{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"# Band Structures and Newton's Method\n",
"\n",
"### Example – Quantum Mechanics\n",
"\n",
"By Øystein Hiåsen, Henning G. Hugdal and Peter Berg \n",
"\n",
"Last edited: April 15 2016\n",
"\n",
"___\n",
"\n",
"One of the first great triumphs of quantum mechanics in applied physics was the electron band theory of solids [[1]](#rsc). The band structure arises due to the periodic potential experienced by the electrons in a solid. In the following, we will use Newton's method to calculate the band structure for the simple Dirac comb potential in one dimension."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Particle in a One-Dimensional Lattice\n",
"\n",
"A simple model for a particle in a periodic lattice, i.e. a crystal, is a periodic potential of delta peaks in one dimension, also known as the Dirac comb potential,\n",
"\n",
"$$\n",
"V(x) = V_0 \\sum_{j}\\delta(x-ja),\n",
"$$\n",
"\n",
"where $a$ is the distance between the delta function peaks. In order to make this seem like a more realistic model, we can imagine that the potential wraps around in a circle that includes a macroscopic amount of peaks, $N \\sim 10^{23}$. This means that the wavefunction of the system has to fulfill periodic boundary conditions,\n",
"\n",
"$$\n",
"\\psi(x+Na)=\\psi(x).\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Bloch's Theorem\n",
"\n",
"Bloch's theorem [[2]](#rsc) states that for a periodic potential \n",
"\n",
"$$\n",
"V(x + a ) = V(x)\n",
"$$\n",
"\n",
"of period $a$, the wavefunction $\\psi(x)$ satisfies \n",
"\n",
"$$\n",
"\\psi(x +a) = e^{iKa} \\psi(x).\n",
"$$\n",
"\n",
"Using periodic boundary conditions as stated above, we find\n",
"\n",
"$$\n",
"\\psi(x+Na) = e^{iNKa} \\psi(x) = \\psi(x).\n",
"$$\n",
"\n",
"Hence, we must have $e^{iKNa} = 1$ which means\n",
"\n",
"$$\n",
"K=\\frac{2\\pi n}{Na}, \\qquad n\\in \\mathbb{Z}.\n",
"$$\n",
"\n",
"For large $N$, which is the case in macroscopic materials, these values of $K$ essentially represent a continuum. \n",
"\n",
"The above implies that one only needs to find $\\psi(x)$ within a single period with respect to $x$, for example $0"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Inline plotting:\n",
"%matplotlib inline\n",
"# Import needed modules:\n",
"from __future__ import division\n",
"import numpy as np\n",
"from numpy import pi\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib.collections as collections\n",
"\n",
"# Set common figure parameters:\n",
"newparams = {'axes.labelsize': 14, 'axes.linewidth': 1, 'savefig.dpi': 300, \n",
" 'lines.linewidth': 1.0, 'figure.figsize': (8, 3),\n",
" 'figure.subplot.wspace': 0.4,\n",
" 'ytick.labelsize': 10, 'xtick.labelsize': 10,\n",
" 'ytick.major.pad': 5, 'xtick.major.pad': 5,\n",
" 'legend.fontsize': 10, 'legend.frameon': False, \n",
" 'legend.handlelength': 1.5}\n",
"plt.rcParams.update(newparams)\n",
"\n",
"# Define default values of the parameters beta, and start and stop values of z.\n",
"DEFAULT_BETA = 10\n",
"DEFAULT_ZSTART = 0\n",
"DEFAULT_ZSTOP = 20\n",
"\n",
"# Define needed functions\n",
"def f(z, beta=DEFAULT_BETA):\n",
" \"\"\"Returns function value f(z) for a given value of beta.\"\"\"\n",
" return np.cos(z) + beta*np.sin(z)/z\n",
"\n",
"def plot_f(beta=DEFAULT_BETA, zstop=DEFAULT_ZSTOP, zstart=DEFAULT_ZSTART, ax=None):\n",
" \"\"\"\n",
" Plots f(z) for a given beta and range of z.\n",
" \"\"\"\n",
" if ax is None:\n",
" ax = plt.gca()\n",
" \n",
" z = np.delete(np.linspace(zstart, zstop, 10000), 0)\n",
