id int64 | question string | answer string | final_answer list | answer_type string | topic string | symbol string |
|---|---|---|---|---|---|---|
401 | A thermistor made of intrinsic silicon material has a resistance value of $500 \Omega$ at 290 K. Assuming the band gap of silicon $E_{\mathrm{q}}=1.12 \mathrm{eV}$ and does not change with temperature, if we assume the carrier mobility remains unchanged, try to estimate the approximate value of the thermistor at 325 K. | [] | Numeric | Semiconductors | ||
402 | The resistivity of intrinsic germanium material with temperature $T$ can be tabulated as follows:
\begin{tabular}{|c|c|c|c|c|}
\hline$T(\mathrm{~K})$ & 385 & 458 & 556 & 714 \\
\hline$\rho(\Omega \cdot \mathrm{~cm})$ & 0.028 & 0.0061 & 0.0013 & 0.00027 \\
\hline
\end{tabular}
Assume $E_{\mathrm{g}}$ is independent of ... | [] | Numeric | Semiconductors | ||
403 | Calculate the resistivity of intrinsic silicon at room temperature (unit $\Omega \cdot \mathrm{~cm}$). It is known that the electron mobility of intrinsic silicon at room temperature is $1350 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, the hole mobility is $500 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, an... | [] | Numeric | Semiconductors | ||
404 | For a certain n-type semiconductor silicon with a doping concentration $N_{\mathrm{D}}=10^{15} \mathrm{~cm}^{-3}$, and a minority carrier lifetime $\tau_{\mathrm{p}}=5 \mu \mathrm{~s}$, if due to external influences all minority carriers are removed (such as near a reverse-biased pn junction), what is the electron-hole... | [] | Numeric | Semiconductors | ||
405 | If the effective mass of holes in germanium $m_{\mathrm{p}}^{*}=0.30 m_{0}(m_{0}$ is the free electron mass), find the hole capture cross-section? | [] | Numeric | Semiconductors | ||
406 | In a piece of p-type semiconductor, there exists a recombination-generation center. When slightly doped, the electrons captured by these centers are reemitted to the conduction band with the same probability as their recombination with holes. Try to find the energy level position of this recombination-generation center... | [] | Equation | Semiconductors | $E_{\mathrm{t}}$: Energy level position of the recombination-generation center (trap level)
$E_{\mathrm{i}}$: Intrinsic Fermi energy level, $E_{\mathrm{i}}=\frac{1}{2}(E_{\mathrm{c}}+E_{\mathrm{v}}-k_{0} T \ln \frac{N_{\mathrm{c}}}{N_{\mathrm{v}}})$
$E_{\mathrm{F}}$: Fermi energy level | |
407 | The number of holes recombined at the surface per unit time per unit surface area; | [] | Numeric | Semiconductors | ||
408 | The number of holes recombined within three diffusion lengths from the surface, per unit time and per unit surface area. | [] | Numeric | Semiconductors | ||
409 | Calculate the bulk lifetime; | [] | Numeric | Semiconductors | ||
410 | Calculate the surface recombination velocity; | [] | Numeric | Semiconductors | ||
411 | If the silicon wafer is uniformly illuminated, and the generation rate of electron-hole pairs is $10^{11} \mathrm{~cm}^{-3} \cdot \mathrm{~s}^{-1}$, what is the hole concentration at the surface? | [] | Numeric | Semiconductors | ||
412 | If the silicon wafer is illuminated and uniformly absorbed by the sample, the generation rate of electron-hole pairs is $10^{11} \mathrm{~cm}^{-3} \cdot \mathrm{~s}^{-1}$. What is the hole current density flowing towards the surface? | [] | Numeric | Semiconductors | ||
413 | A pn junction composed of p-type germanium with a resistivity of $1 \Omega \cdot \mathrm{~cm}$ and n-type germanium with a resistivity of $0.1 \Omega \cdot \mathrm{~cm}$, calculate the built-in potential difference $V_{\mathrm{D}}$ at room temperature (300 K). Given that at the above resistivities, the hole mobility in... | [] | Numeric | Semiconductors | ||
414 | A pn junction composed of p-type germanium with resistivity $1 \Omega \cdot \mathrm{~cm}$ and n-type germanium with resistivity $0.1 \Omega \cdot \mathrm{~cm}$, calculate the width of the depletion region at room temperature (300 K). Given that at these resistivities, the hole mobility in the p region is $\mu_{\mathrm{... | [] | Numeric | Semiconductors | ||
415 | Given a silicon abrupt junction, with resistivities on both sides being $\rho_{\mathrm{n}}=10 \Omega \cdot \mathrm{~cm}$ for $\mathrm{n}-\mathrm{Si}$ and $\rho_{\mathrm{p}}=0.01 \Omega \cdot \mathrm{~cm}$ for $\mathrm{p}-\mathrm{Si}$, and mobilities $\mu_{\mathrm{n}}=100 \mathrm{~cm}^{2}/(\mathrm{V} \cdot \mathrm{s}), ... | [] | Numeric | Semiconductors | ||
416 | Assume a silicon abrupt junction with the impurity concentrations on both sides as $N_{\mathrm{A}}=10^{17} / \mathrm{cm}^{3}, N_{\mathrm{D}}=4.5 \times 10^{15} / \mathrm{cm}^{3}$. The intrinsic carrier concentration of silicon at room temperature is known as $n_{\mathrm{i}} = 1.5 \times 10^{10} / \mathrm{cm}^{3}$, vacu... | [] | Numeric | Semiconductors | ||
417 | Suppose a silicon abrupt junction, with impurity concentrations on either side as $N_{\mathrm{A}}=10^{17} / \mathrm{cm}^{3}, N_{\mathrm{D}}=4.5 \times 10^{15} / \mathrm{cm}^{3}$, when a reverse bias voltage of 10 V is applied, find the value of $x_{\mathrm{D}}$. | [] | Numeric | Semiconductors | ||
