_\mu and E^T_\nu is factored out, the wave functions have the following general form :\psi^S_\mu(t) = \psi^S_\mu \exp(-\tfracE^S_\mu t) :\psi^T_\nu(t) = \psi^T_\nu \exp(-\tfracE^T_\nu t) If the two systems are put closer together, but are still separated by a thin vacuum region, the potential acting on an electron in the combined system is ''U''T + ''U''S. Here, each of the potentials is spatially limited to its own side of the barrier. Only because the tail of a wave function of one electrode is in the range of the potential of the other, there is a finite probability for any state to evolve over time into the states of the other electrode. The future of the sample's state ''μ'' can be written as a linear combination with time-dependent coefficients of \psi^S_\mu (t) and all \psi^T_\nu (t) , :\psi(t) = \psi^S_\mu (t)\,+\,\sum_\nu with the initial condition c_\nu (0)=0 . When the new wave function is inserted into the Schrödinger’s equation for the potential ''U''T + ''U''S, the obtained equation is projected onto each separate \psi^T_\nu (that is, the equation is multiplied by a ^* and integrated over the whole volume) to single out the coefficients c_\nu . All \psi^S_\mu are taken to be ''nearly orthogonal'' to all \psi^T_\nu (their overlap is a small fraction of the total wave functions), and only first order quantities retained. Consequently, the time evolution of the coefficients is given by :\tfracc_\nu (t) = -\tfrac \int\psi^S_\mu \,U_T\, ^* \textrmx\,\textrmy\,\textrmz\,\exp \tfrac(E^S_\mu -E^T_\nu )t. Because the potential ''U''T is zero at the distance of a few atomic diameters away from the surface of the electrode, the integration over ''z'' can be done from a point ''z''o somewhere inside the barrier and into the volume of the tip (''z''>''z''o). If the tunneling matrix element is defined as :M_= \int_\psi^S_\mu \,U_T\, ^* \textrmx\,\textrmy\,\textrmz , the probability of the sample's state ''μ'' evolving in time ''t'' into the state of the tip ''ν'' is :, c_\nu (t), ^2 = , M_, ^2\,\frac . In a system with many electrons impinging on the barrier, this probability will give the proportion of those that successfully tunnel. If at a time ''t'' this fraction was , c_\nu (t), ^2 , at a later time ''t''+d''t'' the total fraction of , c_\nu (t+\textrmt), ^2 would have tunneled. The ''current'' of tunneling electrons at each instance is therefore proportional to , c_\nu (t+\textrmt), ^2-, c_\nu (t), ^2 divided by \textrmt, which is the time derivative of , c_\nu (t), ^2 , :\Gamma_\;\overset\;\frac, c_\nu (t), ^2 = \frac, M_, ^2\,\frac . The time scale of the measurement in STM is many orders of magnitude larger than the typical
femtosecond A femtosecond is a unit of time in the International System of Units (SI) equal to 10 or of a second; that is, one quadrillionth, or one millionth of one billionth, of a second. For context, a femtosecond is to a second as a second is to about 31. ...
time scale of electron processes in materials, and \tfract is large. The fraction part of the formula is a fast oscillating function of (E^S_\mu -E^T_\nu ) that rapidly decays away from the central peak where E^S_\mu =E^T_\nu. In other words, the most probable tunneling process, by far, is the elastic one, in which the electron's energy is conserved. The fraction, as written above, is a representation of the
delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, so :\Gamma_=\tfrac, M_, ^2\,\delta(E^S_\mu -E^T_\nu) . Solid-state systems are commonly described in terms of continuous rather than discrete energy levels. The term \delta(E^S_\mu -E^T_\nu) can be thought of as the
density of states In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
of the tip at energy E^S_\mu , giving :\Gamma_=\tfrac, M_, ^2\,\rho_T(E^S_\mu) . The number of energy levels in the sample between the energies ''\varepsilon'' and ''\varepsilon+\mathrm\varepsilon'' is ''\rho_S(\varepsilon)\mathrm\varepsilon''. When occupied, these levels are spin-degenerate (except in a few special classes of materials) and contain charge ''2e\cdot\rho_S(\varepsilon)\mathrm\varepsilon'' of either spin. With the sample biased to voltage ''V'', tunneling can occur only between states whose occupancies, given for each electrode by the
Fermi–Dirac distribution Fermi–Dirac may refer to: * Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