" ax.set_xlabel(r'$z$')\n",
" ax.set_ylabel(r'$f(z)$')\n",
" pi_start = zstart//pi\n",
" pi_stop = zstop//pi\n",
" ticks = np.arange(pi_start, pi_stop+1)\n",
" ax.set_xticks(ticks*pi)\n",
" fvalues = f(z, beta)\n",
" ticklabel = ['%d $\\pi$' % t if t!=0 else'0' for t in ticks]\n",
" ax.set_xticklabels(ticklabel)\n",
" ax.plot(z, fvalues, label=r'$\\beta = {}$'.format(beta))\n",
"\n",
"def plot_allowed_area(zstop=20, zstart=0, ax=None):\n",
" \"\"\"Plot horizontal lines at y=+/- 1\"\"\"\n",
" if ax is None:\n",
" ax = plt.gca()\n",
" collection=collections.BrokenBarHCollection(xranges=[(zstart,zstop-zstart)],yrange=(-1,2), facecolor='#cccccc' )\n",
" ax.add_collection(collection)\n",
"\n",
"\n",
"# Plot f(z) with beta equal to 5:\n",
"plot_f(beta=5.0)\n",
"plot_allowed_area()\n",
"plt.legend()\n",
"plt.grid()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The above plot shows $f(z)$ for $\\beta=5$ together with a grey area, indicating the allowed values of $f(z)$. The values of $z$ for which the graph is outside the grey area, correspond to forbidden energies, or band gaps. Likewise, we find bands consisting of permissible energy values and separated by band gaps.\n",
"\n",
"We want to calculate the values $z$ for which we have $f(z) \\in [-1,1].$ An interesting feature to note about $f(z)$ is that if $z$ is an integer multiple of $\\pi$, meaning $z=n\\pi , (n\\in \\mathbb{Z})$, then $f(z)$ assumes values that are independent of $\\beta$:\n",
"$$f(n\\pi) = (-1)^n . $$\n",
"This marks the upper bound of a band. Hence, we can focus on finding the lower bounds of $z$ for each band. This will be done using Newton's Method."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Newton's Method\n",
"\n",
"Newton's method is an iterative method for finding the roots of a function by following its tangent line from an initial guess.\n",
"\n",
"In our case, we want to find out when\n",
"\n",
"$$\n",
"f(z) - y_0 = 0,\n",
"$$\n",
"\n",
"given $y_0 = \\pm 1$, where the sign depends on which band we consider.\n",
"\n",
"Given an initial guess for a value $z$ of the root, Newton's method computes the next, hopefully better, approximation of the correct value, according to\n",
"\n",
"$$\n",
"z_{n+1} = z_n - \\frac{f(z_n) - y_0}{f'(z_n)}.\n",
"$$\n",
"\n",
"In the case of our function\n",
"\n",
"$$\n",
"f(z) = \\cos(z) + \\beta \\frac{\\sin(z)}{z},\n",
"$$\n",
"\n",
"we have\n",
"\n",
"$$\n",
"f'(z) = -\\sin(z) + \\beta \\frac{z\\cos(z) - \\sin(z)}{z^2}.\n",
"$$\n",
"\n",
"The derivative, $f'(z)$, and a function returning the next Newton iteration, $z_{n+1}$, are implemented below."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def df(z, beta=DEFAULT_BETA):\n",
" \"\"\"Returns the derivative of f(z).\"\"\"\n",
" return -np.sin(z) + beta*(z*np.cos(z) - np.sin(z))/z**2\n",
"\n",
"def newton_iteration(z_n, beta=DEFAULT_BETA, y0=1):\n",
" \"\"\"One iteration of Newton's method with function f.\n",
" For a given guess of root value, z_n, it returns z_(n+1).\n",
" \"\"\"\n",
" return z_n - (f(z_n, beta) - y0)/df(z_n, beta)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Testing Newton's Method\n",
"\n",
"To check whether the method works as expected, we can test it by trying to locate the start value of $z$ for different bands when starting at the end value, a integer multiple of $\\pi$. The following code will run $N$ iterations of Newton's method with given initial values $z_0$, and plot the points calculated at each iteration."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"def test_newton_iterations(z0, beta=DEFAULT_BETA, y0=1, N=5):\n",