418 | Find the value of the barrier width $x_{\mathrm{D}}$. | [] | Numeric | Semiconductors | ||
419 | Find its maximum electric field, | [] | Numeric | Semiconductors | ||
420 | Find the value of $x_{\mathrm{D}}$ when an external reverse bias of 10 V is applied. | [] | Numeric | Semiconductors | ||
421 | For a silicon $n^+p$ junction, given $x_p = 0.2\ \mu\text{m}$, $L_n = 200\ \mu\text{m}$, $N_A = 10^{15}\ \text{cm}^{-3}$, $n_i = 1.5 \times 10^{10}\ \text{cm}^{-3}$, at room temperature $T = 300\ \text{K}$, find the value of voltage V corresponding to the condition where the barrier recombination current equals the dif... | [] | Numeric | Semiconductors | ||
422 | The relationship between the barrier capacitance $C_{\mathrm{T}}$ of the $\mathrm{p}^{+} \mathrm{n}$ junction made from GaP material and the reverse voltage $V_{\mathrm{R}}$ is measured as follows
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline$V_{\mathrm{R}}(\mathrm{V})$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline$C_{\mathr... | [] | Numeric | Semiconductors | ||
423 | The relationship between the barrier capacitance $C_{\mathrm{T}}$ and reverse voltage $V_{\mathrm{R}}$ of a $\mathrm{p}^{+} \mathrm{n}$ junction made of GaP material is measured as follows
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline$V_{\mathrm{R}}(\mathrm{V})$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline$C_{\mathrm{T}}(\m... | [] | Numeric | Semiconductors | ||
424 | A pn junction diode has the following parameters: $N_{\mathrm{D}}=10^{16} \mathrm{~cm}^{-3}, ~ N_{\mathrm{A}}=5 \times 10^{18} \mathrm{~cm}^{-3}, ~ \tau_{\mathrm{n}}=\tau_{p}=$ $1 \mu \mathrm{~s}, ~ A=0.01 \mathrm{~cm}^{2}$. Assume that the widths on both sides of the junction are much larger than the diffusion lengths... | [] | Numeric | Semiconductors | ||
425 | Zero-bias depletion width; | [] | Numeric | Semiconductors | ||
426 | The thermionic emission current when forward biased at 0.2 V. Assume $\frac{A^{*}}{A}=2.1, A=120 \mathrm{~A} / \mathrm{cm}^{2}$. | [] | Numeric | Semiconductors | ||
427 | Calculate the minority carrier injection ratio. | [] | Numeric | Semiconductors | ||
428 | A metal contacts a uniformly doped $n-Si$ material, forming a Schottky barrier diode. The barrier height on the semiconductor side is known as $qV_{\mathrm{D}}=0.6 \mathrm{eV}, N_{\mathrm{D}}=5 \times 10^{16} \mathrm{~cm}^{-3}$. Calculate the maximum electric field in the semiconductor at the interface under a reverse ... | [] | Numeric | Semiconductors | ||
429 | A metal contacts a uniformly doped $n-Si$ material, forming a Schottky barrier diode. Given the barrier height on the semiconductor side $q V_{\mathrm{D}}=0.6 \mathrm{eV}, N_{\mathrm{D}}=5 \times 10^{16} \mathrm{~cm}^{-3}$, determine the barrier capacitance per unit area under a reverse bias voltage of 5 V. | [] | Numeric | Semiconductors | ||
430 | Ignoring the work function difference between the metal and the semiconductor, what is the voltage $V_{\mathrm{G}}$ on the metal plate when the depletion layer width just reaches its maximum? ($N_{\mathrm{D}}=10^{16} \mathrm{~cm}^{-3}$) | [] | Numeric | Semiconductors | ||
431 | If there is a fixed positive charge at the SiO$_2$-silicon interface, and the measured $V_{\mathrm{T}}=2.6 \mathrm{~V}$, find the amount of fixed positive charge per unit area (neglecting the influence of the work function difference); | [] | Numeric | Semiconductors | $N_{\mathrm{fc}}$: Number of fixed positive charges per unit area at the Si-SiO$_2$ interface. | |
432 | If the above positive charges are uniformly distributed in $\mathrm{SiO}_{2}$, what is the measured $V_{\mathrm{T}}$? (neglecting the effect of work function difference); | [] | Numeric | Semiconductors | ||
433 | Calculate the gate voltage when the substrate surface is intrinsic (when the junction voltage is zero at room temperature). | [] | Numeric | Semiconductors | ||
434 | For the gate-controlled $\mathrm{p}^{+} \mathrm{n}$ diode described in the problem (metallurgical junction area $10^{-3} \mathrm{~cm}^{2}$, overlap area between the gate and n region $10^{-3} \mathrm{~cm}^{2}$, substrate impurity concentration $10^{16} \mathrm{~cm}^{-3}$, oxide thickness $0.2 \mu \mathrm{~m}$, minority... | [] | Numeric | Semiconductors | ||
435 | For the gate-controlled $\mathrm{p}^{+} \mathrm{n}$ diode (with relevant parameters: gate to n-region overlap area $A_s = 10^{-3} \mathrm{~cm}^{2}$, used to calculate the diffusion current; n-region substrate impurity concentration $N_D = 10^{16} \mathrm{~cm}^{-3}$; minority carrier hole lifetime $\tau_p = 1 \mu \mathr... | [] | Numeric | Semiconductors | ||
436 | Estimate the injection ratio between GaAs and $\mathrm{Al}_{0.3} \mathrm{Ga}_{0.7} \mathrm{As}$ at 300 K. | [] | Numeric | Semiconductors | ||
437 | Try to derive the relationship between the absorption coefficient $\alpha$ and the extinction coefficient $\bar{k}$. | [] | Expression | Semiconductors | $\alpha$: Absorption coefficient
