Experiment: Quantum Tunnelling Observed

Contents

The prediction under test
Alpha decay and the Geiger–Nuttall relation (1911, 1928)
Fowler and Nordheim: cold field emission from metals (1928)
Esaki: the tunnel diode and negative differential resistance (1958)
Giaever: tunnelling spectroscopy of the superconducting gap (1960)
Binnig and Rohrer: the scanning tunnelling microscope (1982)
Summary of the evidence

Elementary Quantum Systems makes a prediction that has no classical counterpart at all. A particle whose energy lies below the top of a potential barrier still appears on the far side of it, with a probability that falls exponentially in the width of the barrier and in the square root of its height. Nothing in Newtonian mechanics permits this: a classical particle turns around at the point where its kinetic energy vanishes, and the region beyond is not merely improbable but inaccessible. The quantum statement is not a small correction to that picture; it contradicts it outright.

This chapter reports the measurements. They are unusually varied for a single prediction. The barriers range from the Coulomb repulsion of a heavy nucleus, a few times $10^{-14}\,\mathrm{m}$ across and tens of megaelectronvolts high, to a vacuum gap of a few tenths of a nanometre and a few electronvolts high; the particles are alpha particles, electrons at a metal surface, electrons in a semiconductor band, and quasiparticles in a superconductor; the observables are half-lives, current–field curves, current–voltage curves, and pictures of individual atoms. Between the shortest and the longest alpha-decay half-life lies a factor of about $10^{24}$, and one exponential accounts for it. Two of these experiments, moreover, did not merely confirm the effect: they inverted it into instruments — tunnelling spectroscopy of the superconducting gap and the scanning tunnelling microscope — so that the least intuitive prediction of quantum mechanics is today a routine laboratory tool. The pair tunnelling of Cooper pairs across the same kind of junction is the Josephson effect, measured in Experiment: Superconductivity.

The prediction under test

Every experiment in this chapter measures the same quantity in a different guise, so it is worth writing it once. Let a particle of mass $m$ and energy $E$ meet a one-dimensional barrier $V(z)>E$ between the turning points $z_{1}$ and $z_{2}$. In the forbidden region the Schrödinger equation has no oscillating solution; the wavefunction decays with the local rate

\begin{equation}\tag{91.1} \kappa(z)=\frac{\sqrt{2m\left[V(z)-E\right]}}{\hbar}\ec \end{equation}

and the semiclassical transmission probability of Approximation Methods is

\begin{equation}\tag{91.2} T\simeq\ee^{-G}\ec\qquad G=2\int_{z_{1}}^{z_{2}}\kappa(z)\,\dd z\ep \end{equation}

The whole of what follows is the exponent $G$ evaluated for different barriers. Three features of Equation (91.2) make it testable, and each of the five experiments below turns on one or more of them.

First, $T$ is exponential in the barrier area $\int\kappa\,\dd z$, so a modest change in the barrier produces an enormous change in the rate. That is what makes the Geiger–Nuttall relation as steep as it is, and what makes the tunnel current a length gauge of picometre resolution.

Second, $T$ contains no temperature. A thermally activated process carries the Arrhenius factor $\ee^{-\Delta E/k_{\mathrm{B}}T}$ and dies as the sample is cooled; a tunnelling process does not. Cold emission from a metal and the peak current of a tunnel diode both survive cooling, and that is how they are distinguished from their classical competitors.

Third, the rate depends on the states available on the far side. This is invisible in the one-body formula but decisive in a solid: the current through a junction counts the empty states the particle can arrive in, so a tunnel junction reads out a density of states. Giaever's measurement of the superconducting gap and Esaki's negative differential resistance are both consequences.

Alpha decay and the Geiger–Nuttall relation (1911, 1928)

The oldest evidence for tunnelling was in hand for seventeen years before anyone could read it. Alpha radioactivity presented two facts that classical mechanics could not hold together: the alpha particle leaves the nucleus with an energy far below the top of the electrostatic barrier it must cross, and the time it takes to do so depends on that energy with a violence that no ordinary rate law approaches.

Apparatus

Two independent measurements are combined, both made with the standard radioactive-source technique of the period. The range of the alpha particles from a thin deposit of a separated activity was measured in air at known temperature and pressure, either by the ionization produced in a shallow chamber as an absorbing gap was widened, or by counting scintillations on a zinc-sulfide screen as the screen was withdrawn from the source; the range in air at standard conditions is a monotone function of the emission energy and was used as its proxy. The decay constant of the same activity was obtained from the decay of its ionization current with time, or, for the long-lived parents, from the equilibrium activity of a known mass. The sources were members of the three natural series — uranium, thorium and actinium — separated by the radiochemistry of the day [Geiger:1911].

The second, independent, apparatus is the alpha-scattering geometry of Nuclear Forces and Nuclear Structure: a collimated alpha beam from a strong source, a thin metal foil, and a scintillation screen viewed through a microscope on a rotating arm [Rutherford:1911]. It does not measure a decay rate at all; it measures the height of the barrier, and it is what turns the decay data into a paradox.

