820 KiB
820 KiB
In [1]:
import numpy as np from matplotlib import pyplot as plt from scipy.optimize import fsolve def dr(*x, p = 3): display(np.round(x, p))
In [2]:
A_metal = 4.08 # a_metal = np.array(((1/np.sqrt(2), 1/np.sqrt(2), 0), (1/np.sqrt(2), 0, 1/np.sqrt(2)), (0, 1/np.sqrt(2), 1/np.sqrt(2)))) # dr(a_metal) # a_metal *= A_metal # dr(a_metal) # with a_metal as a: # V_metal = np.dot(a[0], np.cross(a[1], a[2])) # dr(V) # a_ = 2*np.pi/V * np.array((np.cross(a[1],a[2]), np.cross(a[0], a[2]), np.cross(a[0], a[1]))) # dr(a_) # X = np.array((0, 0, 1)) * A # L = np.array((0.5, 0.5, 0.5)) * A # K = np.array((0.75, 0.75, 0)) * A # dr(X, L, K)
In [3]:
TD = 165 Tpl = 1337 a = A_metal * 10**-10
In [4]:
T = np.logspace(1, np.log10(Tpl*1.2), 100) def h1(T): return 50*a*Tpl/T def h2(T): return h1(T)*(TD / T)**5 def h(T): if T > TD: return h1(T) return h2(T) fig, ax = plt.subplots() ax.plot(T, list(map(h1, T)), color='#999', label=r'$\lambda_1(T)$') ax.plot(T, list(map(h2, T)), color='#555', label=r'$\lambda_2(T)$') ax.plot(T, list(map(h, T)), 'k', linestyle=(0,(3,2,1,2)), linewidth=3, label=r'$\lambda(T)$') ax.set_ylim(10**-11, 10) ax.vlines((0.1*TD, TD, Tpl), *ax.get_ylim(), linestyles='--', colors='k') ax.set_xlim(10, T[-1]) ax.legend() ax.set_xscale('log') ax.set_yscale('log') ax.set_xticks((10, TD*0.1, 100, TD, 1000, Tpl)) ax.set_xticklabels(('$10^1$', r'$0.1 \cdot T_D$', '$10^2$', '$T_D$', '$10^3$', '$T_{пл}$')) ax.set_xlabel("T, К") ax.set_ylabel(r"$\lambda, м$") ax.grid(which="both") fig.tight_layout() print(h(0.1*TD), h(TD), h(Tpl))
0.16530181818181822 1.653018181818182e-07 2.04e-08
In [5]:
h_pl = 6.6237 * 10**-34 n0 = 7.08 * 10**29 me = 9.1 * 10**-31 e = 1.6*10**-19 rho = 2.2*10**-8 # td = 7*10**-14 vF = h_pl*(3*(np.pi**2)*n0)**(1/3)/(2*np.pi*me) def tf(T): return h(T)/vF ts273 = me/(rho*(e**2)*n0) td273 = 1/(1/ts273 - 1/tf(273)) # td273 = 7*10**-12 def ts(T, td = td273): return 1/(1/td+1/tf(T)) print('vF', vF) print('h', h(273)) print('tf', tf(273)) print('ts', ts273) print('td', td273) print(A_metal*10**-10/vF) fig, ax = plt.subplots() T = np.logspace(1, np.log10(2*10**4)) ax.plot(T, list(map(tf, T)), color='#999', label=r'$\tau_f(T)$') ax.plot(T, [td273] * len(T), color='#555', label=r'$\tau_d(T)$') ax.plot(T, list(map(ts, T)), 'k', label=r'$\tau_\Sigma(T)$') ax.set_ylim(3*10**-16, 10**-13) ax.vlines((0.1*TD, TD, Tpl), *ax.get_ylim(), linestyles=(0, (1, 10)), colors='k', linewidth=2) ax.set_xlim(10, T[-1]) ax.legend() ax.set_xscale('log') ax.set_yscale('log') ax.set_xticks((10, TD*0.1, 100, TD, 1000, Tpl, 10000)) ax.set_xticklabels(('$10^1$', r'$0.1 \cdot T_D$', '$10^2$', '$T_D$', '$10^3$', '$T_{пл}$', '$10^4$')) ax.tick_params(axis='x', labelrotation = 45) ax.set_xlabel("T, К") ax.set_ylabel(r"$\tau, с$") ax.grid(which="both") fig.tight_layout() for tau_d in (10**-12, 10**-13, 10**-14): print([ts(T, tau_d) for T in (0.1*TD, TD, Tpl)])
