1
00:00:00,240 --> 00:00:08,160
So let us discuss the band structure of graphene, so these words, band structure may sound a bit cryptic

2
00:00:08,160 --> 00:00:08,670
to you.

3
00:00:09,120 --> 00:00:13,320
And for this reason, I will start this section with discussing a bit.

4
00:00:13,620 --> 00:00:17,400
Why do we have band structures and what our band structures?

5
00:00:18,540 --> 00:00:24,840
So first of all, as in almost every notebook, we are going to import the nonpaying module and also

6
00:00:24,840 --> 00:00:26,550
the iPod lip module.

7
00:00:27,210 --> 00:00:32,700
And since I will create and show you several figures here, I will also increase the figure size and

8
00:00:32,700 --> 00:00:35,550
this notebook by writing down this command.

9
00:00:37,200 --> 00:00:43,770
So as I said, I want to start with a simple example that explains to you why we have band structures

10
00:00:43,770 --> 00:00:45,720
and what band structures actually are.

11
00:00:46,440 --> 00:00:53,100
And this is the model of free electrons, which then leads to the quasi free electrons in a periodic

12
00:00:53,100 --> 00:00:53,640
crystal.

13
00:00:54,720 --> 00:00:57,390
So free electrons, you probably know?

14
00:00:57,390 --> 00:00:58,490
I think so.

15
00:00:58,500 --> 00:01:06,060
If you have just an electron in vacuum, for example, you can describe its energy by the kinetic energy,

16
00:01:06,720 --> 00:01:11,880
which is the momentum p squared over two times the mass.

17
00:01:12,360 --> 00:01:18,270
This is basically the same as the kinetic energy, which is am over two times the velocity squared.

18
00:01:19,890 --> 00:01:28,980
So what we can then do is we can express the momentum p by h bar times the wave factor K.

19
00:01:30,060 --> 00:01:34,470
So this is at this point, just some quantity that we have introduced.

20
00:01:35,400 --> 00:01:43,830
So if we write that E is equal to a small square of a 2m times square, then this is actually also the

21
00:01:43,830 --> 00:01:47,010
correct solution of the stationary shooting equation.

22
00:01:47,400 --> 00:01:49,590
If we do not consider any potential.

23
00:01:50,610 --> 00:01:54,640
So you probably know this energy here from classical physics.

24
00:01:54,660 --> 00:01:58,620
So as I said, just particles, just electrons in the vacuum, for example.

25
00:01:59,100 --> 00:02:02,820
But it turns out this is also the correct solution in quantum mechanics.

26
00:02:03,480 --> 00:02:11,610
So here you see the shooting equation, and if we do not consider any potential, it is pretty simple.

27
00:02:11,610 --> 00:02:12,990
It's just this one here.

28
00:02:13,320 --> 00:02:14,400
So you have a pretty.

29
00:02:15,360 --> 00:02:18,870
Then you have two second derivative of the wave function.

30
00:02:19,230 --> 00:02:22,350
You have the energy and you have to wave function again.

31
00:02:23,220 --> 00:02:28,530
And what you tried to do now in quantum mechanics is you try to solve this equation, which means you're

32
00:02:28,530 --> 00:02:35,850
trying to determine at the same time the wave function 5X and also the energy, which will be a number.

33
00:02:37,410 --> 00:02:41,760
And it turns out that this solution here is correct.

34
00:02:41,940 --> 00:02:47,400
It solves this Schrodinger equation if the wave function is a plane wave.

35
00:02:47,910 --> 00:02:56,460
So it's something like a constant times the exponential function with K x as the exponent, so we can

36
00:02:56,460 --> 00:02:57,780
check if this is really true.

37
00:02:58,110 --> 00:02:59,970
So just take this function here.

38
00:03:00,210 --> 00:03:01,110
Put it in here.

39
00:03:01,380 --> 00:03:09,240
So the second derivative of exponential function to the power i.e. K X is the function itself times

40
00:03:09,600 --> 00:03:12,000
the the exponent here.

41
00:03:12,000 --> 00:03:16,770
So we have to calculate the inner derivative, so basically the derivative of the exponent.