$\omega$: Angular frequency
$\bar{k}$: Extinction coefficient
$c$: Speed of light in vacuum
$\pi$: Mathematical constant pi
$\lambda$: Wavelength in vacuum | |
438 | Calculate the number of electron-hole pairs generated per second in the sample; | [] | Numeric | Semiconductors | ||
439 | calculate the increase in the number of electrons in the sample; | [] | Numeric | Semiconductors | ||
440 | When the photocurrent when a 50 V voltage is applied to the sample, calculate photoconductive gain factor. | [] | Numeric | Semiconductors | ||
441 | Given a piece of n-type semiconductor material with a room temperature dark conductivity of $100 \mathrm{~S} / \mathrm{cm}$, when illuminated with light at an intensity of $I=$ $10^{-6} \mathrm{~W} / \mathrm{cm}^{2}$, its absorption coefficient $\alpha=10^{2} / \mathrm{cm}$, the measured ratio of steady-state photocond... | [] | Numeric | Semiconductors | ||
442 | In a p-type silicon with a hole concentration of $10^{16} \mathrm{~cm}^{-3}$, a cold end temperature of $0^{\circ} \mathrm{C}$, and a hot end temperature of $50^{\circ} \mathrm{C}$, assuming long wavelength acoustic wave scattering, calculate the thermoelectric power. | [] | Numeric | Semiconductors | ||
443 | For n-type PoTe with a conductivity of $2000 \mathrm{~S} / \mathrm{cm}$, an electron mobility of $6000 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and an electron effective mass of $0.2 m_{0}$, determine the thermoelectric power factor at room temperature assuming long-wavelength acoustic phonon scattering. | [] | Numeric | Semiconductors | ||
444 | For an n-type PoTe with a conductivity of $2000 \mathrm{~S} / \mathrm{cm}$, electron mobility of $6000 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and an electron effective mass of $0.2 m_{0}$, assuming long-wavelength acoustic scattering, determine the Peltier coefficient at room temperature. | [] | Numeric | Semiconductors | ||
445 | The thermal conductivity of bismuth telluride ($\mathrm{Bi}_{2} \mathrm{Te}_{3}$) is $2.4[\mathrm{~W} /(\mathrm{m} \cdot \mathrm{K})$]. Calculate the percentage contribution of carrier to the thermal conductivity for n-type $\mathrm{Bi}_{2} \mathrm{Te}_{3}$ at $10^{5} \mathrm{~s} / \mathrm{m}$ and $300 \mathrm{~K}$. (A... | [] | Numeric | Semiconductors | ||
446 | Try to find the Seebeck coefficient of intrinsic silicon at room temperature.
Assume the effective masses of electrons and holes are equal, the band gap of silicon is 1.12 eV, and the mobilities of electrons and holes are $0.135 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $0.048 \mathrm{~cm}^{2} /(\mathrm{V} \... | [] | Numeric | Semiconductors | ||
447 | For an indium antimonide sample, the hole concentration at room temperature is 9 times the electron concentration. Calculate the Hall coefficient $R$. Assume at room temperature $b=\mu_{\mathrm{n}} / \mu_{\mathrm{p}}=100, n_{\mathrm{i}}=1.1 \times 10^{16} \mathrm{~cm}^{-3}$. | [] | Numeric | Semiconductors | ||
448 | Hall coefficient of intrinsic material; | [] | Numeric | Semiconductors | ||
449 | intrinsic resistivity; | [] | Numeric | Semiconductors | ||
450 | When $B_{z}=0.1 \mathrm{~Wb} / \mathrm{m}^{2}$, calculate the resistivity of the material considering the scattering of long acoustic waves. | [] | Numeric | Semiconductors | ||
451 | Hall coefficient; | [] | Numeric | Semiconductors | ||
452 | A silicon sample with a conductivity of $0.001 /(\Omega \cdot \mathrm{cm})$ has zero Hall voltage under a weak magnetic field. Assuming the electron mobility $\mu_{\mathrm{n}}=1300 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and the hole mobility $\mu_{\mathrm{p}}=300 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s}... | [] | Numeric | Semiconductors | $n$: Carrier density of electrons in the sample | |
453 | Given the band gap of InSb $E_{\mathrm{g}}=0.15 \mathrm{eV}$, the effective mass of electrons $m_{\mathrm{e}}=0.014 m_{0}$, and the effective mass of holes $m_{\mathrm{h}}=0.18 m_{0}$ (with $m_{0}$ as the inertial mass of the electron). If only electrons are the effective carriers, calculate the Hall coefficient of int... | [] | Numeric | Semiconductors | ||
454 | If a given germanium sample is placed in a magnetic field of $B=0.1$ T, what is $\tan \theta$ when its conductivity is at a minimum? Let $\mu_{\mathrm{n}}=3900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s}), \mu_{\mathrm{p}}=1$ $900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$. | [] | Numeric | Semiconductors | ||
455 | Classified by symmetry type, how many types of point groups are there for Bravais lattices? How many types of space groups are there? How many types of point groups are there for crystal structures? How many types of space groups are there? You should return your answer as a tuple format. | [] | Tuple | Others | ||
456 | Derive the expression for the density of states of the s-band electrons. | [] | Expression | Others | $G(E)$: Density of states of the s-band electrons as a function of energy $E$.