Procedure

For each emitter the range $R$ in air and the decay constant $\lambda$ were tabulated, and $\log\lambda$ plotted against $\log R$. The claim under test was a statistical one about the family: whether the members of a series, which differ in nuclear charge, mass and structure, follow a single relation between how fast they decay and how energetically.

The scattering measurement runs the process backwards. An alpha particle fired at a heavy nucleus is turned around by Coulomb repulsion at the distance of closest approach $d$, fixed by

\begin{equation}\tag{91.3} E=\frac{1}{4\pi\varepsilon_{0}}\frac{2Ze^{2}}{d}\ec \end{equation}

and the angular distribution follows the Coulomb law exactly, with no sign of the nuclear force, so long as $d$ exceeds the nuclear radius. The barrier top is therefore not a theoretical construct: it is measured by the very nuclei that emit the alpha particles.

Observations and data

Geiger and Nuttall found that the members of each natural series fall on a straight line in the $\left(\log R,\log\lambda\right)$ plane, and that the three series give three nearly parallel lines, differing only in intercept [Geiger:1911]. The lines are extraordinarily steep: the observed ranges in air span a factor of about three, from a few centimetres to under a decimetre, while the decay constants span more than twenty powers of ten.

Table 91.1 restates the same fact in modern terms for six familiar emitters, so that the steepness can be read directly. The energies and half-lives are standard tabulated values [Kondev:2021] quoted here for orientation, not measurements of this chapter, and are rounded; Geiger and Nuttall themselves worked with ranges rather than energies.

Emitter	$E_{\alpha}$	half-life
$^{232}$Th	$4.01\,\mathrm{MeV}$	$1.40\times 10^{10}\,\mathrm{yr}$
$^{238}$U	$4.20\,\mathrm{MeV}$	$4.47\times 10^{9}\,\mathrm{yr}$
$^{226}$Ra	$4.78\,\mathrm{MeV}$	$1600\,\mathrm{yr}$
$^{222}$Rn	$5.49\,\mathrm{MeV}$	$3.30\times 10^{5}\,\mathrm{s}$
$^{214}$Po	$7.69\,\mathrm{MeV}$	$1.64\times 10^{-4}\,\mathrm{s}$
$^{212}$Po	$8.78\,\mathrm{MeV}$	$3.0\times 10^{-7}\,\mathrm{s}$

Alpha energy and half-life for six natural alpha emitters. A factor of $2.2$ in energy carries the half-life across some $24$ orders of magnitude — the observation that Equation (91.4) has to explain. Values are standard tabulated ones [Kondev:2021], rounded, and are not drawn from [Geiger:1911], which reports ranges in air.

The scattering data supply the second number. For a heavy nucleus the Coulomb barrier evaluated at the nuclear surface reaches several times the energy of the alpha particles that the same nucleus emits, and the Coulomb scattering law is observed to hold down to distances of closest approach well inside the classical turning point of an emitted alpha [Rutherford:1911].

Phenomenon 91.1 (Emission from below the barrier).

An alpha-emitting nucleus emits alpha particles whose energy lies far below the top of its own Coulomb barrier, as that barrier is measured independently by the scattering of alpha particles from the same nuclei [Rutherford:1911]. There is no classical trajectory by which the particle can leave, and the nucleus decays regardless [Geiger:1911] [Gamow:1928].

Derivation. The inequality is established without any theory of the decay, by the scattering measurement alone. Fire alpha particles of energy $E_{\text{s}}$ at nuclei of the species in question. If the observed angular distribution follows the Coulomb law, the two charges never came closer than the distance $d$ of Equation (91.3) evaluated at $E_{\text{s}}$, and no departure from that law is seen for the fastest natural alpha particles on the heaviest nuclei [Rutherford:1911]. The nuclear surface therefore lies inside that distance, $R_{\mathrm{n}}<d$; and since the Coulomb potential decreases monotonically outward,

\[ V\left(R_{\mathrm{n}}\right)>V\left(d\right)=E_{\text{s}}\ep \]

The barrier top thus exceeds the energy of the fastest alpha particle available for scattering, which in turn exceeds the emission energy of the great majority of the natural emitters of Table 91.1. The classical turning point of an emitted alpha lies outside the nucleus that emitted it, with a forbidden region in between. That the nucleus decays all the same is the observation; the rate at which it does so is Phenomenon 91.2, and both are accounted for by Equation (91.4).

∎

Phenomenon 91.2 (The Geiger–Nuttall relation).

Within each of the three natural radioactive series the decay constant and the range of the emitted alpha particle satisfy $\log\lambda=A+B\log R$ with a slope $B$ common to the three series and an intercept $A$ characteristic of each [Geiger:1911]. Equivalently, the logarithm of the decay constant is linear in the reciprocal square root of the alpha energy. Across the natural emitters a factor of about two in energy moves the half-life by more than twenty orders of magnitude.