42
00:03:17,370 --> 00:03:20,190
And since it's the second derivative, we have to do it twice.

43
00:03:20,910 --> 00:03:29,490
So what we get is I k and both of them squared, so I squared is minus one and k squared is just case

44
00:03:29,490 --> 00:03:29,790
quiet.

45
00:03:30,000 --> 00:03:35,730
So we get for the second derivative minus k squared times the function itself.

46
00:03:37,080 --> 00:03:46,800
So what we have now is minus h bar squared over two m times, K squared times and other minus one from

47
00:03:46,800 --> 00:03:47,430
the ice squared.

48
00:03:47,670 --> 00:03:53,340
So we have H Bar Square over two and Times Square five is equal to ify.

49
00:03:53,940 --> 00:04:00,120
And this is why the only solution for E can then be exactly the prefecture's h by square of a two m

50
00:04:00,120 --> 00:04:01,140
times square.

51
00:04:02,220 --> 00:04:10,770
So you see that our classical solutions, our observations for electrons are also fulfilled and reproduced

52
00:04:10,770 --> 00:04:11,940
in quantum mechanics.

53
00:04:12,570 --> 00:04:18,360
So basically, what this tells us is that the SweeTango equation works just fine and that the quantum

54
00:04:18,360 --> 00:04:22,800
mechanics can re establish the results from classical physics.

55
00:04:23,940 --> 00:04:25,890
But now let's finally get to our example.

56
00:04:26,550 --> 00:04:30,300
So how do we get banned structures or why do we need band structures at all?

57
00:04:31,230 --> 00:04:35,730
Well, to be honest, it's not really that easy to see and not really that easy to explain.

58
00:04:36,210 --> 00:04:42,230
But the reason why these man structures occur is that we have now not a vacuum.

59
00:04:42,240 --> 00:04:49,620
We do not have free electrons, but we have electrons that live in the periodic potential.

60
00:04:50,430 --> 00:04:55,500
So we are considering here a solid or a periodic crystal.

61
00:04:56,310 --> 00:04:58,650
For example, take a piece of iron.

62
00:04:59,460 --> 00:04:59,910
You have.

63
00:04:59,980 --> 00:05:06,520
The iron atoms and the form of periodic letters and of course, the electrons, which are in the letters,

64
00:05:06,880 --> 00:05:10,060
they will feel a certain interaction with the iron atoms.

65
00:05:11,140 --> 00:05:18,160
So it turns out that in order to describe this, it's not that important to know exactly what the potential

66
00:05:18,160 --> 00:05:18,730
looks like.

67
00:05:19,270 --> 00:05:22,600
It's more important to just know the periodicity.

68
00:05:23,470 --> 00:05:29,860
And of course, the periodicity of the letters is the same as for what was already période does.

69
00:05:29,860 --> 00:05:37,000
A T of the potential V is the same as the periodic periodicity of the crystal itself.

70
00:05:38,080 --> 00:05:43,960
So here in this example, I want to do it as simple as possible, so I went to one dimensions.

71
00:05:44,440 --> 00:05:45,670
So what we can say is.

72
00:05:46,950 --> 00:05:50,880
Or Crystal has a certain lattice constant.

73
00:05:51,000 --> 00:05:55,230
So basically the distance between two atoms, which I call a.

74
00:05:55,770 --> 00:06:04,380
And so the potential will have the same value at each of these positions x plus end times a wire and

75
00:06:04,380 --> 00:06:05,960
is just some integer number.

76
00:06:06,920 --> 00:06:12,360
That's pretty clear whenever the electron is directly at the position of an atom.

77
00:06:12,660 --> 00:06:18,180
Then it will have the same potential no matter at which atom it is, because all atoms are the same.

78
00:06:19,650 --> 00:06:25,290
And of course, if we include not as potential in treating equation, it will look a bit different.

79
00:06:25,650 --> 00:06:30,190
And also, this will, of course, change the way functions PHI.

80
00:06:30,210 --> 00:06:31,800
It will change the energies.