vF 3194215.0097415945 h 9.990769230769232e-08 tf 3.1277697964287836e-14 ts 2.282156843862352e-15 td 2.461778939492735e-15 1.2773091315258907e-16 [9.999806768411713e-13, 4.92040452983814e-14, 6.34601653329549e-15] [9.999980676505119e-14, 3.410230365430928e-14, 6.003151637409561e-15] [9.999998067647152e-15, 8.380576583604518e-15, 3.8974325590432934e-15]
In [6]:
k0 = 1.3806*10**-23 L0 = 1/3 *(np.pi*k0/e)**2 print(L0) def sigma(T, td): return e**2*n0*ts(T, td)/me def kappa(T, td): return L0*T*sigma(T, td) T = np.logspace(-10, 3, 1000) for tau_d in (10**-12, 10**-13, 10**-14): fig, (ax1, ax2) = plt.subplots(2, sharex=True) ax1.plot(T, list(map(lambda t: sigma(t, tau_d), T)), 'k') ax2.plot(T, list(map(lambda t: kappa(t, tau_d), T)), 'k') ax2.set_xlim(0, 1000) ax1.set_ylabel(r'$\sigma(T)$') ax2.set_ylabel(r'$\kappa(T)$') ax2.set_xlabel('T, К') ax1.grid() ax2.grid() fig, (ax1, ax2) = plt.subplots(2, sharex=True) colors=('#000', '#999', '#555') for i, p in enumerate((-12, -13, -14)): ax1.plot(T, list(map(lambda t: sigma(t, 10**p), T)), color=colors[i], label=rf'$\sigma(T), \tau_d = 10^{{{p}}}$') ax2.plot(T, list(map(lambda t: kappa(t, 10**p), T)), color=colors[i], label=rf'$\kappa(T), \tau_d = 10^{{{p}}}$') ax2.set_xlim(0, 1000) ax1.set_ylabel(r'$\sigma(T)$') ax2.set_ylabel(r'$\kappa(T)$') ax2.set_xlabel('T, К') ax1.set_yscale('log') ax2.set_yscale('log') ax1.legend() ax2.legend() ax1.grid() ax2.grid() fig.tight_layout() for tau_d in (10**-12, 10**-13, 10**-14): print([(sigma(T, tau_d), np.log10(sigma(T, tau_d))) for T in (0.1*TD, TD, Tpl)]) for tau_d in (10**-12, 10**-13, 10**-14): print([kappa(T, tau_d) for T in (0.1*TD, TD, Tpl)])
2.449482062419317e-08 [(19916977771.00095, 10.299223438658192), (980014813.4330806, 8.991232640328844), (126395912.59634517, 8.10173302991508)] [(1991732415.005714, 9.299230991470994), (679227948.652335, 8.83201554781269), (119566948.12936352, 8.07761114429435)] [(199173587.88625395, 8.2992317467595), (166918982.92584085, 8.222505729795717), (77626577.63312955, 7.89001043959881)] [8049.736164965731, 3960.8723655757485, 4139.412441329386] [804.9876159097746, 2745.19851625431, 3915.7667561536655] [80.49890158778761, 674.6273400143037, 2542.2374397389285]
In [7]:
def rho1(gamma, p): return rho*(1+3/(8*gamma)*(1-p)) def rho2(gamma, p): return rho*(4/(3*gamma*(np.log(1/gamma)+0.423))*(1-p)/(1+p)) gamma = np.logspace(-2, 1, 100) for p in (0, 0.5): fig, ax = plt.subplots() ax.plot(gamma, [rho/rho] * len(gamma), label=r'$\rho$', color='k') ax.plot(gamma, list(map(lambda g: rho1(g, p)/rho, gamma)), color='#999', label=rf'$\rho_1(\gamma, {p})$') ax.plot(gamma, list(map(lambda g: rho2(g, p)/rho, gamma)), color='#555', label=rf'$\rho_2(\gamma, {p})$') gi = np.array((0.023 if p==0 else 0.01, *gamma[::5])) scaler = 0.1 if p == 0 else -0.08 