$N$: Number of identical atoms in the chain.
$\pi$: Mathematical constant pi.
$J_1$: Hopping integral, representing the strength of nearest-neighbor interactions.
$E_0$: On-site energy, representing the energy of an electron in an isolated a... | |
457 | The valence band of a semiconductor material is almost filled with electrons (nearly full band), and the expression for the energy of valence band electrons is $E(k)=-1.016 \times 10^{-34} k^{2}(J)$, where the energy zero point is taken at the top of the valence band. At this time, if the electron at $k=1 \times 10^{6}... | [] | Numeric | Semiconductors | ||
458 | Calculate the mean free path of electrons at room temperature (T=295K). (The density of silver is $10.5 \mathrm{~g} / \mathrm{cm}^{3}$, atomic weight is 107.87, and its resistivity at T=295K is $1.61 \times 10^{-6} \Omega \cdot \mathrm{~cm}$) | [] | Numeric | Others | ||
459 | Phase transition in BCS superconducting systems
Consider the Hamiltonian with parameter $\lambda$, $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$, and the corresponding Gibbs free energy is $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\beta \bar... | [] | Expression | Superconductivity | $\Delta (\lambda)$: Superconducting gap parameter, explicitly dependent on $\lambda$, defined as $\Delta (\lambda) = \lambda \sum_k < C_{-k} C_k >_T$.
$\lambda$: Parameter in the Hamiltonian, interpolating between the normal and superconducting states. | |
460 | Phase transition in BCS superconducting systems
Consider a Hamiltonian with a parameter $\lambda$ given by $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$, with the corresponding Gibbs free energy $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\bet... | [] | Expression | Superconductivity | $\Delta G$: Difference in Gibbs free energy between superconducting and normal states.
$G_S$: Gibbs free energy in the superconducting state.
$G_N$: Gibbs free energy in the normal state.
$g(0)$: Density of states on the Fermi surface.
$\pi$: Mathematical constant pi.
$k_B$: Boltzmann constant.
$\zeta(3)$: Riemann zeta... | |
461 | Phase transition in BCS superconducting systems
Consider a Hamiltonian with parameter $\lambda$ given by $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$, and the corresponding Gibbs free energy is $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\bet... | [] | Expression | Superconductivity | $\Delta S$: Change in entropy, defined as $S_\mathrm{S}-S_\mathrm{N}$.
$g(0)$: Density of states at the Fermi surface.
$\pi$: Mathematical constant pi.
$k_B$: Boltzmann constant.
$\zeta(3)$: Riemann zeta function evaluated at 3.
$T_c$: Critical temperature for the superconducting transition.
$T$: Temperature of the sys... | |
462 | Phase transition in BCS superconducting system
Consider a Hamiltonian with parameter $\lambda$ given by $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$, the corresponding Gibbs free energy is $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\beta \ba... | [] | Expression | Superconductivity | $\Delta c$: Discontinuity in the electronic specific heat.
$T$: Temperature of the system.
$\Delta S$: Change in entropy.
$T_c$: Critical temperature.
$g(0)$: Density of states at the Fermi surface.
$k_B$: Boltzmann constant. | |
463 | Phase transition in BCS superconducting system
Consider a Hamiltonian $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$ with parameter $\lambda$, the corresponding Gibbs free energy is $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\beta \bar{H}_{\ma... | [] | Expression | Superconductivity | $H_c (T)$: Critical magnetic field at temperature $T$
$H_c (0)$: Critical magnetic field at absolute zero temperature
$\gamma$: Euler-Mascheroni constant
$\zeta (3)$: Riemann zeta function evaluated at 3
$T$: Temperature
$T_c$: Critical temperature | |
464 | London Theory
Superconductors have two properties:
\begin{itemize}
\item[(i)] The DC resistance disappears when $T < T_c$, and a resistance-free supercurrent exists, which is the ideal conductivity of the superconductor.
\item[(ii)] The Meissner effect, a weak magnetic field cannot penetrate the interior o... | [] | Equation | Superconductivity | $\Lambda$: London penetration depth parameter, defined as $\Lambda = \frac{m}{n_se^2}$
$\mathbf{j}_s$: Supercurrent density, the current density of superconducting electrons
$t$: Time
$\boldsymbol{E}$: Electric field | |
465 | London theory
Superconductors have two properties:
\begin{itemize}
\item[(i)] When $T < T_c$, the DC resistance disappears, and there exists a resistance-free supercurrent, which is the ideal conductivity of a superconductor.