f, are not the same, that is, when either one or the other is occupied, but not both. That will be for all energies ''\varepsilon'' for which f(E_F-eV+\varepsilon)-f(E_F+\varepsilon) is not zero. For example, an electron will tunnel from energy level E_F-eV in the sample into energy level E_F in the tip (''\varepsilon=0''), an electron at E_F in the sample will find unoccupied states in the tip at E_F+eV (\varepsilon=eV), and so will be for all energies in-between. The tunneling current is therefore the sum of little contributions over all these energies of the product of three factors: ''2e\cdot\rho_S(E_F-eV+\varepsilon)\mathrm\varepsilon'' representing available electrons, f(E_F-eV+\varepsilon)-f(E_F+\varepsilon) for those that are allowed to tunnel, and the probability factor ''\Gamma'' for those that will actually tunnel. : I_t = \frac\int_^ (E_F -eV + \varepsilon) - f(E_F + \varepsilon)\, \rho_S (E_F - eV + \varepsilon) \, \rho_T (E_F + \varepsilon) \, , M, ^2 \, d \varepsilon . Typical experiments are run at a liquid helium temperature (around 4 K) at which the Fermi level cut-off of the electron population is less than a millielectronvolt wide. The allowed energies are only those between the two step-like Fermi levels, and the integral becomes : I_t = \frac \int_^ \rho_S (E_F - eV + \varepsilon) \, \rho_T (E_F + \varepsilon) \, , M, ^2 \, d \varepsilon . When the bias is small, it is reasonable to assume that the electron wave functions and, consequently, the tunneling matrix element do not change significantly in the narrow range of energies. Then the tunneling current is simply the convolution of the densities of states of the sample surface and the tip, : I_t \propto \int_^ \rho_S (E_F - eV + \varepsilon) \, \rho_T (E_F + \varepsilon) \, d \varepsilon . How the tunneling current depends on distance between the two electrodes is contained in the tunneling matrix element :M_= \int_\psi^S_\mu \,U_T\, ^* \textrmx\,\textrmy\,\textrmz . This formula can be transformed so that no explicit dependence on the potential remains. First, the U_T\, ^* part is taken out from the Schrödinger equation for the tip, and the elastic tunneling condition is used so that :M_= \int_ \left(^* E_\mu \psi^S_\mu+\psi^S_\mu\tfrac\tfrac^* \right) \textrmx\,\textrmy\,\textrmz . Now E_\mu\, is present in the Schrödinger equation for the sample, and equals the kinetic plus the potential operator acting on \psi^S_\mu . However, the potential part containing ''U''S is on the tip side of the barrier nearly zero. What remains, :M_= -\tfrac \int_ \left(^* \tfrac - \tfrac^* \right) \textrmx\,\textrmy\,\textrmz can be integrated over ''z'' because the integrand in the parentheses equals \partial_z \left( ^* \, \partial_z \psi^S_\mu - \, \partial_z ^* \right) . Bardeen's tunneling matrix element is an integral of the wave functions and their gradients over a surface separating the two planar electrodes, :M_= \tfrac \int_ \left( \tfrac^* - ^* \tfrac \right) \textrmx\,\textrmy . The exponential dependence of the tunneling current on the separation of the electrodes comes from the very wave functions that ''leak'' through the potential step at the surface and exhibit exponential decay into the classically forbidden region outside of the material. The tunneling matrix elements show appreciable energy dependence, which is such that tunneling from the upper end of the ''eV'' interval is nearly an order of magnitude more likely than tunneling from the states at its bottom. When the sample is biased positively, its unoccupied levels are probed as if the density of states of the tip is concentrated at its Fermi level. Conversely, when the sample is biased negatively, its occupied electronic states are probed but the spectrum of the electronic states of the tip dominates. In this case it is important that the density of states of the tip is as flat as possible. The results identical to Bardeen's can be obtained by considering adiabatic approach of the two electrodes and using the standard time-dependent perturbation theory. This leads to
Fermi's golden rule In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of ...
for the transition probability \Gamma_ in the form given above. Bardeen's model is for tunneling between two planar electrodes and does not explain scanning tunneling microscope's lateral resolution. Tersoff and Hamann used Bardeen's theory and modeled the tip as a structureless geometric point. This helped them disentangle the properties of the tip—which are hard to model—from the properties of the sample surface. The main result was that the tunneling current is proportional to the local density of states of the sample at the Fermi level taken at the position of the center of curvature of a spherically-symmetric tip (''s''-wave tip model). With such a simplification, their model proved valuable for interpreting images of surface features bigger than a nanometer, even though it predicted atomic-scale corrugations of less than a picometer. These are well below the microscope's detection limit and below the values actually observed in experiments. In sub-nanometer resolution experiments, the convolution of the tip and sample surface states will always be important, to the extent of the apparent inversion of the atomic corrugations that may be observed within the same scan. Such effects can only be explained by modeling of the surface and tip electronic states and the ways the two electrodes interact from
first principles In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nuan ...
.