Derivation. Outside the nuclear radius $R_{\mathrm{n}}$ the alpha particle of charge $2e$ sees only the Coulomb field of the daughter nucleus of charge $Ze$,

\[ V(r)=\frac{1}{4\pi\varepsilon_{0}}\frac{2Ze^{2}}{r}\ec \]

which exceeds the emission energy $E$ everywhere inside the outer turning point $b$ defined by $V(b)=E$. Substituting in Equations (91.1) and (91.2) and writing $b=2Ze^{2}/ \left(4\pi\varepsilon_{0}E\right)$,

\[ G=\frac{2}{\hbar}\int_{R_{\mathrm{n}}}^{b} \sqrt{2m\left(\frac{2Ze^{2}}{4\pi\varepsilon_{0}r}-E\right)}\,\dd r =\frac{2\sqrt{2mE}}{\hbar}\int_{R_{\mathrm{n}}}^{b} \sqrt{\frac{b}{r}-1}\,\dd r\ep \]

The remaining integral is elementary. With $x=R_{\mathrm{n}}/b$,

\begin{equation}\tag{91.4} G=\frac{2b\sqrt{2mE}}{\hbar} \left[\arccos\sqrt{x}-\sqrt{x\left(1-x\right)}\right]\ep \end{equation}

For a barrier much wider than the nucleus, $x\to0$, the bracket tends to $\pi/2$ and

\begin{equation}\tag{91.5} G\longrightarrow2\pi\eta\ec\qquad \eta=\frac{1}{4\pi\varepsilon_{0}}\frac{2Ze^{2}}{\hbar v}\ec\qquad v=\sqrt{\frac{2E}{m}}\ep \end{equation}

The decay constant is the number of assaults on the barrier per unit time, $f$, times the probability that one succeeds, so

\[ \ln\lambda=\ln f-G\ec \]

and since $\eta\propto Z/\sqrt{E}$ the logarithm of the decay constant is linear in $ZE^{-1/2}$. Within a single series $Z$ varies only slightly, and the statement collapses to the empirical relation of Phenomenon 91.2, the range in air being a monotone function of $E$. The prefactor $f$ is a nuclear frequency of order $v/2R_{\mathrm{n}}$ and varies by a factor of a few across the whole family; it cannot produce twenty orders of magnitude, and it does not need to, because $G$ does.

∎

The size of $G$ is what carries the argument. For the alpha decay of $^{238}$U the velocity parameter is $v/c\approx0.047$ and the daughter charge $Z=90$, so $\eta\approx28$ and the point-Coulomb exponent $2\pi\eta$ of Equation (91.5) is of order $170$; the finite nuclear radius, through the bracket of Equation (91.4), roughly halves it. An exponent of order $10^{2}$ that scales as $E^{-1/2}$ changes by tens when the energy changes by a factor of two, and $\ee^{-G}$ therefore changes by tens of orders of magnitude. This is the first quantitative success of quantum mechanics inside the nucleus, and it was obtained twice over within months: by Gamow [Gamow:1928] and, independently, by Gurney and Condon [Gurney:1928] [Gurney:1929].

Interpretation

The two phenomena above are irreconcilable in classical mechanics and are two readings of one formula in quantum mechanics. Phenomenon 91.1 says the particle is where it cannot be; Phenomenon 91.2 says the rate at which it gets there is exponential in the barrier, which is the signature of Equation (91.2) and of nothing else. No rate law built on a classical escape over the barrier can reproduce the observed steepness with a reasonable nuclear temperature, and the alternative of a barrier low enough to be climbed is excluded by the scattering measurement that fixes its height.

Two consequences reach far beyond this chapter. Run the same penetration factor backwards and it governs the fusion of light nuclei at stellar temperatures, where the thermal energies are likewise far below the Coulomb barrier: this is the Gamow-peak argument of Stellar Structure and Nucleosynthesis, first drawn by Atkinson and Houtermans within a year of Gamow's paper [Atkinson:1929]. And the exponential sensitivity is the reason alpha half-lives are useful chronometers over geological time, since a rate this steep in energy is correspondingly insensitive to everything else, including the chemical and thermal environment.

A caution belongs here about attribution. Gurney and Condon's short note in Nature [Gurney:1928] and Gamow's paper [Gamow:1928] appeared within weeks of one another, and the full Gurney–Condon treatment followed the next year [Gurney:1929]. The qualitative insight — that radioactive decay is barrier penetration and needs no trigger — is common to both; the explicit evaluation of Equation (91.4) and its confrontation with the Geiger–Nuttall systematics is Gamow's.

Primary references

[Geiger:1911] for the empirical relation; [Rutherford:1911] for the independent measurement of the barrier; [Gamow:1928] [Gurney:1928] [Gurney:1929] for the explanation; [Atkinson:1929] for the stellar inverse. The energies and half-lives of Table 91.1 are standard tabulated values quoted at second hand and are not taken from any of these papers.

Fowler and Nordheim: cold field emission from metals (1928)

A metal emits electrons when it is hot: the thermionic current is the tail of the Fermi distribution spilling over the work function, and it vanishes as the metal is cooled. A metal also emits electrons when it is cold, if the field at its surface is strong enough — and that current does not vanish as the metal is cooled. The two emissions obey different laws, and the difference is the whole of the evidence.