\item[(ii)] Meissner effect, a weak magnetic field cannot penetrate inside a bul... | [] | Expression | Superconductivity | $\boldsymbol{B}(z)$: Magnetic field as a function of depth $z$
$\boldsymbol{B}(0)$: Magnetic field at the surface of the superconductor ($z=0$)
$z$: Spatial coordinate representing depth into the superconducting sample
$\lambda_L$: London penetration depth, defined by $\lambda_L^2 = \frac{mc^2}{4\pi n_s e^2}$ | |
466 | Pippard Theory
In superconductors, within the coherence length $\xi_0 = \frac{\hbar v_F}{\pi \Delta(0)}$, there exists a correlation of electron motion, hence a perturbing potential acting at one point will inevitably affect the velocity of superconducting electrons and current density within the spatial scale of $\... | [] | Expression | Superconductivity | $\mathbf{j}(\mathbf{q})$: Supercurrent density in momentum space
$c$: Speed of light
$\lambda_L^2(0)$: London penetration depth squared at zero temperature
$\mathbf{A}(\mathbf{q})$: Vector potential in momentum space | |
467 | Pippard Theory
In superconductors, there is a correlation of electron motion within the coherence length $\xi_0 = \frac{\hbar v_F}{\pi \Delta(0)}$, thus a perturbative potential acting at one point will inevitably affect the velocity and current density of superconducting electrons within the spatial scale of $\xi_0$... | [] | Expression | Superconductivity | $\mathbf{j}(\mathbf{q})$: Supercurrent density in momentum space
$c$: Speed of light
$K(0)$: Kernel function $K(q)$ evaluated at $q=0$
$q$: Wave vector or momentum variable
$\xi_0$: Coherence length, defined as $\frac{\hbar v_F}{\pi \Delta(0)}$
$\mathbf{A}(\mathbf{q})$: Vector potential in momentum space
$\lambda_L^2(0... | |
468 | The Current in Superconductors
According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation it is given by:
\begin{align}
\hat{\mathbf{j}}(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\n... | [] | Expression | Superconductivity | $\mathbf{j}_2(\mathbf{r})$: Current density $\mathbf{j}_2(\mathbf{r})$.
$n$: Electron density.
$e$: Elementary charge.
$m$: Mass of the charge carrier.
$c$: Speed of light.
$\mathbf{A}(\mathbf{r})$: Magnetic vector potential in real space.
$\pi$: Mathematical constant pi.
$\lambda_L(0)$: London penetration depth at zer... | |
469 | Current in Superconductors
According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation it is:
\begin{align}
\hat{\mathbf{j}}(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\nabla \Psi) - ... | [] | Expression | Superconductivity | $c$: Speed of light.
$\pi$: Mathematical constant pi.
$\lambda_L(0)$: London penetration depth at zero temperature.
$\mathbf{A}(\mathbf{q})$: Fourier transform of the vector potential.
$\epsilon_+$: Shorthand for single-particle energy $\epsilon_{\mathbf{k}+\mathbf{q}/2}$.
$\epsilon_-$: Shorthand for single-particle en... | |
470 | Current in Superconductors
According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation, it is expressed as:
\begin{align}
\hat{\mathbf{j}}(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\... | [] | Expression | Superconductivity | $\lambda_L^2(0)$: Square of the London penetration depth at zero temperature.
$\xi_+$: Quasiparticle energy $\xi_{\mathbf{k}+\mathbf{q}/2} = \sqrt{\epsilon_{+}^2 + \Delta^2}$.
$\xi_-$: Quasiparticle energy $\xi_{\mathbf{k}-\mathbf{q}/2} = \sqrt{\epsilon_{-}^2 + \Delta^2}$.
$\epsilon_+$: Single-particle energy $\epsilon... | |
471 | Meissner Effect in Superconductors
Assume the current in a superconductor follows $\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})$. If $K(0)=0$, it indicates the absence of the Meissner effect in the superconductor; otherwise, it exists.
At the microscopic level, the current in a BCS superc... | [] | Expression | Superconductivity | $K_n(q)$: Kernel function for a normal conductor. | |
472 | Meissner Effect in Superconductors
Assume the current in a superconductor follows $\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})$. If $K(0)=0$, it can be shown that there is no Meissner effect in the superconductor, otherwise it exists.
Microscopically, the BCS superconducting current can ... | [] | Expression | Superconductivity | $K(q)$: Kernel function relating current density to vector potential, orientation-independent
$\lambda_L(0)$: London penetration depth at zero temperature | |
473 | Meissner effect in superconductors
Assuming the current in a superconductor follows $\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})$, if $K(0)=0$, it can be stated that the superconductor does not exhibit the Meissner effect, otherwise it does.
Microscopically, BCS superconductor current can b... | [] | Expression | Superconductivity | $K(q)$: Orientation-independent kernel function describing the current response in a superconductor.
$\lambda_L(0)$: London penetration depth at zero temperature.
$q$: Magnitude of the wave vector.
$\xi_0$: Coherence length of the superconductor, defined as $\xi_0 = \frac{\hbar v_F}{\pi \Delta(0)}$.
