Thinking in Waves Episode 17 ／ Part VIII · Differentiating the effective constants
Mass → speed → mass
Effective mass ──
the reciprocal of mass was a curvature
Expand the dispersion relation \(\omega(k)\) about \(k=0\) and you get ──
mass at order 0, speed at order 1, and mass again at order 2.
★ One word was pointing at different features of the same curve.
In Episode 6 mass was a cutoff frequency. In Episode 16 the speed of light was a slope.
Both were places on the same curve \(\omega(k)\). So let us keep going.
01Expand the dispersion relation
| Order | Taylor coefficient | Physical meaning | Where it appeared |
|---|---|---|---|
| 0 | \(\omega_0\) | ★ rest mass (cutoff \(mc^2/\hbar\)) | Episode 6 |
| 1 | \(d\omega/dk\) | ★ group velocity (effective light speed) | Episode 16 |
| 2 | \(d^2\omega/dk^2\) | ★ reciprocal effective mass \(\hbar/m^*\) | this episode |
| 4 | \(d^4\omega/dk^4\) | relativistic correction, non-parabolicity | §02 |
The heart of §01
mass → speed → mass
Orders 0 and 2 are both called "mass". One word, two different features of a curve.
Written out, the definition of effective mass is ──
$$\frac{1}{m^*}=\frac{1}{\hbar^2}\frac{d^2E}{dk^2}$$
★ The reciprocal of mass is the curvature of the dispersion relation.
02Check it with relativity first ── orders 0 and 2
| \(v/c\) | Exact \(E\) [keV] | to order 2 | error | to order 4 | error |
|---|---|---|---|---|---|
| 0.001 | 510.9992 | 510.9992 | \(1.3\times10^{-13}\) | 510.9992 | \(0\) |
| 0.01 | 511.0245 | 511.0245 | \(1.3\times10^{-9}\) | 511.0245 | \(6.3\times10^{-14}\) |
| 0.1 | 513.5733 | 513.5798 | \(1.3\times10^{-5}\) | 513.5732 | \(6.4\times10^{-8}\) |
| 0.3 | 535.6725 | 536.2681 | \(1.1\times10^{-3}\) | 535.6433 | \(5.4\times10^{-5}\) |
| 0.5 | 590.0508 | 596.1654 | \(1.0\times10^{-2}\) | 589.0682 | \(1.7\times10^{-3}\) |
Conclusion of §02
Up to \(v/c=0.1\), order 2 alone is good to \(10^{-5}\).
Ordinary physics gets away with "mass is a constant" because the parabolic approximation works so well.
★ Conversely ── effective mass stops being constant when you leave the parabola.
In semiconductors that actually happens at high fields (non-parabolicity).
03Electrons in a solid ── effective mass is "reluctance to hop"
\(t\): hopping strength to a neighbour; \(a\): lattice spacing
★ The easier the hop, the lighter ── effective mass is just "reluctance to move"
| \(t\) [eV] | \(a\) [nm] | \(m^*/m_e\) (analytic) | (numerical derivative) | difference |
|---|---|---|---|---|
| 0.1 | 0.30 | 4.233313 | 4.233313 | \(8.6\times10^{-9}\) |
| 0.5 | 0.30 | 0.846663 | 0.846663 | \(7.8\times10^{-9}\) |
| 1.0 | 0.30 | 0.423331 | 0.423331 | \(7.8\times10^{-9}\) |
| 3.0 | 0.30 | 0.141110 | 0.141110 | \(8.9\times10^{-9}\) |
§03 ── at the top of the band, mass goes negative
Expand the same formula at \(k=\pi/a\) (the band top):
band bottom \(k=0\): \(m^*/m_e=+0.4233\) ／ band top \(k=\pi/a\): \(m^*/m_e=-0.4233\)
★ Push it and it goes backwards ── that is a hole. Semiconductor engineering's "p-type" is a name for this negative curvature.
In Episode 10 it was "the parity of the order sets the arrow of time". Here ── the sign of the second derivative sets the direction of the force.