Apparatus

A sharply pointed metal cathode — a wire etched to a tip of radius of order $100\,\mathrm{nm}$, tungsten being the usual choice — facing an anode in a sealed and evacuated tube. The geometry is the point of the apparatus: the field at a tip of radius $r$ held at potential $V$ is of order $V/r$, so a few kilovolts across a centimetre gap becomes several volts per nanometre at the apex, some four orders of magnitude larger than the mean field in the tube. The emitted current is collected at the anode and measured over several decades with an electrometer; the cathode can be heated independently, so that the emission can be followed as a function of temperature at fixed field.

Procedure

The emitted current $I$ is recorded against the applied voltage, and converted to a current density $J$ and a surface field $F$ through the tip geometry. Two tests are then applied. The first is the shape of the curve: the prediction is that $\ln\left(J/F^{2}\right)$ is linear in $1/F$, a form now universally called a Fowler–Nordheim plot, with a slope fixed by the work function alone. The second is the temperature dependence at fixed field, which separates the two mechanisms decisively: thermionic emission is exponential in $-1/T$ and field emission is, to first approximation, independent of $T$.

Observations and data

Cold metals emit copiously once the surface field reaches the order of $10^{9}\,\mathrm{V}/\mathrm{m}$, that is a few volts across a nanometre, at temperatures where thermionic emission is unmeasurably small. Plotted as $\ln\left(J/F^{2}\right)$ against $1/F$ the measured curves are straight over several decades of current, and the slopes are consistent with work functions of a few electronvolts — the same work functions that the photoelectric threshold of The Photon: Photoelectric and Compton Effects measures on the same metals. Fowler and Nordheim analysed the existing cold-emission data on exactly this basis and obtained the law from the new wave mechanics [Fowler:1928].

Phenomenon 91.3 (Cold field emission).

A metal at room temperature or below emits electrons into vacuum when the field at its surface reaches the order of $10^{9}\,\mathrm{V}/\mathrm{m}$, although no electron in it has the energy to surmount the work function. The emitted current density obeys

\begin{equation}\tag{91.6} J=\frac{a\,F^{2}}{\phi} \exp\left(-\frac{4\sqrt{2m_{\mathrm{e}}}\,\phi^{3/2}} {3\hbar eF}\right)\ec \end{equation}

so that $\ln\left(J/F^{2}\right)$ is linear in $1/F$ with a slope fixed by the work function $\phi$ alone, and the current is essentially independent of temperature [Fowler:1928].

Derivation. Put the metal surface at $z=0$, the electron gas filled to the Fermi level, and the vacuum level a work function $\phi$ above it. An applied field $F$ tilts the vacuum potential, so that an electron at the Fermi level sees

\[ V(z)-E_{\mathrm{F}}=\phi-eFz\qquad\left(z>0\right)\ec \]

a triangular barrier of height $\phi$ and base $\phi/eF$. The barrier does not have to be surmounted, only crossed, and Equation (91.2) gives

\[ G=\frac{2}{\hbar}\int_{0}^{\phi/eF} \sqrt{2m_{\mathrm{e}}\left(\phi-eFz\right)}\,\dd z =\frac{4\sqrt{2m_{\mathrm{e}}}}{3\hbar eF}\,\phi^{3/2}\ep \]

Multiplying $\ee^{-G}$ by the rate at which electrons of the degenerate gas arrive at the surface — an integral over the occupied states which is algebraic, not exponential, in $F$ and contributes the factor $F^{2}/\phi$ — gives Equation (91.6). Taking logarithms,

\[ \ln\frac{J}{F^{2}}=\ln\frac{a}{\phi} -\frac{4\sqrt{2m_{\mathrm{e}}}\,\phi^{3/2}}{3\hbar e}\cdot\frac{1}{F}\ec \]

which is the straight line the experiment plots. Note what is absent: the temperature never entered, because the electrons that cross are those at the Fermi level, not those in the thermal tail. Note also the field required. The exponent falls to order $10$ only when $F$ reaches some $10^{9}\,\mathrm{V}/\mathrm{m}$ for a work function of a few electronvolts, which is why cold emission is a phenomenon of sharp points and thin gaps and was never seen from a flat electrode.

∎

Interpretation

This is the first tunnelling experiment in which the barrier is under the experimenter's control. In alpha decay the barrier is whatever the nucleus provides; here the height is chosen by choosing the metal and the width is set by turning a voltage knob, and the predicted functional form in both variables can be checked. The straightness of the Fowler–Nordheim plot tests the exponent's dependence on $F$; the slope tests its dependence on $\phi^{3/2}$ against an independent photoelectric measurement of $\phi$; and the absence of temperature dependence excludes the thermionic alternative outright.

The historical position of [Fowler:1928] is worth stating plainly. It is one of the earliest quantitative applications of the new quantum mechanics to a solid, and it rests on Sommerfeld's Fermi-statistics treatment of the electron gas [Sommerfeld:1928], published the same year. The pre-quantum theories of cold emission, which sought to lower the barrier until classical escape became possible, could not survive a measurement of $\phi$ by any other route.

Two long descendants follow. The field-emission electron source, being cold, bright and small, became the standard gun of high-resolution electron microscopy; and the recognition that a vacuum gap is simply a barrier whose width can be varied continuously is the seed of Section 91.6.