$K(0)$: Value of th... | |
474 | London penetration depth at finite temperatures
The London equation is a significant equation describing superconductors, reflecting the perfect diamagnetism of superconductors and can be written as
\begin{equation}
\mathbf{j}_s(\mathbf{r}) = - \frac{c}{4\pi}\frac{1}{\lambda_L^2}\mathbf{A}(\mathbf{r}),
\end{e... | [] | Equation | Superconductivity | $K_2(q, T)$: Linear response coefficient of the diamagnetic current
$q$: Wave number
$T$: Temperature
$\pi$: Mathematical constant pi
$n$: Number density of superconducting electrons
$e$: Elementary charge
$m$: Mass of the charge carrier
$c$: Speed of light | |
475 | Ginzberg Landau Theory
Ginzburg and Landau proposed using a complex quantity $\psi(r)$ to describe the 'effective wave function' of superconducting electrons, with charge $e^*$ and mass $m^*$, and the corresponding system free energy density and free energy are:
\begin{equation}
f_s = f_n + \alpha(T) |\psi(\mat... | [] | Equation | Superconductivity | $m$: Mass, appearing in the Ginzburg-Landau equation (corresponds to $m^*$ in the problem statement).
$i$: Imaginary unit.
$\hbar$: Reduced Planck's constant.
$\nabla$: Nabla operator.
$e$: Charge, appearing in the Ginzburg-Landau equation (corresponds to $e^*$ in the problem statement).
$c$: Speed of light.
$\mathbf{A... | |
476 | Ginzburg-Landau Theory
Ginzburg and Landau proposed using a complex variable $\psi(r)$ to describe the "effective wave function" of superconducting electrons, with charge $e^*$ and mass $m^*$. The corresponding system free energy density and free energy are:
\begin{equation}
f_s = f_n + \alpha(T) |\psi(\mathbf{... | [] | Expression | Superconductivity | $\psi_0$: spatially uniform solution for the effective wave function, representing a superconducting state
$\alpha(T)$: temperature-dependent coefficient in the Ginzburg-Landau free energy density
$\beta(T)$: temperature-dependent coefficient in the Ginzburg-Landau free energy density
$T$: temperature
$T_c$: critical t... | |
477 | Ginzburg Landau Theory
Ginzburg and Landau propose using a complex quantity $\psi(r)$ to describe the 'effective wave function' of superconducting electrons, with charge $e^*$ and mass $m^*$. The corresponding system free energy density and free energy are:
\begin{equation}
f_s = f_n + \alpha(T) |\psi(\mathbf{r... | [] | Expression | Superconductivity | $\lambda(T)$: Penetration depth, temperature-dependent.
$m$: Mass of superconducting electrons (used in one form of the final answer, corresponding to $m^*$ in the problem statement).
$c$: Speed of light in vacuum.
$\pi$: Mathematical constant pi, approximately 3.14159.
$e$: Charge of superconducting electrons (used in... | |
478 | Ginzburg-Landau Theory
Ginzburg and Landau proposed using a complex quantity $\psi(r)$ to describe the "effective wave function" of superconducting electrons, with charge $e^*$ and mass $m^*$; the corresponding system free energy density and free energy are:
\begin{equation}
f_s = f_n + \alpha(T) |\psi(\mathbf{... | [] | Expression | Superconductivity | $\Phi$: Magnetic flux, $\Phi = \int_S \mathbf{B} \cdot d\mathbf{S}$
$n$: Integer multiple, quantum number
$\Phi_0$: Magnetic flux quantum, $\Phi_0 = \frac{2\pi\hbar c}{e^*}$
$\pi$: Mathematical constant pi
$\hbar$: Reduced Planck's constant
$c$: Speed of light
$e^*$: Charge of superconducting electrons
$\mathbf{B}$: Ma... | |
479 | Hubbard Model in Narrow Bandwidth
Consider the following single band Hubbard model,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c,c^\dagger$ are the annihilation and creat... | [] | Equation | Strongly Correlated Systems | $G_{kk'}^\sigma(\omega)$: Off-diagonal elements of the single-particle Green's function in Bloch representation. | |
480 | Hubbard Model under Narrow Bandwidth — Green's Function Analysis
Consider the following single-band Hubbard model,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c, c^\dagger... | [] | Expression | Strongly Correlated Systems | $E_{k\sigma}^{(1,2)}$: Combined notation for the two branches of the energy spectrum (poles of the Green's function) for momentum $k$ and spin $\sigma$.
$E_k$: Single-particle energy for momentum $k$ in the non-interacting system.
$U$: On-site Coulomb repulsion strength.
$T_0$: A parameter in the Green's function, repr... | |
481 | Hubbard Model in Narrow Band - Green's Function Analysis
Consider the following single-band Hubbard model,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c,c^\dagger$ are the... | [] | Equation | Strongly Correlated Systems | $G_{k}^{\sigma}(\omega)$: Single-particle Green's function for momentum $k$ and spin $\sigma$ at energy $\omega$.
$U$: On-site Coulomb repulsion energy.
$\omega$: Energy (or frequency).
$E_k$: Energy of the band electron with momentum $k$. | |
482 | Hubbard Model in the Narrow Band Limit—Green's Function Analysis
Consider the single-band Hubbard model as follows,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c,c^\dagger... | [] | Equation | Strongly Correlated Systems | $\rho_\sigma (\omega)$: Density of states for spin $\sigma$ at energy $\omega$
$U$: On-site Coulomb repulsion energy (Hubbard U)
$\omega$: Energy or frequency
$D(\omega)$: Density of states of a single band $E_k$ at energy $\omega$
$\sigma$: Spin index | |
483 | Narrow-band Hubbard model—Green's function analysis
Consider the following single-band Hubbard model,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c,c^\dagger$ are the... | [] | Equation | Strongly Correlated Systems | $G_k^\sigma(\omega)$: Single-particle Green's function for momentum $k$ and spin $\sigma$ at frequency $\omega$.