04Real materials ── four orders of magnitude in mass
| Material | \(m^*/m_e\) | Cyclotron frequency @1 T | Use |
|---|---|---|---|
| Free electron (vacuum) | 1 | 28.0 GHz | reference |
| Silicon (conduction band) | 0.26 | 108 GHz | integrated circuits |
| Germanium | 0.12 | 233 GHz | early transistors |
| Gallium arsenide | 0.067 | 418 GHz | fast devices, lasers |
| Indium antimonide | 0.014 | 2.00 THz | ★ 1/70 of a free electron |
| Heavy-fermion systems (\(\mathrm{UPt_3}\) etc.) | \(\sim200\) | 140 MHz | ★ 200 times a free electron |
★ Contrast with Episode 4
There, "\(c\) is a defined value and cannot be differentiated".
Here, effective mass changes by four orders of magnitude depending on the material ── far beyond differentiating; it can be designed.
Where is the difference? \(c\) is part of the unit system; \(m^*/m_e\) is a dimensionless ratio.
★ Episode 7's conclusion again ── what you can move is always the dimensionless quantity.
05Graphene ── zero mass and a light speed of \(c/300\)
★ The dispersion relation of a massless particle (\(E=pc\) with \(c\) replaced by \(v_F\))
Something interesting happens. Compute this world's fine-structure constant ──
The heart of §05 ── strongly coupled QED on a benchtop
\(\alpha_{\text{graphene}}=2.2\) ── greater than one.
In Episode 5 the vacuum's \(\alpha\) ran between 1/137 and 1/129. Here it is 2.2 ── ★ a non-perturbative regime sitting on a desk.
The reason is visible in the formula ── \(\alpha\propto1/c\), so dividing the effective light speed by 300 multiplies the coupling by 300.
Where light is slow, electromagnetism is strong.
Massless, and yet a cyclotron mass is definable ── \(m_c=E_F/v_F^2\).
| Fermi level \(E_F\) [eV] | \(m_c/m_e\) |
|---|---|
| 0.01 | 0.00176 |
| 0.05 | 0.00879 |
| 0.10 | 0.01759 (a common experimental condition) |
| 0.30 | 0.05277 |
Mass proportional to energy ── a hallmark of a relativistic dispersion relation. What gives a massless particle its apparent "mass" is kinetic energy alone.
★ A desktop version of the next episode's theme (99 % of the proton's mass is kinetic energy).
06Giving the photon a mass ── inside a superconductor
Episode 6 said "mass = cutoff frequency". Run it backwards ── manufacture a cutoff and the photon grows a mass.
| Material | \(\lambda_L\) [nm] | Photon mass [kg] | Energy [eV] |
|---|---|---|---|
| Aluminium | 16 | \(2.20\times10^{-35}\) | 12.33 |
| Lead | 37 | \(9.51\times10^{-36}\) | 5.33 |
| Niobium | 39 | \(9.02\times10^{-36}\) | 5.06 |
| YBCO (high-\(T_c\)) | 150 | \(2.35\times10^{-36}\) | 1.32 |
The heart of §06 ── the Higgs mechanism in miniature
Inside niobium, the photon has a mass of 5.06 eV.
This is the Higgs mechanism, scaled down ── a superconducting condensate gives the photon mass; the Higgs field gives the W and Z theirs. The same single formula.
Check it:
W boson: mass 80.377 GeV, range \(\hbar/(M_Wc)=2.455\times10^{-18}\) m
Photon in niobium: mass 5.06 eV, range \(3.900\times10^{-8}\) m
Mass ratio \(1.588\times10^{10}\) ／ length ratio \(1.589\times10^{10}\) ／ ratio of ratios \(=0.999934\)
★ Confirmation that "the weak force is weak" and "a superconductor expels magnetic field" are bound by one formula.
07Is the photon in vacuum really massless?
§07 ── back to Episodes 3 and 4
The vacuum is "possibly a waveguide with a cutoff somewhere beyond a period of an hour". Nothing more can be said.
★ That is ── "the photon is massless" is the claim that the cutoff frequency lies below the observation window.
Exactly the structure of Episode 3's "a physical constant = a component with period longer than the window". The low-frequency side is always "not yet visible".
08What the word "mass" points at
| Name | Definition | What it really is | Where |
|---|---|---|---|
| Rest mass | \(\omega_0=mc^2/\hbar\) | intercept of the dispersion relation | Ep. 6 |
| Effective mass \(m^*\) | \(\hbar/(d^2\omega/dk^2)\) | ★ curvature of the dispersion relation | this ep. |
| Cyclotron mass | \(\hbar k/(d\omega/dk)\) | ratio of slope to position | §05 |
| Photon mass in a medium | \(\hbar/(\lambda c)\) | reciprocal decay length | §06 |
| Longitudinal / transverse mass | direction-dependent curvature | anisotropic crystals | ── |
Conclusion of §08
"Mass" is not one quantity but a name for a way of reading a dispersion relation.