Gallery of STM images

File:Scanning tunneling microscope (STM) 250 nm by 250 nm image of one-atom-thick silver islands grown on palladium (111) surface.png, One-atom-thick silver islands grown on terraces of the (111) surface of palladium. Image size is 250 nm by 250 nm. File:Atomic resolution Au100.JPG, The characteristic reconstruction fringes on the (100) surface of gold are 1.44
nanometers 330px, Different lengths as in respect to the molecular scale. The nanometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: nm) or nanometer (American and British English spelling differences#-re ...
wide and consist of six atomic rows that sit on top of five rows of the crystal bulk. Image size is approximately 10 nm by 10 nm. File:Chiraltube.png, A 7 nm long part of a single-walled
carbon nanotube A scanning tunneling microscopy image of a single-walled carbon nanotube Rotating single-walled zigzag carbon nanotube A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers. ''Single-wall carbon na ...
. File:Silicium-atomes.png, Atoms on the surface of a crystal of
silicon carbide Silicon carbide (SiC), also known as carborundum (), is a hard chemical compound containing silicon and carbon. A semiconductor, it occurs in nature as the extremely rare mineral moissanite, but has been mass-produced as a powder and crystal sin ...
(SiC) are arranged in a hexagonal lattice and are 0.3 nm apart. File:Cens nanomanipulation3d Trixler.jpg, STM nanomanipulation of PTCDA molecules on
graphite Graphite () is a crystalline form of the element carbon. It consists of stacked layers of graphene. Graphite occurs naturally and is the most stable form of carbon under standard conditions. Synthetic and natural graphite are consumed on large ...
to inscribe the logo of the
Center for NanoScience The Center for NanoScience (CeNS) was founded in 1998 at the Ludwig-Maximilians-University (LMU) in Munich. Its aim is to promote, bundle and join interdisciplinary research in the field of nanoscience The nanoscopic scale (or nanoscale) ...
(CeNS), Munich.


Early invention

An earlier invention similar to Binnig and Rohrer's, the ''Topografiner'' of R. Young, J. Ward, and F. Scire from the
NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
, relied on field emission. However, Young is credited by the Nobel Committee as the person who realized that it should be possible to achieve better resolution by using the tunnel effect.