81
00:06:32,100 --> 00:06:34,650
And this is what we are going to discuss next.

82
00:06:35,700 --> 00:06:39,960
So corresponding to my explanation, I will create now a figure.

83
00:06:40,410 --> 00:06:42,960
And of course, I could have created this figure in advance.

84
00:06:42,960 --> 00:06:49,680
But I think it's a nice exercise to which we can do together to practice on how to create figures.

85
00:06:50,700 --> 00:06:55,170
So what we're trying to do here is we are trying to generate a crystal.

86
00:06:55,200 --> 00:06:58,410
So basically just some dots that represent atoms.

87
00:06:59,640 --> 00:07:05,770
And then we are going to plot the potential V above or in the same plot.

88
00:07:05,790 --> 00:07:06,390
You could say.

89
00:07:07,800 --> 00:07:12,610
So the first thing that I'm going to do is I'm going to generate an X-ray.

90
00:07:13,050 --> 00:07:16,160
So these will be our values on the x axis.

91
00:07:16,170 --> 00:07:21,510
And this corresponds to the values of our coordinate x that enters the potential, for example.

92
00:07:22,590 --> 00:07:26,880
And as we do, it's most often for these plots.

93
00:07:26,880 --> 00:07:30,270
We will be generated by writing and pitot and then space.

94
00:07:30,960 --> 00:07:33,480
And now we have to decide which range we want to use.

95
00:07:34,050 --> 00:07:40,440
And of course, I tried this already before, and I think if we go, if we want to plot three atoms.

96
00:07:40,600 --> 00:07:46,620
So basically going from minus three to plus three, it looks good if we have a bit of a margin here.

97
00:07:46,620 --> 00:07:50,250
So I go from minus three point five to plus three point five.

98
00:07:50,820 --> 00:07:53,730
And we take one hundred and one points in between.

99
00:07:54,180 --> 00:07:56,250
So that's the function will then look smooth.

100
00:07:57,900 --> 00:08:08,340
So now we have our X-ray and what we can do next is we can just plot the the atom positions and we can

101
00:08:08,340 --> 00:08:10,800
do this by writing, Please don't get her.

102
00:08:12,360 --> 00:08:13,790
And then we just write.

103
00:08:16,170 --> 00:08:17,730
Let me do it like this, and we just write.

104
00:08:20,880 --> 00:08:28,830
Our MP darts in space, so we create a new land space and it goes from minus three to three.

105
00:08:29,340 --> 00:08:34,590
We have seven points, so we have basically here a list starting from minus three and then minus two

106
00:08:34,590 --> 00:08:36,059
minus one zero one two three.

107
00:08:36,630 --> 00:08:41,429
And then for the Y coordinates, I say the microphone should just be zero.

108
00:08:41,940 --> 00:08:47,400
So most easy would be to just right and zeros and then the number of points that we want to have.

109
00:08:48,600 --> 00:08:54,060
And you see, we have here our chain of atoms and we have plotted seven of these atoms.

110
00:08:55,380 --> 00:08:59,730
So now we can use our X-ray to plot on top of the potential.

111
00:09:00,690 --> 00:09:02,520
So I will write peeled dot plots.

112
00:09:05,140 --> 00:09:10,930
And now we wrote X Ray, and we plot to potential.

113
00:09:11,830 --> 00:09:16,660
Of course, we don't know what the potential looks like exactly, but here I want to just use a function

114
00:09:16,660 --> 00:09:18,370
which gets to periodicity, right?

115
00:09:18,490 --> 00:09:23,920
So I write x ray and then we multiply by two point.

116
00:09:25,270 --> 00:09:27,100
So I think this should work.

117
00:09:27,640 --> 00:09:28,060
Yes.

118
00:09:28,690 --> 00:09:31,300
So you see now, OK, it's a bit choppy, actually.

119
00:09:31,300 --> 00:09:33,250
Some maybe increase the number here.

120
00:09:34,180 --> 00:09:35,200
Yeah, it looks much better.

121
00:09:35,890 --> 00:09:42,880
So you see we have a cosine function and the potential at the position of these atoms is always at the

122
00:09:42,880 --> 00:09:43,420
minimum.