" \"\"\"Returns the N first iterations of the Newton's method for f(z) given intial value z0.\"\"\"\n",
" zn = np.zeros(N+1)\n",
" zn[0] = z0\n",
" for i in range(N):\n",
" zn[i+1] = newton_iteration(zn[i], beta, y0)\n",
" return zn\n",
" \n",
"def insert_annotations(zn, values, ax=None):\n",
" \"\"\"Code for putting labels near the iteration points\n",
" Arguments:\n",
" zn Iteration points\n",
" values Function values f(z) at points zn\n",
" ax Axes instance\n",
" \"\"\"\n",
" if ax is None:\n",
" ax = plt.gca()\n",
" previous_ytext = 0\n",
" previous_z = 10e9\n",
" for i,z in enumerate(zn):\n",
" if abs(z-previous_z) < 0.3:\n",
" ytext = previous_ytext + 10\n",
" else:\n",
" ytext = 4\n",
" ax.annotate(\"z{}\".format(i), xy=(z,values[i]), textcoords='offset points', xytext=(0, ytext))\n",
" previous_ytext = ytext\n",
" previous_z = z\n",
" \n",
"def plot_tangents(zlist, beta=DEFAULT_BETA, ax=None):\n",
" \"\"\"Plot the tangents of f(z) on the points in zlist.\"\"\"\n",
" if ax is None:\n",
" ax = plt.gca()\n",
" zstart, zstop = ax.get_xlim()\n",
" z = np.linspace(zstart, zstop, 1000)\n",
" for zn in zlist:\n",
" y = df(zn, beta)*(z - zn) + f(zn, beta)\n",
" ax.plot(z, y)\n",
"\n",
"def purge_last_equal_iterations(iterations, precision=0.005):\n",
" \"\"\"Remove the last points that are practically equal\"\"\"\n",
" ind = np.where(np.abs( iterations - iterations[-1] )< precision)\n",
" return iterations[:(ind[0][0]+1)]\n",
"\n",
"def scale_axis_pi(ax=None):\n",
" \"\"\"Changes the x-axis to show ticks on integer values of pi.\"\"\"\n",
" if ax is None:\n",
" ax = plt.gca()\n",
" zstart, zstop = ax.get_xlim()\n",
" pi_start = zstart//pi\n",
" pi_stop = zstop//pi\n",
" ticks = np.arange(pi_start, pi_stop+1)\n",
" ax.set_xticks(ticks*pi)\n",
" ticklabel = [r'%d $\\pi$' % t if t!=0 else'0' for t in ticks]\n",
" ax.set_xticklabels(ticklabel)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
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Yv6ix2Kvqhaq6DEBEJolI26OsV66q/1DVZ/wdZEPontmdF0a9wL9X/ZtHFz6KW91+23eT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\n8SIrNYGTWsSSn98HVWXL3jI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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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YlTELN5MbU+Kn8FbyW3i5eRkdTQjhYByu2ANW4CGt9fdKqQBgu1JqtdY6w+hg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\ncfkvF6vViru7I24DXr3DFYf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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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w/Xwhhz7bjW+MmQ+cGvc8PUkXDEnn3ZW7eMa+hevG9fZ0OH5NRMiMyiQzKpMr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\nYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"beta = 7\n",
"z0_list = np.array([1,2,3])\n",
"\n",
"for i, z0 in enumerate(z0_list):\n",
" y0 = -1 if z0%2==0 else 1\n",
" plt.figure()\n",
" \n",
" iterations = test_newton_iterations((z0)*pi, beta=beta, y0=y0, N=5)\n",
" iterations = purge_last_equal_iterations(iterations)\n",
" values = f(iterations, beta)\n",
" \n",
" plot_f(zstop=5.5*pi, beta=beta)\n",
" plt.plot(iterations, values, '*')\n",
" insert_annotations(iterations, values)\n",
" \n",
" scale_axis_pi()\n",
" plot_allowed_area()\n",
" plot_tangents(iterations[:1], beta)\n",
" \n",
" plt.ylim([-4,12])\n",
" plt.xlim([0,5.5*pi])\n",
" plt.title(r'$z_0={}\\pi$, $\\beta={}$, $y={}$'.format(z0, beta, y0))\n",