ax.scatter(gi[gi > 0.55], list(map(lambda g: rho1(g, p)/rho + scaler/(g-0.55)**(0.5), gi[gi > 0.55])), marker='o', c='k') ax.scatter(gi[gi < 0.55], list(map(lambda g: rho2(g, p)/rho - scaler/(0.55-g)**(0.5), gi[gi < 0.55])), marker='x', c='k') # ax.scatter(d[::11], list(map(lambda d: rho1(d, p) if d/h(Tpl) > 0.55 else rho2(d, p), d[::11]))) ax.set_ylim(0, 15) ax.set_xlim(np.min(gamma), np.max(gamma)) ax.set_xscale('log') ax.set_xticklabels(ax.get_xticks()) ax.set_xlabel(r'$\gamma$') ax.set_ylabel(fr'$\rho(\gamma, {p})/\rho$') ax.legend() ax.grid(which="both") fig.tight_layout() print(rho1(3, 0)/rho, 3*h(Tpl))
/tmp/ipykernel_3418/4170067680.py:31: UserWarning: FixedFormatter should only be used together with FixedLocator ax.set_xticklabels(ax.get_xticks())
1.125 6.12e-08
In [8]:
mn = 0.0133*me mp = 0.6*me EG = 0.17 * e print('mn', mn) print('mp', mp) def EF(T): return 0.5*EG+3/4*k0*T*np.log(mp/mn) fig, ax = plt.subplots() ax.plot(T, list(map(EF, T)), 'k', label='$E_F(T)$') ax.plot(T, [EF(0)]*len(T), 'k', linestyle='--', label=r'$0.5 \cdot E_G$') ax.plot(T, list(map(lambda t: k0*t*1.6, T)), color='#999', label='$E_T(T)$') ax.set_xlabel("T, К") ax.set_ylabel("E, Дж") ax.legend() ax.set_xlim(np.min(T), np.max(T)) ax.set_ylim(0, ax.get_ylim()[1])
mn 1.2103e-32 mp 5.46e-31
Out[8]:
(0.0, 5.569410568277869e-20)
In [9]:
def fa(En, T): return 1/(np.exp((En-EF(T))/(k0*T))+1) fig, ax = plt.subplots() En = np.linspace(0, EF(300)*2, 1000) print(EF(50), EF(250), EF(300)) styles = {300: '-', 250: '--', 50: '-.'} for t in (300, 250, 50): En = np.linspace(0, EF(t)*2, 1000) ax.plot(En-EF(t), list(map(lambda E: fa(E, t), En)), 'k' + styles[t], label=f'T = {t} К') ax.set_ylabel("$f(E_n)$") ax.set_xlabel("$E_n - E_F(T)$") ax.legend() ax.grid()
1.5572100270608516e-20 2.3460501353042574e-20 2.5432601623651087e-20
In [10]:
def EFa(T): return EF(0)*(1-np.pi**2/12*(k0*T/EF(0))**2) def g(Eps): return 2*np.pi*(2*mn)**(3/2)/h_pl**3*Eps**(1/2) def N(Eps): return g(Eps)*fa(Eps, 300) fig, ax = plt.subplots() Eps = np.linspace(0, 10*EFa(300), 1000) ax.plot(Eps, list(map(g, Eps))) ax.plot(Eps, list(map(N, Eps))) # ax.plot(Eps, list(map(lambda x: np.exp(x), Eps))) # ax.plot(Eps, list(map(lambda x: np.exp(x)/(0.3*np.exp(x)+1), Eps))) Ec = 4.665*e print(EF(0)/e, EF(50)/e, EF(300)/e, EF(Tpl)/e, Ec) print(4.835*e+EF(300), k0*300) # if True: # T = np.linspace(1, 1000) # ax.plot(T, list(map(lambda t: (EF(t))/(k0*t), T))) # ax.set_yscale('log') # def Nc(T): # return # print(Eg/(k0*np.log(Nc(0)*Nv(0)/Nd**2)))
0.085 0.09732562669130322 0.15895376014781928 0.41458725772544797 7.464e-19 7.990326016236512e-19 4.1417999999999996e-21
In [11]:
def n0(T): return 2*((2*np.pi*mn*k0*T)/h_pl**2)**(3/2)*np.exp((EF(T)-EG)/(k0*T)) def p0(T): return 2*((2*np.pi*mp*k0*T)/h_pl**2)**(3/2)*np.exp((-EF(T))/(k0*T)) fig, ax = plt.subplots() T = np.linspace(50, 1000, 1000) ax.plot(T**-1, list(map(n0, T)), c='#999', linestyle=(7, (7, 7)), label='$n(T)$') ax.plot(T**-1, list(map(p0, T)), 'k', linestyle=(0, (7, 7)), label='$p(T)$') ax.set_yscale('log') ax.set_xlim(0, 10**-2) ax.set_ylim(10**19, 10.0**24) ax.legend() ax.grid(which='both') ax.set_xlabel('$T^{-1}$')
Out[11]:
Text(0.5, 0, '$T^{-1}$')
In [12]:
Eg = 0.03 * e Nd = 10**22 def nd(T): return (2*Nd)/(1+np.sqrt(1+(8*Nd/(2*(2*np.pi*mn*k0*T/h_pl**2)**(3/2)))*np.exp(Eg/(k0*T)))) def nd1(T): return np.sqrt(1/2*Nd*2*((2*np.pi*mn)/h_pl**2)**(3/2))*(k0*T)**(3/4)*np.exp(-Eg/(2*k0*T)) def n(T): return n0(T) + nd(T) Ts = fsolve(lambda x: nd1(x[0]) - n(x[0]), [1/0.007])[0] Ti = fsolve(lambda x: nd1(x[0]) - Nd, [400])[0] print(Ts, Ti) fig, ax = plt.subplots() ax.set_xlim(0, 0.013) ax.set_ylim(10.0**20, 2.3*10.0**23) ax.plot(T**-1, list(map(n0, T)), 'k--', label='$n_0(T)$') ax.plot(T**-1, list(map(nd, T)), 'k-.', label='$n_d(T)$') ax.plot(T**-1, list(map(nd1, T)), '-.', c='#888', label='$n_{d1}(T)$') ax.plot(T**-1, [Nd]*len(T), 'k', linestyle='dotted', label='$n_{d2}(T) = N_d$') ax.plot(T**-1, list(map(n, T)), 'k', label='n(T)') ax.vlines([1/Ts, 1/Ti], *ax.get_ylim(), colors='k', linestyles=(0, (5, 5)), linewidth=1) ax.annotate(r'$\dfrac{1}{T_s}$', (1/Ts-0.0005, 1.3*10.0**20)) ax.annotate(r'$\dfrac{1}{T_i}$', (1/Ti-0.0005, 1.3*10.0**20)) ax.text(1/Ts+(ax.get_xlim()[1]-1/Ts)/2, 1.3*10.0**20, 'I') ax.text(1/Ti+(1/Ts-1/Ti)/2, 1.3*10.0**20, 'II') ax.text(1/Ti/2, 1.3*10.0**20, 'III') ax.set_yscale('log') ax.legend() ax.grid(which='both') ax.set_xlabel('$T^{-1}$') fig.tight_layout()
147.95825436672482 365.6284735670228
In [13]:
fig, ax = plt.subplots() ax.plot(T, list(map(n0, T)), 'k--', label='$n_0(T)$') ax.plot(T, list(map(nd, T)), 'k-.', label='$n_d(T)$') ax.plot(T, list(map(nd1, T)), '-.', c='#888', label='$n_{d1}(T)$') ax.plot(T, [Nd]*len(T), 'k', linestyle='dotted', label='$n_{d2}(T) = N_d$') ax.plot(T, list(map(n, T)), 'k', label='n(T)') ax.vlines([Ts, Ti], *ax.get_ylim(), colors='k', linestyles=(0, (5, 5)), linewidth=1) ax.text(Ti+(ax.get_xlim()[1]-Ti)/2, 1.3*10.0**21, 'III') ax.text(Ts+(Ti-Ts)/2, 1.3*10.0**21, 'II') ax.text(Ts/2, 1.3*10.0**21, 'I') ax.annotate('$T_s = $' + str(np.round(Ts)), (Ts+10, 1.3*10.0**20)) ax.annotate('$T_i = $' + str(np.round(Ti)), (Ti+10, 1.3*10.0**20)) ax.set_yscale('log') ax.legend() ax.grid(which='both') ax.set_xlabel('$T, К$') # ax.set_xlim(0, 0.013) ax.set_ylim(10.0**20, 2.3*10.0**23) fig.tight_layout()
In [14]:
def mun(T): return 76000*10**-4*(T/300)**-2 def mup(T): return 5000**10**-4*(T/300)**-2.7 fig, ax = plt.subplots() ax.plot(T, list(map(mun, T)), c='#999', linestyle=(7, (7, 7)), label=r'$\mu_n(T)$') ax.plot(T, list(map(mup, T)), c='#000', linestyle=(7, (7, 7)), label=r'$\mu_p(T)$') ax.set_yscale('log') ax.legend() ax.set_xlim(50, 1000) ax.set_ylim(ax.get_ylim()[0], 10**3) ax.set_xlabel('T, К') ax.grid(which='both') print(mun(77))