123
00:09:43,780 --> 00:09:49,240
So it would be good for the electron to be positioned right at the atom position and then in between,

124
00:09:49,630 --> 00:09:50,890
the potential is positive.

125
00:09:51,640 --> 00:09:52,570
So this makes sense.

126
00:09:52,570 --> 00:09:56,050
And also it's of course, reasonable that we have this periodicity.

127
00:09:57,250 --> 00:09:59,230
So now this plot looks a bit ugly.

128
00:09:59,230 --> 00:10:02,890
So let me try to improve a bit the visuals here.

129
00:10:03,340 --> 00:10:10,990
So we did this already several times, so I will just write and try to talk a bit while I do this.

130
00:10:10,990 --> 00:10:18,370
So first of all, we must create a figure and then add a subplot because what I want to do is I want

131
00:10:18,370 --> 00:10:19,960
to change the aspect ratio.

132
00:10:20,650 --> 00:10:25,690
And apparently this is only possible if we create such a figure with a subplot.

133
00:10:26,200 --> 00:10:31,300
So once again, this one means we have a grid of one time small plots and this is the first plot.

134
00:10:31,390 --> 00:10:35,770
So, yeah, it's a bit useless because we only are going to use one plot.

135
00:10:37,210 --> 00:10:42,190
So I think this is a bit tedious about Python two always having to write this just to set the aspect

136
00:10:42,190 --> 00:10:42,510
ratio.

137
00:10:42,520 --> 00:10:44,350
But it is what it is.

138
00:10:44,470 --> 00:10:47,440
And now we can finally write, said aspect ratio.

139
00:10:48,190 --> 00:10:49,730
And you see how it looks.

140
00:10:49,780 --> 00:10:51,040
I think it looks now much better.

141
00:10:52,360 --> 00:11:01,750
And not the only thing that I'm going to do is I'm going to write X label, which is the position which

142
00:11:01,750 --> 00:11:04,570
is X, and the unit here is eight.

143
00:11:05,380 --> 00:11:11,690
So we are plotting multiples of eight, which is two letters constant, and the white label is just

144
00:11:11,710 --> 00:11:14,490
a potential three.

145
00:11:16,330 --> 00:11:17,890
And I think this is a really nice figure.

146
00:11:18,640 --> 00:11:23,650
I mean, it's still pretty simplistic, but it allows us to understand what I explained before.

147
00:11:24,670 --> 00:11:28,870
So we have no our potential v of X, which fulfills this condition.

148
00:11:29,320 --> 00:11:35,050
It has the same value no matter which position we look at if we shift the position by multiples of the

149
00:11:35,050 --> 00:11:35,860
letters constant.

150
00:11:36,490 --> 00:11:41,950
For example, we are here, we are shifting by two letters constants than we are here, and it's the

151
00:11:41,950 --> 00:11:42,670
same value.

152
00:11:44,380 --> 00:11:44,860
All right.

153
00:11:45,310 --> 00:11:51,880
So now we must explore how do the wave functions and especially how do the ion energies change?

154
00:11:52,990 --> 00:11:59,410
So the way functions, I can tell you, they've become periodic block functions, so I will not go into

155
00:11:59,410 --> 00:12:00,220
the details here.

156
00:12:00,220 --> 00:12:05,590
But it turns out that these functions are, of course, also periodic in space, which I think is quite

157
00:12:05,590 --> 00:12:05,980
clear.

158
00:12:06,340 --> 00:12:11,590
If the potential is periodic, then the wave function is also periodic, but they are also periodic

159
00:12:11,590 --> 00:12:12,750
in K.

160
00:12:13,630 --> 00:12:18,450
So here we have the solution of plane waves e to the power ICAC's.

161
00:12:19,030 --> 00:12:24,760
And so what happens here is we get another factor here, which is some function that's periodic in X

162
00:12:24,910 --> 00:12:35,470
and in K. So the only thing that we care about is the energies, and I will show you how this changes.