" plt.grid()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"From the above plots, one can see that the convergence of Newton's method is highly dependent on the initial guess, $z_0$. For example, in the last plot we see that Newton's method converges towards the end value $z=2\\pi$ of the previous band instead of the start value of the current band. To obtain the desired convergence, it is crucial to avoid undesired stationary points.\n",
"\n",
"The below function is implemented in order to run Newton's method up to a given precision."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"PRECISION = 0.000001\n",
"MAX_NEWTON_ITERATIONS = 30 # Making sure we don't get an infinite loop.\n",
"\n",
"def run_newton(z_start, y0, beta=DEFAULT_BETA):\n",
" \"\"\"Runs Newton's method to calculate the lower value z with a given precision.\"\"\"\n",
" z_new = z_start\n",
" iteration = 0\n",
" while(abs(f(z_new, beta) - y0) > PRECISION):\n",
" iteration += 1\n",
" assert iteration <= MAX_NEWTON_ITERATIONS, \"The method has run for max iterations={}\".format(MAX_NEWTON_ITERATIONS)\n",
" z_new = newton_iteration(z_new, beta, y0)\n",
" return z_new"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This function will be used to calculate the lower bound of each of the first ten bands. \n",
"Since we know the end value of each band, it might be a good idea to begin the search from values close to, but slightly smaller than those end values. This is what the following code does, including a plot of the final band structure. "
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"def calculate_bands(beta=DEFAULT_BETA, num_bands=10, start_band=0):\n",
" \"\"\"Returns the z-values of the beginning and end of each band.\n",
" \n",
" The values are formatted as a 2xN matrix where N is the number of bands.\n",
" \"\"\"\n",
" band_numbers = np.arange(start_band, num_bands+start_band)\n",
" end_points = (band_numbers+1)*pi # The known end points\n",
" start_points = np.zeros(num_bands) # Initializing the array of start points\n",
" previous_end_point = 0\n",
" \n",
" # Loop through each band and find the start point of the band\n",
" # by guessing a value close to the end point.\n",
" for n in range(num_bands):\n",
" band_number = band_numbers[n]\n",
" end_point = end_points[n]\n",
" z0 = end_point - 0.2*pi # Initial guess\n",
" y0 = (-1)**(band_number) # The value of f(z) we are looking for.\n",
" \n",
" # Find new possible start_point\n",
" possible_start_point = run_newton(z0, y0, beta)\n",
" # If not between end points try a new initial guess z0\n",
" while(not (previous_end_point < possible_start_point < end_point) ):\n",
" z0 -= pi*0.1\n",
" assert (z0 > previous_end_point) # Make sure the initial guess is between end points\n",
" possible_start_point = run_newton(z0, y0, beta)\n",
" \n",
" start_points[n] = possible_start_point\n",
" previous_end_point = end_point\n",
" \n",
" # Assert that all end points comes after the start_points\n",
" assert (end_points > start_points).all(), \"Not all end points are after start points\"\n",
" return np.array([start_points, end_points]) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using the above function, the width of the energy bands are calculated."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Band widths:\n",
" [ 1.35764238 1.65766308 1.88877155 2.0668647 2.20564343 2.31548818\n",
" 2.40389076 2.47619049 2.53620199 2.58668506]\n"
]
}
],
"source": [
"b = calculate_bands(start_band=2)\n",