115.36515432619332
In [15]:
def sigma(T): return e*n(T)*mun(T) + e*p0(T)*mup(T) def sigma0(T): return e*n0(T)*mun(T) + e*p0(T)*mup(T) fig, ax = plt.subplots() ax.plot(T, list(map(sigma, T)), 'k', label=r'$\sigma(T)$') ax.plot(T, list(map(sigma0, T)), 'k--', label=r'$\sigma_0(T)$') ax.set_yscale('log') ax.set_xlim(50, 1000) ax.set_xlabel('T, К') ax.set_ylabel(r'$\sigma(T), См$') ax.grid(which='both')
In [16]:
chi = 4.665*e def Phin_s(T): return chi + EF(T) def Nc(T): return 2*(mn*k0*T/(2*np.pi*h_pl**2))**(3/2) def EF_p(T): return (EG+Eg)/2 + 1/2 * k0 * T * np.log(Nd/(2*Nc(T))) def Phin_p(T): return chi + k0*T*np.log(Nc(T)/Nd) for T in (0.1*TD, TD, Tpl): print(EF(T), Phin_s(T), EF_p(T), Phin_p(T)) print(EF(T)/e, Phin_s(T)/e, EF_p(T)/e, Phin_p(T)/e)
1.4250793089300812e-20 7.606507930893009e-19 1.689112988123594e-20 7.444598420029438e-19 0.08906745680813007 4.75406745680813 0.10556956175772461 4.6528740125183985 2.0107930893008098e-20 7.665079308930082e-19 2.0977349450367212e-20 7.348663187534218e-19 0.1256745680813006 4.7906745680813 0.13110843406479505 4.5929144922088865 6.633396123607168e-20 8.127339612360717e-19 2.7366739214353514e-20 7.108719797749712e-19 0.41458725772544797 5.079587257725447 0.17104212008970945 4.4429498735935695
In [17]:
ephik = -0.1*e A = 4*np.pi*e*me*k0**2/h_pl**3 lims = {300: (10, 18), 250: (10, 18), 50: (17, 23)} for T in (300, 250, 50): js = A*T**2*np.exp(-ephik/(k0*T)) def j(U): return js*(np.exp(e*U/(k0*T))-1) fig, ax = plt.subplots() U = np.logspace(-5,0, 1000) ax.plot(U, list(map(j, U)), 'k', label=r'$j(U)$, Прямое смещение') ax.plot(U, list(map(lambda x: -j(-x), U)), 'k--', label=r'$-j(-U)$, Обратное смещение') print((10**-3)/j((10**-3))) ax.set_yscale('log') ax.set_xscale('log') ax.set_ylim(10.**lims[T][0], 10.**lims[T][1]) ax.set_xlim(np.min(U), np.max(U)) ax.set_xlabel('$U, В$') # ax.set_ylabel(r'$j(U)$') ax.legend() ax.grid(which='both')
4.937450473004247e-15 2.7255474336441733e-15 1.0971095076797715e-22
In [18]:
PHI = 4.75*e epsilon0 = 8.85*10**-12 epsilon = 17.72 for T in (300,): v = np.sqrt(3*k0*T/mn) E = mn*v**2/2 print(v, E) def W(D): return -h_pl*np.log(D)/(2*(PHI-chi-E)) def N(W): return 2*epsilon0*epsilon*ephik/(W*e)**2 for D in (0.1, 0.01): print(W(D)) print(N(W(D)))
1013231.4758388158 6.212699999999999e-21 1.0322873634795202e-13 -1.8395677158075775e+34 2.0645747269590405e-13 -4.5989192895189437e+33
In [19]:
def n(T): return 1/(3*np.pi**2)*(2*np.pi*mn/h_pl**2)**(3/2)*(EF(T)) def p(T): return 1/(3*np.pi**2)*(2*np.pi*mp/h_pl**2)**(3/2)*(EG-EF(T)) fig, ax = plt.subplots() T = np.linspace(50, 1000, 1000) ax.plot(T**-1, np.array(list(map(n, T)))/10**20) ax.plot(T**-1, np.array(list(map(p, T)))/10**20) # ax.set_xlim(0, 10**-2) # ax.set_ylim(4*10**11, 10.0**12) ax.set_yscale('log') ax.grid(which='both')