Primary references

[Fowler:1928]; the underlying electron-gas statistics [Sommerfeld:1928]. The empirical finding that the logarithm of the cold-emission current is linear in the reciprocal field predates the theory and is due to Millikan and his collaborators [Millikan:1928]; it is quoted here at second hand through [Fowler:1928], which analyses that data.

Esaki: the tunnel diode and negative differential resistance (1958)

Tunnelling in a solid need not cross a vacuum gap. In a semiconductor the forbidden energy gap is itself a barrier, and a p–n junction narrow enough turns it into one a carrier can cross. What Esaki found in doing so was a current that falls as the voltage rises — a negative differential resistance, impossible for any junction whose carriers move by drift and diffusion.

Apparatus

Germanium p–n junctions doped degenerately on both sides, so that the Fermi level lies inside the conduction band on the n-side and inside the valence band on the p-side. Degenerate doping is what makes the experiment: the depletion layer of a junction narrows as the doping rises, and at the impurity concentrations used the junction is of order $10\,\mathrm{nm}$ wide — thin enough for Equation (91.2) to give an appreciable transmission across the forbidden gap. The junctions were made by alloying, contacted for four-terminal measurement, and mounted in a cryostat so that the characteristic could be taken at several temperatures [Esaki:1958].

Procedure

The current–voltage characteristic is recorded through forward and reverse bias, at room temperature and on cooling. Three features are looked for. In forward bias, the ordinary junction current is thermal injection over the built-in barrier and rises monotonically and exponentially; any departure from monotonicity is disqualifying for that mechanism. In reverse bias, an ordinary junction passes only a small saturation current until avalanche breakdown; a tunnelling junction conducts freely from the smallest reverse bias. And on cooling, the thermal current must collapse while a tunnel current must not.

Observations and data

In forward bias the current rises steeply from zero to a peak at a bias of a few tens of millivolts, then falls with increasing voltage to a valley at a few hundred millivolts, and only then resumes the ordinary exponential climb of a diode. Between peak and valley the differential resistance $\dv{V}{I}$ is negative. In reverse bias the junction conducts immediately, with no threshold. The peak current is weakly dependent on temperature while the current beyond the valley follows the strongly temperature-dependent thermal law, so that cooling the sample makes the negative-resistance region more pronounced rather than less [Esaki:1958].

Phenomenon 91.4 (Negative differential resistance in a degenerate junction).

A p–n junction degenerate on both sides and of order $10\,\mathrm{nm}$ wide passes, in forward bias, a current that rises to a peak at a bias of a few tens of millivolts and then decreases with increasing voltage before rising again: over a finite range of bias the differential resistance is negative. The effect is weakly temperature-dependent, unlike the thermal injection current that dominates beyond the valley [Esaki:1958].

Derivation. Elastic tunnelling conserves energy, so a carrier crosses only into an empty state at its own energy; the current is therefore proportional to the overlap, in energy, between occupied states on one side and empty states on the other, weighted by the transmission $\ee^{-G}$ of the junction.

Degeneracy on both sides is what creates such an overlap. On the n-side the conduction band is occupied up to a Fermi level lying above its edge; on the p-side the valence band is empty down to a Fermi level lying below its edge. At zero bias the two Fermi levels coincide, the forward and reverse flows cancel, and no net current flows. A forward bias $V$ raises the whole n-side ladder by $eV$, opening a window of width $eV$ in which occupied conduction states on the n-side face empty valence states on the p-side at the same energy: the tunnel current grows with $V$.

It cannot grow indefinitely. As the n-side rises further, its conduction-band edge is lifted past the p-side valence-band edge, and an electron in the window then faces the forbidden gap of the p-side, where there are no states at its energy at all. The overlap shrinks to zero, and with it the tunnel current. A current that rises from zero, is positive in between, and returns to zero must pass through a maximum and then decrease — and a decreasing current at increasing voltage is negative differential resistance. Beyond the cut-off, only the ordinary thermal injection current of a p–n junction remains, and its exponential climb produces the valley and the recovery.

The reverse-bias behaviour is the same argument with the roles exchanged: reverse bias always increases the overlap between occupied valence states and empty conduction states, so the current rises monotonically from zero bias with no threshold, which is what distinguishes this junction from an ordinary rectifier.

∎

Derivation pending.

Magnitude of the interband tunnel current: the transmission of a carrier across the forbidden gap of a uniformly biased semiconductor, the imaginary wavevector inside the gap obtained by continuing the two-band dispersion relation, and the resulting exponential dependence of the peak current on the gap, the effective mass and the junction field

Interpretation

The observation of Phenomenon 91.4 is decisive because negative differential resistance is not a quantitative anomaly but a qualitative impossibility for transport by drift and diffusion, in which carrier flow increases with the driving field by construction. Its appearance requires a mechanism in which raising the bias can remove the final states, and interband tunnelling is such a mechanism because it is elastic and the semiconductor gap is empty of states. The temperature behaviour closes the argument: a thermally activated process cannot become more prominent on cooling.