163
00:12:36,460 --> 00:12:44,710
So to get an idea on what the ion energies look like is we could just say, OK, we don't really care

164
00:12:44,710 --> 00:12:46,810
so much about the actual potential here.

165
00:12:46,810 --> 00:12:55,480
We just care about the periodicity and we take our free electrons which have this energy and position

166
00:12:55,480 --> 00:12:58,750
this parabola at every position of the atoms.

167
00:13:00,100 --> 00:13:02,920
And we know that we have this periodicity in X and K.

168
00:13:02,920 --> 00:13:06,940
So this means we can also do the same thing in K.

169
00:13:07,660 --> 00:13:12,190
So yeah, sounds a bit difficult, but here I'm going to show you what's going to happen.

170
00:13:13,060 --> 00:13:22,210
So first of all, we are going to create and carry so some number of cape points, which is again a

171
00:13:22,210 --> 00:13:23,230
linear space.

172
00:13:25,030 --> 00:13:30,370
So remember, once again, the key value basically determines the momentum of the electrons because

173
00:13:30,370 --> 00:13:32,350
p is equal to each bar.

174
00:13:32,380 --> 00:13:33,070
Time is key.

175
00:13:34,150 --> 00:13:42,280
So here we could start, for example, from PI and go to point sorry, like this.

176
00:13:44,350 --> 00:13:52,840
And then we generate an energy ari, which is one half times.

177
00:13:53,050 --> 00:13:58,720
Or actually it is H Bach Square over two m Times Square.

178
00:13:59,020 --> 00:14:03,140
But for simplicity, here I will use H Bar and both of them will be equal.

179
00:14:03,240 --> 00:14:03,840
All to one.

180
00:14:04,800 --> 00:14:07,830
So this means we just have one half times square.

181
00:14:08,220 --> 00:14:16,560
So Kerry Square, and now we can plot this is, of course, really easy.

182
00:14:16,620 --> 00:14:25,650
We just write guilty to the plot and we write Kerry come up and are great.

183
00:14:27,240 --> 00:14:28,590
So this is our parabola.

184
00:14:29,610 --> 00:14:33,420
So now we can, of course, gift the axis some label.

185
00:14:33,570 --> 00:14:43,170
So I copy this and I write this here the way factor, which is proportional to the momentum of the electrons

186
00:14:46,500 --> 00:14:50,250
in units of one over, actually, but it's not so important.

187
00:14:51,300 --> 00:14:53,970
And then we have on the y axis the energy.

188
00:14:58,990 --> 00:15:07,030
And now what happens is I said it was already that the energy will be also periodic in key space.

189
00:15:09,850 --> 00:15:19,780
So here on this axis, we have the key factor on the value of Kate and the energy will be periodic and.

190
00:15:20,980 --> 00:15:25,780
And it will be periodic at the unit of two PI over eight.

191
00:15:26,080 --> 00:15:32,590
So this means when we shift this or when we go to PT. one minus two PI, we will see the same parabola.

192
00:15:33,580 --> 00:15:35,230
So this is what I'm going to do now.

193
00:15:35,860 --> 00:15:45,100
First of all, let me increase here the range for which we are plotting, and now I'm going to add more

194
00:15:45,100 --> 00:15:46,000
of these parabolas.

195
00:15:46,780 --> 00:15:57,040
But first of all, let me change that star to black and then we can just copy this paste and just right

196
00:15:57,040 --> 00:16:00,160
here, plus two times pipe.

197
00:16:02,340 --> 00:16:06,960
And you see, now we have a second parabola shifted by two pi over eight.

198
00:16:08,490 --> 00:16:12,270
And this happens, of course, for all multiples of two PI.

199
00:16:13,080 --> 00:16:16,950
So of course, I cannot do all of them here, but let me just do a few.

200
00:16:17,340 --> 00:16:21,930
So minus two pi and we do the same thing for four PI.

201
00:16:24,850 --> 00:16:32,010
Do the same thing for six pie, and I'd say, let's also do it for eight pie and we have quite a few

202
00:16:32,460 --> 00:16:33,240
parabolas.

