Run this chapter’s code in Google Colab (no installation).
Slides 110–113: single quantum well, quantised levels, two-dimensional density of states, worked example on the levels of a GaAs/AlGaAs well, multiple quantum wells.
A layer of a narrow-gap semiconductor thinner than about 20 nm, sandwiched between wide-gap barriers, confines electrons along \(x\) (slide 110). The motion along \(x\) is quantised, while the motion in the plane (\(y\), \(z\)) stays free (slide 111):
\[ E_n=E_c+\frac{h^2n^2}{8m_e^*d^2}+\frac{\hbar^2k_y^2}{2m_e^*}+\frac{\hbar^2k_z^2}{2m_e^*},\qquad n=1,2,3,\dots \]
The first term is the infinite-well result. Each level \(E_n\) starts a sub-band whose density of states per unit area is constant, \(g_{2D}=m^*/\pi\hbar^2\). The total DOS is a staircase that touches the bulk \(\sqrt E\) curve at each step (slide 110c). Carriers collect at the bottom of the lowest sub-band, so the emission is narrower and occurs at
\[ h\nu=E_{g1}+E_1+E_1' \]
(electron plus hole confinement energies). The well width becomes a design parameter for the wavelength.
For a well of finite depth \(\Delta E_c\) the wavefunction leaks into the barriers and the levels are lower. They follow from matching \(\psi\) and \(\psi'/m^*\) at the interfaces:
\[ \text{even: }\ \frac{k}{m_w}\tan\frac{kd}{2}=\frac{\kappa}{m_b},\qquad \text{odd: }\ -\frac{k}{m_w}\cot\frac{kd}{2}=\frac{\kappa}{m_b},\qquad k=\frac{\sqrt{2m_wE}}{\hbar},\ \ \kappa=\frac{\sqrt{2m_b(\Delta E_c-E)}}{\hbar}. \]
A finite well always has at least one bound state, however thin. Make the well narrow (2–3 nm): \(E_1\) approaches the top, most of the wavefunction sits in the barriers, and the infinite-well formula becomes useless.
GaAs well between Al₀.₄Ga₀.₆As barriers, \(\Delta E_c=0.30\) eV, \(m_e^*=0.067m_e\), \(d=12\) nm. Find \(E_1\), \(E_2\) for an infinite well and compare with the finite-well values 0.022, 0.088 and 0.186 eV quoted on the slide.
d, V0 = 12e-9, 0.30
pd.DataFrame({
"infinite well (slide: 0.039, 0.156, 0.351)": sl.qw_infinite([1, 2, 3], d, 0.067),
"finite well, m* = 0.067 everywhere": sl.qw_finite(d, V0, 0.067)[:3],
"finite well, barrier mass 0.0919 (Al₀.₄Ga₀.₆As)": sl.qw_finite(d, V0, 0.067, 0.0919)[:3],
"finite well, m* = 0.080 everywhere": sl.qw_finite(d, V0, 0.080)[:3],
"slide (finite)": [0.022, 0.088, 0.186]}, index=["n = 1", "n = 2", "n = 3"]).round(4)| infinite well (slide: 0.039, 0.156, 0.351) | finite well, m* = 0.067 everywhere | finite well, barrier mass 0.0919 (Al₀.₄Ga₀.₆As) | finite well, m* = 0.080 everywhere | slide (finite) | |
|---|---|---|---|---|---|
| n = 1 | 0.0390 | 0.0256 | 0.0242 | 0.0222 | 0.022 |
| n = 2 | 0.1559 | 0.1006 | 0.0960 | 0.0875 | 0.088 |
| n = 3 | 0.3508 | 0.2161 | 0.2104 | 0.1902 | 0.186 |
The infinite-well numbers match the slide. The finite-well values quoted on the slide are about 15 % lower than a direct solution with \(m^*=0.067m_e\), even with the heavier barrier mass of Al₀.₄Ga₀.₆As. They correspond to an effective mass close to \(0.08m_e\) (non-parabolicity raises the mass of confined electrons, which may be the origin). The ratio infinite/finite is 1.5 for the textbook parameters instead of the quoted 1.8. Either way the conclusion holds: the third level is bound in the real well but above the barrier in the infinite model.