The mechanism itself was not new in 1958 — Zener had proposed interband tunnelling as the cause of dielectric breakdown a quarter of a century earlier [Zener:1934] — but the breakdown context gave no clean signature, since a runaway current can be produced in several ways. Esaki's contribution was to make a junction narrow enough that tunnelling dominates at low bias, where its consequences are unmistakable, and to read off from the shape of the characteristic that the current is a density-of-states overlap.

The device consequences were immediate and are treated with the band theory of Electrons in Solids: Band Theory and the junction physics of Semiconductors: a tunnel current involves no storage of minority carriers and no transit time across a drift region, so the tunnel diode responds far faster than a conventional junction, and a negative resistance is an amplifier and an oscillator. For this chapter its importance is evidential — it is tunnelling observed in a solid, at millivolt energies, with the density of states of the far side written into the shape of the curve.

Primary references

[Esaki:1958]; the interband tunnelling mechanism [Zener:1934].

Giaever: tunnelling spectroscopy of the superconducting gap (1960)

Esaki's junction reads out the states available on the far side of a barrier. Giaever turned that incidental feature into the measurement itself: if the tunnel current counts final states, then a junction with a superconductor on one side is a spectrometer for the superconducting energy gap — a quantity of order a millielectronvolt, far below anything optical spectroscopy could then reach.

Apparatus

A thin-film sandwich junction evaporated on a glass substrate. An aluminium strip is deposited first and its surface oxidized in air or in an oxygen glow discharge to grow an insulating film of aluminium oxide a few nanometres thick; a lead strip is then evaporated across it, the crossing defining the junction area. The oxide is the barrier; its thickness is what sets the junction resistance, and it must be thin enough to pass a measurable current and free enough of pinholes to pass it by tunnelling rather than through metallic shorts. Contacts at the four ends of the two strips allow a four-terminal measurement, which keeps the lead resistances out of the answer. The assembly sits in a liquid-helium cryostat, and a solenoid supplies a magnetic field large enough to drive the lead film normal [Giaever:1960].

The choice of metals is deliberate. Lead becomes superconducting at about $7.2\,\mathrm{K}$, aluminium at about $1.2\,\mathrm{K}$. At $4.2\,\mathrm{K}$ the lead is superconducting and the aluminium is not, so the junction has one superconducting and one normal electrode; and the superconductivity can be switched off in place, without changing the junction, by applying the magnetic field.

Procedure

The current–voltage characteristic of the junction is recorded in two states of the same sample: with the lead film normal, and with it superconducting. The comparison is the experiment. Any feature that appears in the second curve and not in the first belongs to the superconducting state and not to the barrier, the geometry or the contacts — an internal control of exactly the kind an oxide barrier demands, since a pinhole or a thermal effect would show in both. The measurement is then repeated at several temperatures between the base temperature and the transition, so that the temperature dependence of any threshold can be followed.

Observations and data

With both films normal the junction is ohmic: current proportional to voltage over the range of interest, which is itself the signature of tunnelling through a thin barrier rather than conduction through metal bridges. With the lead superconducting the characteristic changes completely at low bias. The current is strongly suppressed — reduced to a small residual set by the temperature — until the bias exceeds a threshold, above which the curve rises steeply and rejoins the normal line, displaced from it by roughly the threshold voltage. The threshold is of order a millivolt, corresponding to an energy of order $1\,\mathrm{meV}$. On warming, the threshold shrinks and vanishes at the transition temperature of the lead film [Giaever:1960].

Phenomenon 91.5 (A tunnel junction measures the superconducting gap).

In a normal-metal–insulator–superconductor tunnel junction the current is suppressed below a threshold bias and the differential conductance above it maps out the excitation spectrum of the superconductor. The threshold measures the energy gap directly; for lead it corresponds to an energy of the order of a millielectronvolt, and it closes continuously as the temperature approaches the transition temperature [Giaever:1960].

Derivation pending.

The tunnel current between a normal metal and a superconductor: the elastic-tunnelling current as the convolution of the two densities of states with the difference of the two Fermi functions [Bardeen:1961], the BCS quasiparticle density of states with its square-root edge at the gap, and the consequence that the differential conductance at low temperature is proportional to that density of states evaluated at the bias energy

The scale is set independently by the theory of Superconductivity and Superfluidity. The weak-coupling BCS relation between the gap and the transition temperature, $2\Delta\left(0\right)=3.53\,k_{\mathrm{B}}T_{\mathrm{c}}$, gives about $2.2\,\mathrm{meV}$ for a transition temperature of $7.2\,\mathrm{K}$; lead is a strong-coupling superconductor and its actual gap lies somewhat above the weak-coupling value. The measured threshold is of that order, which is the point: a prediction of the theory published three years earlier [Bardeen:1957] was confronted with a direct measurement of the quantity it predicts.

Interpretation

Two results are obtained at once, and they support each other. The first is about tunnelling: the current through the junction is not a property of the barrier alone but of the states on either side of it, because switching the far electrode between two states of the same metal — with the barrier untouched — changes the characteristic qualitatively. The second is about superconductivity: the excitation spectrum has a gap, of the predicted size, closing at the transition temperature.