"print('Band widths:\\n', b[1]-b[0])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The following code will plot the energy bands for a given $\\beta$."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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SUeFQxpvzt4czfrryPkdrM0oWNSU93/L4pe5IzXsT9CboK9QWeP31rBX3XznM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\npgqMMZ7hqqQAWAu8BfjBII69VFWPVDm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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def plot_bands(bands, ax=None):\n",
" \"\"\"Plots the bands given a list of tuples containing beginning and end positions of the band.\"\"\"\n",
" if ax is None:\n",
" ax=plt.gca()\n",
" xranges = zip(bands[0], bands[1]-bands[0])\n",
" ylim = ax.get_ylim()\n",
" collection=collections.BrokenBarHCollection(xranges=xranges, yrange=ylim, facecolor='#bb9999')\n",
" ax.add_collection(collection)\n",
" return ax\n",
"\n",
"beta = DEFAULT_BETA\n",
"zstop = 30\n",
"num_bands= int(zstop//pi)\n",
"\n",
"bands = calculate_bands(beta=beta, num_bands=num_bands) \n",
"plot_f(beta=beta, zstop=zstop)\n",
"plot_bands(bands)\n",
"plt.ylim([-2,2])\n",
"plt.title(r\"Energy bands for $\\beta=$ %d \" % beta);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we see, the band widths increase with increasing $z$. Let's calculate the lowest allowed energy value:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Lowest allowed value of z = 2.6277.\n"
]
}
],
"source": [
"print(r'Lowest allowed value of z = %.4f.' % bands[0,0])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We know that $z$ is related to $k$ by\n",
"\n",
"$$\n",
"z=ka,\n",
"$$\n",
"\n",
"and that $k$ is related to the energy $E$ by \n",
"\n",
"$$\n",
"k = \\frac{\\sqrt{2mE}}{\\hbar}.\n",
"$$\n",
"\n",
"Setting $\\hbar=m=a=1$ for simplicity, we obtain\n",
"\n",
"$$\n",
"E = \\frac{z^2}{2}.\n",
"$$\n",
"\n",
"Hence, the lowest allowed electron energy in the crystal is $E_0 = 3.5703$. In this case $\\beta=10$, which corresponds to a delta function strength $V_0=10$. Similarly, one can calculate the range of allowed energies in each band.\n",
"\n",
"It is also interesting to investigate how the band widths depend on $\\beta$."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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76bjhx2NouRx2lxuHx4Pd7cHhLoRu7G7PeJ7Lg93txuFyYx/Lc2G2zOzfrhL6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\nLyLDbOsK4+9KTeAi8IVtWOdzESlFHp0LDfQqS2y1BwpdliURKQ2sAUY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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def plot_bandwidths(betas, num_bands):\n",
" \"\"\"Plots the bandwidths against band_number for different betas.\"\"\"\n",
" \n",
" for beta in betas:\n",
" bands = calculate_bands(beta, num_bands)\n",
" width = bands[1,:] - bands[0,:]\n",
" plt.plot(width, label=str(beta))\n",
" plt.title(r'Band widths for different values of $\\beta$')\n",
" plt.xlabel('Band number')\n",
" plt.ylabel('Band width')\n",
" plt.legend(loc=4)\n",
" \n",
"plot_bandwidths(betas=[5,10,20,30,40,50], num_bands=60)\n",
"plt.plot([0,60],[pi, pi],':');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We see that the width of the bands increases towards $\\pi$. When the band width is equal to $\\pi$, it means that the bands are in fact connected and can be regarded as one continuous band.\n",
"\n",
"___\n",
"\n",
"## References:\n",
"\n",
"[1]: Hemmer, P. C. _Kvantemekanikk_. Tapir Akademisk Forlag, 2005\n",
"
\n",
"[2]: Griffiths, D. J. _Introduction to Quantum Mechanics_. Pearson Education, 2004."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.5.1"
}
},
"nbformat": 4,
"nbformat_minor": 1
}