Other related techniques

Many other microscopy techniques have been developed based upon STM. These include
photon scanning microscopy The operation of a photon scanning tunneling microscope (PSTM) is analogous to the operation of an electron scanning tunneling microscope, with the primary distinction being that PSTM involves tunneling of photons instead of electrons from the sa ...
(PSTM), which uses an optical tip to tunnel photons; scanning tunneling potentiometry (STP), which measures electric potential across a surface;
spin polarized scanning tunneling microscopy Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
(SPSTM), which uses a
ferromagnetic Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
tip to tunnel spin-polarized electrons into a magnetic sample;
multi-tip scanning tunneling microscopy Multi-tip scanning tunneling microscopy (Multi-tip STM) extends ''scanning tunneling microscopy'' (STM) from imaging to dedicated electrical measurements at the nanoscale like a ″multimeter at the nanoscale″. In materials science, nanoscience, ...
which enables electrical measurements to be performed at the nanoscale; and
atomic force microscopy Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the op ...
(AFM), in which the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
caused by interaction between the tip and sample is measured. STM can be used to manipulate atoms and change the topography of the sample. This is attractive for several reasons. Firstly the STM has an atomically precise positioning system which enables very accurate atomic scale manipulation. Furthermore, after the surface is modified by the tip, the same instrument can be used to image the resulting structures. IBM researchers famously developed a way to manipulate
xenon Xenon is a chemical element with the symbol Xe and atomic number 54. It is a dense, colorless, odorless noble gas found in Earth's atmosphere in trace amounts. Although generally unreactive, it can undergo a few chemical reactions such as the ...
atoms adsorbed on a
nickel Nickel is a chemical element with symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and ductile transition metal. Pure nickel is chemically reactive but large pieces are slow to ...
surface. This technique has been used to create electron ''corrals'' with a small number of adsorbed atoms, and observe
Friedel oscillations Friedel oscillations, named after French physicist Jacques Friedel, arise from localized perturbations in a metallic or semiconductor system caused by a defect in the Fermi gas or Fermi liquid. Friedel oscillations are a quantum mechanical analo ...
in the electron density on the surface of the substrate. Aside from modifying the actual sample surface, one can also use the STM to tunnel electrons into a layer of electron beam
photoresist A photoresist (also known simply as a resist) is a light-sensitive material used in several processes, such as photolithography and photoengraving, to form a patterned coating on a surface. This process is crucial in the electronic industry. T ...
on the sample, in order to do
lithography Lithography () is a planographic method of printing originally based on the immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by the German a ...
. This has the advantage of offering more control of the exposure than traditional
electron beam lithography Electron-beam lithography (often abbreviated as e-beam lithography, EBL) is the practice of scanning a focused beam of electrons to draw custom shapes on a surface covered with an electron-sensitive film called a resist (exposing). The electron b ...
. Another practical application of STM is atomic deposition of metals (gold, silver, tungsten, etc.) with any desired (pre-programmed) pattern, which can be used as contacts to nanodevices or as nanodevices themselves.


See also

*
Scanning probe microscopy Scan may refer to: Acronyms * Schedules for Clinical Assessment in Neuropsychiatry (SCAN), a psychiatric diagnostic tool developed by WHO * Shared Check Authorization Network (SCAN), a database of bad check writers and collection agency for bad ...
*
Atomic force microscope Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the diffr ...
*
Electrochemical scanning tunneling microscope The electrochemical scanning tunneling microscope (EC-STM) is a scanning tunneling microscope that measures the structures of surfaces and electrochemical reactions in solid-liquid interfaces at atomic or molecular scales. Development Electrochemi ...
*
Microscopy Microscopy is the technical field of using microscopes to view objects and areas of objects that cannot be seen with the naked eye (objects that are not within the resolution range of the normal eye). There are three well-known branches of micr ...
*
Electron microscope An electron microscope is a microscope that uses a beam of accelerated electrons as a source of illumination. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have a hi ...
*
Multi-tip scanning tunneling microscopy Multi-tip scanning tunneling microscopy (Multi-tip STM) extends ''scanning tunneling microscopy'' (STM) from imaging to dedicated electrical measurements at the nanoscale like a ″multimeter at the nanoscale″. In materials science, nanoscience, ...
*
IBM (atoms) IBM in atoms was a demonstration by IBM scientists in 1989 of a technology capable of manipulating individual atoms. A scanning tunneling microscope was used to arrange 35 individual xenon atoms on a substrate of chilled crystal of nickel to spel ...


References


Further reading

* * * * * * * * * * * * *


External links


A scanning tunelling microscope filmed during operation by an electron microscope


WeCanFigureThisOut.org

* ttp://toutestquantique.fr/en/microscopy/ Animations and explanations on various types of microscopes including electron microscopes(Université Paris Sud) {{DEFAULTSORT:Scanning Tunneling Microscope Scanning probe microscopy Swiss inventions German inventions Microscopes 1981 introductions Articles containing video clips 20th-century inventions