The methodological gain is larger than either. A tunnel junction whose differential conductance is proportional to a density of states is a spectrometer of resolution set by temperature, and at a few kelvin that resolution is a fraction of a millielectronvolt — three orders of magnitude finer than optical spectroscopy of the period. Tunnelling spectroscopy went on to measure the phonon structure imprinted on the superconducting gap, and so to confirm that the pairing interaction is the lattice one.

This experiment also sets the stage for the next step in the same apparatus. Replace the normal electrode with a second superconductor and the single-particle current acquires a larger threshold [Giaever:1960b]; but a second current, carried by pairs and flowing at zero bias, appears alongside it. That is the Josephson effect, predicted in [Josephson:1962] and observed in [Anderson:1963a], and it is measured in Experiment: Superconductivity rather than here, because its content is the macroscopic phase rather than barrier penetration as such.

Primary references

[Giaever:1960]; the gap prediction [Bardeen:1957]; the pair tunnelling that follows from the same junction [Josephson:1962] [Anderson:1963a].

Binnig and Rohrer: the scanning tunnelling microscope (1982)

The last experiment in this chapter is the one in which tunnelling stops being the thing measured and becomes the thing measuring. The transmission through a vacuum gap falls by about an order of magnitude for every $0.1\,\mathrm{nm}$ of extra width. Read forwards, that is a prediction to be tested; read backwards, it is a displacement transducer of picometre sensitivity, and holding the current constant while a tip is scanned holds the gap constant to the same precision.

Apparatus

A sharp metal tip mounted on a piezoelectric tripod, which translates it in three orthogonal directions with sub-picometre response to an applied voltage, facing the sample across a vacuum gap of a few tenths of a nanometre. A coarse approach mechanism — in the original instrument a piezoelectric walker, since the gap must be closed from millimetres to angstroms without collision — brings the tip within tunnelling range. A bias of order a volt is applied between tip and sample and the tunnel current, of order a nanoampere, is amplified and compared with a set point; the error signal drives the vertical piezo through a feedback loop. The whole assembly is in ultra-high vacuum and, critically, isolated from vibration: the first instrument floated its stage magnetically above a superconducting bowl, and later designs used stacked springs with eddy-current damping [Binnig:1982].

The vibration requirement follows from the very sensitivity that makes the instrument work. If the current is to be held to a few percent, the gap must be held to a few picometres, and a building vibrates by far more than that.

Procedure

Two measurements are made with the same hardware. The first tests the prediction: with the feedback loop open, the tunnel current is recorded as the tip is advanced or retracted by a known piezoelectric displacement, and $\ln I$ is plotted against the displacement. The slope gives $2\kappa$ of Equation (91.1), and hence an apparent barrier height that can be compared with the work functions of the two electrodes — a control that also detects contamination, since an adsorbed layer lowers the apparent barrier far below the clean-surface value.

The second is imaging. The feedback loop is closed, holding the current and therefore the gap constant, and the tip is raster-scanned across the surface. The voltage applied to the vertical piezo, recorded as a function of the lateral position, is the image: a contour of constant tunnel current, which is a contour of constant gap over a homogeneous surface.

Observations and data

The tunnel current across a vacuum gap is measurable and reproducible, and it varies exponentially with the gap width at a rate corresponding to an apparent barrier height of a few electronvolts — the order of a metal work function, as Equation (91.1) requires. Scanned images resolve monatomic steps, a few tenths of a nanometre in height, as clean terraces, and resolve surface reconstructions on the metal and intermetallic surfaces examined [Binnig:1982]. Within a year the same instrument resolved the seven-by-seven reconstruction of the silicon (111) surface in real space [Binnig:1983], ending a long-standing crystallographic controversy.

Phenomenon 91.6 (Vacuum tunnelling made an instrument).

A current flows between two electrodes separated by a vacuum gap that no electron of the available energy can cross classically, and its logarithm is linear in the gap width with a slope corresponding to a barrier of the order of the work function. Holding that current constant while a tip is scanned therefore holds the gap constant to a few picometres, which is sufficient to image the topography of a surface at the atomic scale [Binnig:1982].

Derivation. The gap is a barrier of height equal to the work function $\phi$ and width equal to the separation $d$, so Equation (91.1) gives a decay rate $\kappa=\sqrt{2m_{\mathrm{e}}\phi}/\hbar$ that does not depend on $d$ at all, and Equation (91.2) gives $I\propto\ee^{-2\kappa d}$: the whole width dependence of the current is that of the squared amplitude of the decaying solution at the far electrode. This is the calculation carried out in Elementary Quantum Systems, and its numerical content is what the instrument exploits. For a clean metal of work function about $4\,\mathrm{eV}$ the rate is $\kappa\approx10^{10}\,/\mathrm{m}$, so that $2\kappa d\approx2$ for $d=0.1\,\mathrm{nm}$ and the current changes by roughly a factor $\ee^{-2}$ per tenth of a nanometre. Inverting the relation gives the instrument: holding $I$ to $1\,\mathrm{\%}$ holds $d$ to $\delta d=\delta I/\left(2\kappa I\right)$, which is about half a picometre. What this one-dimensional argument does not supply is the relation between the current and the surface it is imaging, and that is what the image requires.