x = np.linspace(-20e-9, 20e-9, 800)
lev = sl.qw_finite(d, V0, 0.067)
fig, ax = plt.subplots(figsize=(7.5, 4))
ax.plot(x*1e9, np.where(abs(x) <= d/2, 0, V0), color=sl.INK, lw=2)
for n, E in enumerate(lev):
psi = sl.qw_wavefunction(x, E, d, V0, 0.067, parity=n % 2)
ax.plot(x*1e9, E + 0.035*psi, label=f"E_{n+1} = {E*1e3:.1f} meV")
ax.hlines(E, -d/2*1e9, d/2*1e9, color=sl.INK2, lw=0.8, ls=":")
for E in sl.qw_infinite([1, 2, 3], d, 0.067):
ax.hlines(E, -d/2*1e9, d/2*1e9, color=sl.PALETTE[7], lw=1, ls="--")
ax.set(xlabel="x (nm)", ylabel="E − E_c (eV)", ylim=(-0.02, 0.4)); ax.legend(fontsize=8, loc="upper right"); plt.show()
The emitted photon energy is \(E_{g1}+E_1+E_1'\). For GaAs/Al₀.₃Ga₀.₇As (\(\Delta E_g=0.374\) eV, split 67:33 between CB and VB, heavy-hole mass \(0.5m_e\)):
dEg = 1.247*0.3; dEc, dEv = 0.67*dEg, 0.33*dEg
widths = np.linspace(2e-9, 25e-9, 60)
E1 = np.array([sl.qw_finite(w, dEc, 0.067, 0.067 + 0.083*0.3)[0] for w in widths])
H1 = np.array([sl.qw_finite(w, dEv, 0.50, 0.50 + 0.14*0.3)[0] for w in widths])
Eph = 1.424 + E1 + H1
fig, ax = plt.subplots(figsize=(7.5, 3.6))
ax.plot(widths*1e9, sl.wavelength_from_eV(Eph)*1e9, label="quantum well")
ax.axhline(sl.wavelength_from_eV(1.424)*1e9, color=sl.INK2, ls="--", lw=1.2, label="bulk GaAs")
ax.set(xlabel="well width d (nm)", ylabel="emission wavelength (nm)"); ax.legend(); plt.show()
E = np.linspace(0, 0.4, 2000)
bulk = sl.dos_3d(E, 0.067)
step = 0.067*sl.me/(np.pi*sl.hbar**2*d)*sl.qe # m^-3 eV^-1 per sub-band
stair_fin = step*np.sum([E >= e for e in sl.qw_finite(d, V0, 0.067)], axis=0)
stair_inf = step*np.sum([E >= e for e in sl.qw_infinite(np.arange(1, 6), d, 0.067)], axis=0)
fig, ax = plt.subplots(figsize=(7.5, 3.6))
ax.plot(E*1e3, bulk/1e24, label="bulk (3D)", color=sl.INK2, ls="--")
ax.plot(E*1e3, stair_inf/1e24, label="QW, infinite-well levels", lw=1.4)
ax.plot(E*1e3, stair_fin/1e24, label="QW, finite-well levels", lw=2.2)
ax.axvline(V0*1e3, color=sl.GRID, lw=2); ax.text(V0*1e3 + 3, 0.5, "top of the well", fontsize=8)
ax.set(xlabel="E − E_c (meV)", ylabel="g(E) (10²⁴ m⁻³ eV⁻¹)"); ax.legend(fontsize=8); plt.show()
The step-like DOS puts a finite number of states right at the band edge instead of zero. Carriers pile up at \(E_1\), the spectrum narrows, and less carrier density is needed to reach a given emission rate (or optical gain in lasers, Ch. 4).
One thin well holds a limited number of carriers. At high current it saturates, carriers overflow into the barriers and the light output becomes sublinear. Spreading the current over \(N\) wells delays saturation. A minimal model gives each well a light output \(P_w=I_w/(1+I_w/I_\text{sat})\):
I = np.linspace(0, 10, 300) # in units of the saturation current of one well
fig, ax = plt.subplots(figsize=(7, 3.6))
for N in (1, 3, 5, 10):
ax.plot(I, N*(I/N)/(1 + I/N), label=f"{N} well{'s' if N > 1 else ''}")
ax.plot(I, I, ":", color=sl.INK2, lw=1.2, label="ideal, linear")
ax.set(xlabel="current I / I_sat", ylabel="light output (a.u.)", ylim=(0, 10)); ax.legend(fontsize=8); plt.show()
In real InGaN MQWs the holes are heavy and travel poorly, so most of the light comes from the well nearest the p side. The extra wells help less than this model suggests. This is one of the active research topics around “efficiency droop”.
scipy.linalg.eigh_tridiagonal) for an arbitrary potential: a graded well, two coupled wells (splitting into bonding/antibonding states), or a superlattice (minibands, compare with the Kronig–Penney model of Section 1.2).nextnano and the open-source Schrödinger–Poisson solver aestimo (Python) for real heterostructures.