Calculator
Example data table
| k (1/Å) | E (eV) | Comment |
|---|---|---|
| 0.00 | 1.4200 | Band edge reference |
| 0.02 | 1.4226 | Near-parabolic region |
| 0.04 | 1.4304 | Use points close to extremum |
| 0.06 | 1.4434 | Farther points can bias curvature |
Formula used
In a parabolic approximation near a band extremum, the dispersion satisfies: E(k) ≈ E0 + (ħ² k²)/(2 m*). Taking the second derivative gives: d²E/dk² = ħ² / m*, so m* = ħ² / (d²E/dk²).
For fitted data, this calculator performs a quadratic least-squares fit E(k)=a k² + b k + c, then uses d²E/dk² = 2a. Negative curvature yields a negative electron effective mass; hole mass is commonly taken as |m*|.
How to use this calculator
- Select a method: curvature input or parabolic fit.
- If using curvature, enter d²E/dk² and choose its units.
- If using fitting, enter 3–7 nearby k and E points.
- Optionally add longitudinal and transverse masses for ellipsoidal summaries.
- Press Calculate to view results above the form, then export CSV or PDF.
Professional article
Effective Mass in Band Theory
In a crystal, carriers respond to forces as if they have an effective inertia set by the band structure. This m* links acceleration to force and lets models replace periodic motion with a band‑edge parameter. It appears in mobility, diffusion, and density‑of‑states estimates used in device design.
Curvature Is the Key Data
Near a minimum or maximum, many bands are close to parabolic. The curvature d²E/dk² sets how quickly energy rises with wave‑vector: flatter bands mean heavier carriers. The calculator converts inputs like eV·Å² or eV·nm² into SI for consistency.
Core Relationship and Units
For E(k) ≈ E0 + ħ²k²/(2m*), differentiation gives m* = ħ²/(d²E/dk²). With E in joules and k in 1/m, d²E/dk² has units of J·m². Outputs are shown in kg and as the ratio m*/m0 for quick benchmarking.
Typical Semiconductor Ranges
Values vary widely: GaAs electrons are about 0.067 m0, while silicon electrons are often near 0.26 m0 in simplified models. Holes are commonly heavier, frequently ~0.3 to >1 m0 depending on direction and mixing. In many 2D semiconductors, masses span roughly 0.1 to 1 m0. In narrow bands, effective masses can exceed several m0, reducing mobility strongly.
Why Sign Matters for Electrons and Holes
A conduction minimum has positive curvature and positive electron m*. A valence maximum has negative curvature, giving a negative electron m*; hole calculations usually use |m*|. This tool reports the sign and the hole‑mass magnitude for clear interpretation.
Using k–E Fitting for Data
If curvature is unknown, enter several (k, E) points from spectroscopy or simulations. A quadratic least‑squares fit E(k)=ak²+bk+c is used, then d²E/dk²=2a. Use 3–7 points close to the extremum, and avoid non‑parabolic regions. Keeping k values balanced around the extremum can improve stability.
Anisotropy: Longitudinal and Transverse Masses
Some valleys are ellipsoidal, so direction matters. With longitudinal ml and transverse mt (in m0 units), the calculator estimates mDOS=(ml·mt²)^(1/3) and mcond=3/(1/ml+2/mt). These summaries support carrier statistics and conductivity modeling.
Practical Reporting and Export
For traceable workflows, record the chosen method, units, curvature sign, and any fitted points. Export CSV for spreadsheets or a PDF summary for lab notebooks and documentation. Exports capture outputs needed for verification.
FAQs
What is an effective mass, in simple terms?
It is a band‑structure parameter that relates force to acceleration near a band edge, letting carriers behave like free particles with a modified inertia.
Why can m* be negative?
At a valence‑band maximum the curvature is negative, so the electron effective mass from ħ²/(d²E/dk²) is negative. Hole models typically use the magnitude |m*|.
Which method should I use: curvature or fitting?
Use curvature input when you already know d²E/dk² from theory or literature. Use fitting when you have nearby k–E points from measurement or simulation and want the local curvature.
How many k–E points are best for fitting?
Three points is the minimum, but 5–7 points close to the extremum is usually better. Avoid points far away where the band becomes non‑parabolic.
What units does the calculator expect?
You can enter curvature in eV·Å², eV·nm², or J·m². For fitting, choose k units (1/Å, 1/nm, 1/m) and energy in eV or J.
What are mDOS and mcond in the anisotropy option?
They summarize ellipsoidal bands using longitudinal ml and transverse mt. mDOS affects carrier statistics, while mcond is often used for conductivity and mobility approximations.
Why do my results change when I include distant points?
A quadratic fit assumes a parabolic region. Distant points may sample non‑parabolic dispersion, skewing the curvature. Use points tightly clustered around the band edge for a more local m*.
Accurate inputs produce reliable masses for practical semiconductor modeling.