$n_{\bar{\sigma}}$: Particle number operator for an electron at site $i$ with spin $\bar{\sigma}$ (opposite spin). In the context of the Green's function, it often represents the expectation value $\langle n... | |
484 | Anderson s-d exchange model and Green's function equation of motion
In 1961, Anderson proposed the s-d mixing model, suggesting that to discuss the formation of local magnetic moments by transition metal impurity atoms in a non-magnetic metal matrix, two factors must be considered: First, similar to the formation of i... | [] | Equation | Strongly Correlated Systems | $\\omega$: Frequency.
$E_{k\\sigma}$: Energy of an electron in state $k$ with spin $\sigma$, $E_{k\\sigma} = E_k + \\sigma \\mu_B h$.
$a_{kk'\\sigma}$: Shorthand notation for the Green's function $\\ll C_{k\\sigma} | C_{k'\\sigma}^\\dagger \\gg_\\omega$.
$\\delta_{k, k'}$: Kronecker delta function.
$V_{kd}$: Electron t... | |
485 | Anderson s-d exchange model and Green's function equation of motion
In 1961, Anderson proposed the s-d mixing model, suggesting that when discussing the formation of local magnetic moments by transition metal impurity atoms in a non-magnetic metal matrix, it is essential to consider two factors: Firstly, similar to t... | [] | Equation | Strongly Correlated Systems | $\omega$: Frequency.
$E_{d\sigma}$: Energy of a d-orbital electron with spin $\sigma$, defined as $E_{d\sigma} = E_d + \sigma \mu_B h$.
$a_\sigma$: Symbol for the Green's function $\ll d_\sigma | d_\sigma^+ \gg_\omega$.
$U$: Coulomb interaction strength among d-shell electrons.
$c_\sigma$: Symbol for $\ll n_{d\bar{\sig... | |
486 | Anderson s-d exchange model and Green's function equations of motion
In 1961, Anderson proposed the s-d mixing model, arguing that to discuss the formation of localized magnetic moments by transition metal impurity atoms in a non-magnetic metallic matrix, two factors must be considered simultaneously: First, similar ... | [] | Equation | Strongly Correlated Systems | $\omega$: Frequency.
$E_{k\sigma}$: Energy of an s-electron with momentum $k$ and spin $\sigma$.
$a_{k\sigma}$: Symbol representing the mixed Green's function $\ll C_{k\sigma} | d_\sigma^+ \gg_\omega$.
$V_{kd}$: s-d mixing interaction strength, representing the matrix element for electron transfer between s and d state... | |
487 | Anderson s-d exchange model and Green's function equation of motion
In 1961, Anderson proposed the s-d mixing model, suggesting that discussing the formation of local magnetic moments of transition-metal impurity atoms in a non-magnetic metal matrix must simultaneously consider two factors: First, similar to the form... | [] | Equation | Strongly Correlated Systems | $\omega$: Energy or frequency in Fourier space.
$E_{k'\sigma}$: Energy of an s-orbital electron with wave vector $k'$ and spin $\sigma$, following the general definition $E_{k\sigma} = E_k + \sigma \mu_B h$.
$a_{k'\sigma}$: Symbol representing the mixed Green's function $\ll d_\sigma | C_{k'\sigma}^+ \gg_\omega$.
$V_{k... | |
488 | In 1961, Anderson proposed the s-d mixing model, where he considered that the formation of localized magnetic moments by transition metal impurity atoms in non-magnetic metal matrices must take into account two factors: first, similar to the formation of intrinsic magnetic moments in free atoms, the Coulomb interaction... | [] | Equation | Strongly Correlated Systems | $\omega$: Frequency in the Fourier transform.
$E_{d\sigma}$: Energy of an electron in d-orbital state with spin $\sigma$.
$U$: Coulomb interaction strength between d-shell electrons.
$n_{d\bar{\sigma}}$: Number operator for electrons in d-orbital state with spin $\bar{\sigma}$. In the final approximate equation, it rep... | |
489 | Anderson s-d exchange model and Green's function equation of motion
In 1961, Anderson proposed the s-d mixing model. He considered that the discussion of the formation of a localized magnetic moment by transition metal impurity atoms in non-magnetic metallic matrices must account for two factors: Firstly, similar to t... | [] | Expression | Strongly Correlated Systems | $\delta_{kk'}$: Kronecker delta, equal to 1 if $k=k'$ and 0 otherwise.
$\omega$: Frequency in the Fourier transformed Green's function.
$E_{k\sigma}$: Energy of an s-orbital electron with wave vector $k$ and spin $\sigma$, $E_{k\sigma} = E_k + \sigma \mu_B h$.