Read the intercept and you get rest mass; the curvature, effective mass; the decay length, a photon mass in a medium. All features of the same curve \(\omega(k)\).
Exercises
- In a heavy-fermion system the electron mass is 200×. Did the electron get heavier?
Show answer
No. The electron is the same; what changed is how the dispersion relation bends.
When \(f\) electrons mix strongly with conduction electrons, the band becomes extremely flat near the Fermi level ── a flat band means small curvature means large effective mass.
★ In §03's language, it is "hard to hop": the \(t\) in \(m^*=\hbar^2/(2ta^2)\) is small. What is heavy is not the electron but the environment it sits in. - What does \(\alpha=2.2\) in graphene mean, in Episode 5's terms?
Show answer
In Episode 5, \(\alpha\) ran with \(\ln E\), from 1/137 to 1/129 in vacuum ── always far below one, so perturbation theory works.
In graphene \(\alpha_g=\alpha\,c/v_F=2.2\), so the expansion parameter exceeds one; summing Feynman diagrams does not converge.
★ In real graphene, though, screening reduces the effective \(\alpha_g\) below one, and whether it is truly strongly coupled is debated. What we computed is the bare, unscreened value. - How could "the photon is massless" be falsified?
Show answer
By finding a cutoff frequency. Concretely ──
① look for a frequency below which electromagnetic waves stop propagating (a vacuum waveguide cutoff)
② look for a distance at which Coulomb's law departs from \(1/r^2\) (a Yukawa deviation)
③ check whether planetary or galactic magnetic fields decay faster than expected
★ All three are one direction: go to lower frequency / longer distance ── a concrete form of Episode 3's "\(\omega=0\) is out of reach". So the bound can keep falling, but zero can never be proved.
Summary mass → speed → mass
Expanding the dispersion relation at \(k=0\) gives rest mass at order 0, group velocity at order 1, and the reciprocal effective mass at order 2. One word pointing at different features of one curve.
Checked against relativity, order 2 alone is good to \(10^{-5}\) up to \(v/c=0.1\) ── "mass is a constant" works because the parabolic approximation is so good.
In solids, \(m^*=\hbar^2/(2ta^2)\). The easier the hop the lighter, and at the top of a band it goes negative (holes). Real materials span 0.014 to 200 ── four orders; mass is a design parameter.
Graphene is a world of zero mass and effective light speed \(c/300\), where \(\alpha_{\text{graphene}}=2.19\) ── where light is slow, electromagnetism is strong.
Inside a superconductor the photon grows a mass (5.06 eV in niobium). Against the W boson, the mass ratio and length ratio agree to \(10^{-4}\) ── "the weak force is weak" and "magnetic field is expelled" are one formula.
★ The reciprocal of mass is the curvature of the dispersion relation.
And "the photon is massless" is the claim that its cutoff lies below the observation window.
Of the proton's 938 MeV, the quark masses sum to just 9 MeV ── 99 % is confined kinetic energy.
★ In wave language ── a wave confined in a box has a lowest frequency. That was the mass. And mass runs, too.
This document is Episode 17 of the series "Thinking in Waves". The computation is in kenshou/yukou.js. On the numbers: the tight-binding \(m^*\) is a one-dimensional nearest-neighbour model, and we confirmed the analytic and numerically differentiated values agree to \(10^{-8}\) (internal consistency of the model, not a value for a real material). The effective masses in the table are representative and vary with direction and doping (silicon is anisotropic, 0.98 longitudinal and 0.19 transverse; 0.26 is the conductivity effective mass). The 200 for heavy fermions varies greatly with material and temperature. The London penetration depth depends strongly on temperature and sample and varies in the literature ── the table uses \(T\to0\) representative values, and the "photon mass" derived from them is our own conversion. \(\alpha_{\text{graphene}}=2.19\) is the bare, unscreened value; the effective coupling is believed to be smaller. The photon-mass bound \(10^{-18}\) eV is a representative published figure, with values from \(10^{-14}\) to \(10^{-27}\) eV depending on method (and on how strong an assumption one accepts). ── To print, use your browser's Print → Save as PDF.