∎

Derivation pending.

The tunnel current as a probe of the local density of states: the transfer-Hamiltonian expression for the current between two weakly coupled electrodes in terms of the overlap of their surface wavefunctions [Bardeen:1961], and its reduction, for a tip modelled by a single spherically symmetric orbital, to the local density of states of the sample at the position of the tip and at the bias energy [Tersoff:1985]

Interpretation

The measurement of the apparent barrier height is the physics; the images are its proof by demonstration. An exponential current–distance relation with a decay rate matching the work function establishes that the transport is vacuum tunnelling and not field emission, contact conduction, or an adsorbate bridge — each of which would give a different slope or no exponential at all. That the same relation, once established, yields atomic-scale images is a much stronger statement than any single I–d curve, because it requires the exponential to hold locally, reproducibly, and over a whole surface.

It is worth stating what has happened here in the terms of Section 91.1. The exponential sensitivity that made tunnelling seem an exotic curiosity in alpha decay — a rate too small to matter, too steep to control — is the same property that makes the scanning tunnelling microscope work. Steepness is precision when the exponent is a length you wish to measure. The most counterintuitive prediction in Elementary Quantum Systems became, within sixty years of being written down, the finest length gauge in surface physics and the basis of an entire family of scanned-probe instruments.

Primary references

[Binnig:1982].

Summary of the evidence

Experiment	Barrier	Decisive observation
Geiger–Nuttall 1911, explained by Gamow and by Gurney and Condon 1928	Coulomb repulsion of the daughter nucleus; tens of $\mathrm{MeV}$ high, a few times $10^{-14}\,\mathrm{m}$ wide	half-life exponential in $ZE^{-1/2}$, spanning some $24$ orders of magnitude while the energy changes by a factor of two
Fowler–Nordheim 1928	work function of a metal surface tilted by an applied field; a few $\mathrm{eV}$ high, of order $10^{-9}\,\mathrm{m}$ wide	$\ln\left(J/F^{2}\right)$ linear in $1/F$ with slope fixed by $\phi^{3/2}$, and no temperature dependence
Esaki 1958	forbidden gap of a degenerate germanium junction; of order $1\,\mathrm{eV}$ high, of order $10^{-8}\,\mathrm{m}$ wide	negative differential resistance in forward bias, impossible for drift-diffusion transport
Giaever 1960	aluminium-oxide film a few nanometres thick	current suppressed below a threshold of order $1\,\mathrm{meV}$ that closes at the transition temperature of the lead film
Binnig–Rohrer 1982	vacuum gap of a few tenths of a nanometre; a few $\mathrm{eV}$ high	current exponential in the gap at the work-function rate; constant- current scanning images monatomic steps

Tests of barrier penetration reported in this chapter. The barrier column gives its physical origin and rough scale; the last column gives the observable that no classical mechanism reproduces. The five span some seven orders of magnitude in barrier height and five in barrier width, and one exponential describes all of them.

The entries in Table 91.2 have almost nothing in common at the level of the apparatus. They were performed seventy years apart, by nuclear physicists, vacuum physicists, semiconductor engineers, low-temperature physicists and surface scientists, on barriers differing by seven orders of magnitude in height. What they share is Equation (91.2): in every case the measured quantity is exponential in $\int\kappa\,\dd z$ across a region the particle classically cannot enter, with the mass, the height and the width entering exactly as Equation (91.1) prescribes and the temperature not entering at all.

That last point deserves the final word, because it is what makes the case unavoidable. Every classical account of these observations has to supply the particle with the missing energy from somewhere, and the only available reservoir is thermal. But the alpha half-life of a nucleus is insensitive to the temperature of the sample; cold field emission persists at temperatures where thermionic emission is unmeasurable; the tunnel diode's peak current strengthens relative to its surroundings on cooling; Giaever's threshold and Binnig and Rohrer's images are obtained at liquid-helium temperatures and below. The classically forbidden region is entered, and it is not entered by borrowing heat. It is entered because the wavefunction of The Postulates of Quantum Mechanics does not vanish there, and Equation (91.2) says by how much.

Emitter	$E_{\alpha}$	half-life
$^{232}$Th	\(4.01\,\mathrm{MeV}\)	\(1.40\times 10^{10}\,\mathrm{yr}\)
$^{238}$U	\(4.20\,\mathrm{MeV}\)	\(4.47\times 10^{9}\,\mathrm{yr}\)
$^{226}$Ra	\(4.78\,\mathrm{MeV}\)	\(1600\,\mathrm{yr}\)
$^{222}$Rn	\(5.49\,\mathrm{MeV}\)	\(3.30\times 10^{5}\,\mathrm{s}\)
$^{214}$Po	\(7.69\,\mathrm{MeV}\)	\(1.64\times 10^{-4}\,\mathrm{s}\)
$^{212}$Po	\(8.78\,\mathrm{MeV}\)	\(3.0\times 10^{-7}\,\mathrm{s}\)