203
00:16:36,540 --> 00:16:43,710
So you see, of course, here I would have I would need to have more parabolas here, but you can see

204
00:16:43,710 --> 00:16:45,150
pretty nicely what's happening here.

205
00:16:45,690 --> 00:16:53,370
So we have all of these parabolas which are shifted by K, which basically is due to the periodicity

206
00:16:53,370 --> 00:16:54,030
of the letters.

207
00:16:54,420 --> 00:16:57,990
So every atom position we get, we have an electron.

208
00:16:58,410 --> 00:17:05,130
And this means and in case bays, we have these parabolas, which are shifted by multiples of two pi

209
00:17:05,160 --> 00:17:05,670
over eight.

210
00:17:06,990 --> 00:17:12,210
So no, I told you that this party is not really reasonable because we missed the parabola instead of

211
00:17:12,210 --> 00:17:13,619
further to the left and to the right.

212
00:17:14,160 --> 00:17:25,670
So I would say, let's zoom in here in this range and we can do this by using the commands plus thoughts.

213
00:17:26,130 --> 00:17:26,829
Excellent.

214
00:17:28,770 --> 00:17:31,110
And we will limit this to minus.

215
00:17:31,680 --> 00:17:41,160
I can just copy here minus one point five PI and we're going to 1.5 Pi and the and the energy I will

216
00:17:41,160 --> 00:17:48,030
also limit and I will go from zero to sixty five.

217
00:17:49,470 --> 00:17:55,290
So you see, now the band structure looks much better or yeah, I don't know, it looks better, but

218
00:17:55,290 --> 00:17:57,420
at least we have now this.

219
00:17:58,200 --> 00:18:00,750
Yeah, we do not have this artifact of the left and the right.

220
00:18:01,950 --> 00:18:04,350
Maybe for the moment, let me increased as a bit.

221
00:18:06,040 --> 00:18:09,980
And here you can nicely see that we really have some periodicity.

222
00:18:10,850 --> 00:18:15,660
So the band structure is still periodic with respect to two pi over.

223
00:18:15,950 --> 00:18:19,460
So here, here and here it's periodic.

224
00:18:20,090 --> 00:18:25,880
So basically, this means it is sufficient to just look at the band structure in this range here from

225
00:18:25,880 --> 00:18:28,910
minus PI over eight to plus PI over eight.

226
00:18:30,200 --> 00:18:39,200
And to visualize this, let me just add a few lines here so we can do this by writing minus and P Pi

227
00:18:40,130 --> 00:18:43,640
to minus and P Pi.

228
00:18:44,930 --> 00:18:49,340
This is the change of the X coordinate, which doesn't change at all, and the line goes from minus

229
00:18:49,640 --> 00:18:51,620
f sorry from zero to sixty five.

230
00:18:52,340 --> 00:18:54,110
And I will plot this graph.

231
00:18:55,160 --> 00:18:56,540
So you see, we have now this line.

232
00:18:57,140 --> 00:18:58,010
Let's add another one

233
00:19:02,840 --> 00:19:03,800
four plus point.

234
00:19:05,180 --> 00:19:08,780
And let's change the plot range to 1.5.

235
00:19:10,340 --> 00:19:11,030
So here we are.

236
00:19:11,510 --> 00:19:16,430
This is now the cell, or we call this in physics, the burial zone.

237
00:19:16,640 --> 00:19:18,160
But it's not so important what we call it.

238
00:19:18,180 --> 00:19:24,050
But this is like some characteristic range in K. that is able to describe all of the electrons.

239
00:19:25,160 --> 00:19:32,540
So we have now this relation of free electrons that are in a periodic lattice and we have this relation

240
00:19:32,540 --> 00:19:37,310
of the energy with respect to K, which is once again proportional to the momentum.

241
00:19:39,410 --> 00:19:42,590
So so far, they are still free electrons.

242
00:19:42,590 --> 00:19:45,110
They are still a quadratic their energy.

243
00:19:45,110 --> 00:19:50,600
It still depends quadratic on the momentum, quadratic on the velocity.