$V_{kd}$: s-d mixing matrix element (hybridization strength... | |
490 | Please analyze the conservation laws during the electron-phonon interaction process. You should return your answer as an equation. | [] | Equation | Superconductivity | $\hbar$: Reduced Planck's constant
$\mathbf{k}'$: Final wavevector of an electron
$\mathbf{k}$: Initial wavevector of an electron
$\mathbf{q}$: Wavevector of a phonon
$\mathbf{K}_n$: Reciprocal lattice vector | |
491 | Please provide the expression for the transition probability of the system from the initial to the final state during electron-phonon interaction (considering the long-time limit). You should return your answer as an equation. | [] | Equation | Superconductivity | $W(i \rightarrow f)$: Transition probability from initial state $i$ to final state $f$
$\hbar$: Reduced Planck's constant
$|f\rangle$: Final state of the system
$H_{ep}$: Hamiltonian for electron-phonon interaction
$|i\rangle$: Initial state of the system
$\delta$: Dirac delta function
$E_f$: Energy of the final state
... | |
492 | Please provide the energy conservation relation in the electron-phonon interaction process and explain the specific relationship between electron energy and phonon energy before and after scattering. We only consider the case that the electron absorbs a phonon. You should return your answer as an equation. | [] | Equation | Superconductivity | $\varepsilon_{\mathbf{k} + \mathbf{q}}$: Energy of the electron after absorbing a phonon, with its wave vector changed from $\mathbf{k}$ to $\mathbf{k} + \mathbf{q}$.
$\varepsilon_{\mathbf{k}}$: Energy of the electron before scattering.
$\hbar$: Reduced Planck's constant.
$\omega_{\mathbf{q}}$: Frequency of the absorbe... | |
493 | Find the equation of motion for $Q_q$, and discuss how to derive the phonon frequency correction You should return your answer as an equation. | [] | Equation | Superconductivity | $\ddot{Q}_{\mathbf{q}}$: Second time derivative of the normal coordinate $Q_{\mathbf{q}}$.
$\Omega_{\mathbf{q}}$: Frequency of perturbed LA phonons for wave vector $\mathbf{q}$.
$Q_{\mathbf{q}}$: Normal coordinate of phonons for wave vector $\mathbf{q}$.
$M_{-\mathbf{q}}$: Coupling coefficient for electron-phonon inter... | |
494 | Calculate the ionic density fluctuations $\rho^i_q$ produced by lattice vibrations | [] | Expression | Superconductivity | $\rho_{\mathbf{q}}^i$: Fourier component of ionic density fluctuation with wave vector $\mathbf{q}$
$i$: Imaginary unit
$N$: Number density of ions (number of ions per unit volume)
$M$: Mass of a single ion
$\mathbf{e}_{\mathbf{q}}$: Polarization vector of phonons with wave vector $\mathbf{q}$
$\mathbf{q}$: Wave vector... | |
495 | Using linear response theory, calculate the relationship between $\rho_q$ and $\rho^i_q$ | [] | Expression | Superconductivity | $\rho_{\mathbf{q}}$: Fourier component of the electron charge density at wave vector $\mathbf{q}$
$\epsilon(\mathbf{q})$: Static dielectric function at wave vector $\mathbf{q}$
$\rho_{\mathbf{q}}^i$: Fourier component of the ionic charge density at wave vector $\mathbf{q}$ | |
496 | Under the long-wavelength approximation, considering the electron screening effect, derive the dispersion relation between the LA phonon angular frequency $\omega_{\mathbf{q}}$ and the wave vector $q$, and specify its form $\omega_{\mathbf{q}}$ with respect to $q$. Hint: You can use the Thomas-Fermi dielectric function... | [] | Equation | Superconductivity | $\omega_{\mathbf{q}}$: LA phonon angular frequency for wave vector $\mathbf{q}$.
$c_L$: Longitudinal sound velocity.
$q$: Magnitude of the wave vector $\mathbf{q}$. | |
497 | With the known dispersion relation of LA phonons in the form $\omega_{\mathbf{q}} = c_L q$, please provide the specific expression for the speed of sound of LA phonons $c_L$ (i.e., the Bohm-Staver speed of sound formula), and explain each physical quantity. You should return your answer as an equation. | [] | Equation | Superconductivity | $c_L$: Speed of sound of LA phonons
$m$: Electron mass
$M$: Ion mass
$v_F$: Fermi velocity | |
498 | When electrons move within an ionic crystal, they cause relative displacements between positive and negative ions, forming a local polarization field. This polarization, accompanying the electron motion, excites LO phonons, leading to the renormalization of the electron ground state energy and effective mass, forming a... | [] | Expression | Superconductivity | $m^*$: Effective mass of the polaron.
$m$: Effective mass of the band electron.
$\alpha$: Dimensionless electron-LO phonon coupling constant, defined as $\alpha = \frac{e^2}{2\hbar\omega_L} \left( \frac{2m\omega_L}{\hbar} \right)^{1/2} \left( \frac{1}{\epsilon_\infty} - \frac{1}{\epsilon_0} \right)$. | |
499 | Please calculate the average number of virtual phonons excited around the electron. | [] | Expression | Superconductivity | $\langle N_{\text{ph}} \rangle$: Average number of virtual phonons excited around the electron
$\alpha$: Fröhlich coupling constant | |
500 | Analyze the situation near the Fermi surface and describe the interaction when the attractive potential is greater than the screened Coulomb potential, and elaborate on the approximation method of BCS theory. | [] | Expression | Superconductivity | $V_{\text{net}}$: Net potential, sum of the attractive electron-phonon mediated potential and the repulsive screened Coulomb potential.
$V_{\mathbf{k}_1,\mathbf{q}}$: Interaction coefficient/potential for electron-phonon mediated interaction.
$e$: Elementary charge of an electron.
$q$: Magnitude of the momentum transfe... |
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