244
00:19:51,020 --> 00:19:52,670
So, yeah, nothing really new here.

245
00:19:53,540 --> 00:20:01,580
But the thing is the potential does not only provide the periodicity, but it also provides some perturbation.

246
00:20:02,150 --> 00:20:09,020
So you see for the tweeting equation we have included here does value V and of course, it will do something

247
00:20:09,020 --> 00:20:09,920
to the energies.

248
00:20:10,790 --> 00:20:17,300
And this is a bit hard to understand without me having to talk for an hour, which I don't want.

249
00:20:17,690 --> 00:20:19,520
So I'm going to tell you what's going to happen.

250
00:20:20,240 --> 00:20:25,310
So the potential break some symmetry in the system.

251
00:20:25,670 --> 00:20:31,610
And whenever this happens in a physical system, the general is degeneracy will split up.

252
00:20:32,510 --> 00:20:38,630
So this means all of these points were two bands across the bands will actually split up.

253
00:20:39,380 --> 00:20:41,030
I'm going to show you what this means.

254
00:20:41,960 --> 00:20:44,350
So I have prepared this already in advance.

255
00:20:44,360 --> 00:20:48,800
So let me just copy here a few lines of code that you can also write down if you want.

256
00:20:49,400 --> 00:20:53,960
And let me run this, so I'm adding here for more functions.

257
00:20:56,160 --> 00:20:58,620
And you see, these are these RET functions here.

258
00:20:59,460 --> 00:21:04,380
So this is what is going to happen when you consider now the actual potential.

259
00:21:05,250 --> 00:21:10,470
So of course, this may not be the absolute correct solution, but it describes qualitatively what is

260
00:21:10,470 --> 00:21:11,280
going to happen.

261
00:21:12,360 --> 00:21:17,820
So these crossings here, they will split up due to due to the potential.

262
00:21:18,300 --> 00:21:23,700
And so the initially quadratic band dispersions become more something like cosine functions.

263
00:21:24,780 --> 00:21:33,180
And also, what you can see is that now the bands are the parabolas do not go from to not go from zero

264
00:21:33,180 --> 00:21:34,050
to infinity.

265
00:21:34,470 --> 00:21:37,110
But we have really some well-defined bands here.

266
00:21:37,200 --> 00:21:44,790
So these red bands, they just oscillate in some range and between the bands, we even have band gaps

267
00:21:45,120 --> 00:21:47,010
where there are no states anymore.

268
00:21:47,910 --> 00:21:55,410
So this means due to this periodicity and due to this potential, the relation of energy to the momentum

269
00:21:55,890 --> 00:21:57,600
has changed drastically.

270
00:21:58,650 --> 00:22:01,560
And these curves that we see here.

271
00:22:01,860 --> 00:22:04,740
So maybe let me for a moment copy.

272
00:22:05,040 --> 00:22:07,770
So a common piece so that we don't see them anymore.

273
00:22:09,840 --> 00:22:18,600
So this year is called a band structure, and it is really important for describing materials.

274
00:22:18,990 --> 00:22:24,420
And it allows us to determine if a material is, for example, a metal or an insulator.

275
00:22:24,780 --> 00:22:28,410
And it's really, really different for different materials.

276
00:22:29,430 --> 00:22:36,390
And in the following, we are going to consider graphene and we will calculate the actual band structure

277
00:22:36,390 --> 00:22:37,200
of graphene.

278
00:22:37,560 --> 00:22:40,860
And I think it's really a very interesting topic.

279
00:22:42,300 --> 00:22:48,210
So to close the section, let me restore the initial parabolas so that you have this nice figure in

280
00:22:48,210 --> 00:22:48,780
your notebook.

281
00:22:49,680 --> 00:22:54,810
And yeah, I wrote it down here as a main effect of the potential of the band crossing split up.

282
00:22:55,350 --> 00:23:02,190
But still, it's very important to realize that the periodicity has to be conserved, so it's still

283
00:23:02,190 --> 00:23:05,370
periodic with respect to to pay your way.

284
00:23:05,730 --> 00:23:09,420
So for here and here, the band structure is exactly the same.