lecture 10_ch16_dynamics of bloch electrons.pdf
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Dynamics of Bloch Electrons( h f )(Ch 16 of CMP)
• Classical Electron Dynamics• Classical Electron Dynamics– Drude model– Hall effect
• Semiclassical Electron Dynamics – Wave packet of Bloch electrons– Velocity and Effective mass– Bloch oscillationNoninteracting electrons in an Electric field– Noninteracting electrons in an Electric field
• Quantizing Semiclassical Dynamics – Wannier‐Stark LaddersWannier‐Stark Ladders– de Hass‐van Alphen effect
1
Classical Electron DynamicsDrude model
Based on the free electron model, assume that an electron i lli i ith b bilit it ti f 1/
Drude model
experiences a collision with a probability per unit time of 1/. The probability of collision in any infinitesimal time interval dt is dt/ where is called the relaxation timeElectrons under external electric and magnetic fields obey interval dt is dt/, where is called the relaxation time.
dt
.vmv eEc
vB me
( ) (1 ) ( ) ( )dtp t dt p t F t dt
( ) ( )dp p t F tdt
The relaxation time describes the damping; without external fields, the velocity of electrons is
dt
/0
tv t v e 14 15~ 10 to 10 sec
2
In the presence of an electric field,v /tmB A Be
.vmv eE m
/tmB A Bee eE m
eA E Assume a solution of the form
/ ,tv t A Be
00v t v
A Em
Assume a solution of the form
/0
te ev t E v E em m
0
0ev E B
m
m m
If the density of mobile electrons is n, in response to , h d
E
2the current density is j nev 2
.ne Em
2
.ne Thus the electrical conductivity is j E
3
m
Consider that an electron moves along the x axis in a uniform
Classical Hall Effect
Consider that an electron moves along the x axis in a uniform magnetic field along the z axis and electric field along the x axis.
B
E
.v vmv eE e B m
c
In a plate‐shaped specimen, a transverse electric field is developed in the steady state because current can not flow out.
4
.v vmv eE e B mc
ˆB Bz
v v If B=0In the steady state,
0 xmv 0.y xx
v veE e B mc
0 .j E
2
0ne
m 0 0.yx
y y
vvmv eE e B mc
eB After multiplying by nem
m
2
x x c yne E nev nev
m
c mcm
0 x x c yE j j m
0 y c x yE j j 2ne E nev nev
0 x x c yj j
5
y yy c x yE nev nev
m
j E
j E
Conductivity and resistivity become 2nd rank tensors, i.e.
,j E .j E
1 BMagnetoresistance is
0 0
1
1
c
xx xy
.xy yxB
nec
B
10 0
1yx yy c
ceBmc
The Hall coefficient is defined as1 ,HR
nec
d di t f t i l t th d it fdepending on no parameters of materials except the density of carriers.
A measurement of the Hall coefficient determines the sign of charge carriers.
6
Semiclassical Electron Dynamics
Velocity of Bloch electronsVelocity of Bloch electrons
For free electrons,2 2
,2
kE mv k
2m
v k 1 E
1
Emv k
2v k
m k E
k
2
2 2
1 1m k
E
Are these equations valid for Bloch electrons?
The wave functions of Bloch electrons are ,ik rnk nkr e u r
2 2 .2 nk nkr r
m
E
which obeys
7
2 ik rnke u r
ik r ik r
nk nkike u r e u r
ik r ik rnk nkik ike u r e u r
2 22ik r ik k
2ik r ik rnk nkike u r e u r
2iknke ik k u r
2
2 22ik k u r u r E 2
2 nk nkik k u r u rm
E
Under the action of weak static fields, of Bloch electrons k
,slightly increases to .k k
The Hamiltonian is
2 22ˆ k k k k H
2 22k k i k k k k
m
H
2 2i k k k
8
2 2i k k k
2
ˆ ˆ 2 22k k k i k k k
m
H H
kTo the first
order of 2m korder of
We now apply the perturbation method by using the perturbation Hamiltonian
21ˆ k k i H
using the perturbation Hamiltonian 1 .k k k im H
The first‐order correction to the energy is
2
1k nk nku r k k i u r
m E
ik rk ku r e r
m nk nk
2
ik r ik rnk nke r k i e r
mk
m ik r ik r ik r
nk nk nki e r e r rk e
ˆnk nk nkr k P r
m E
9
m
1 ˆ1k n
nkn kr
kP r
m
E
v
k mThis is the group velocity of the wave packet of Bloch electrons.
kdv v 1 E21 k E
Effective mass
kdvdt k t
v
1 nkvk
E
1 nk
tkk
k
E
2
2
1 nk kvk k
E
1v k M
k k
2
-12
1 nk
k k
M
E
M is called the effective mass tensor. k k
Electrons have negative effective mass are interpreted as “holes” with positive charge.
10
are interpreted as holes with positive charge.
The perturbation Hamiltonian is 1ˆ ˆ.k k Pm H
(2)ˆnk nk nk nkr k P r
m E E
2(2)
(0) (0)
ˆ ˆnk n k n k nk
nk
k P k P
EE E
(0) (0)nkn n nk n km
E E
2
-1 1 nkM
E 12
nk
k k
M
ˆ ˆ1 1 P P c c2 (0) (0)
1 1 nk n k n k nk
n n nk n k
P Pm m
E E
c.c.
The deviation of effective mass from m arises from the virtual transitions between energy bands.
11
gy
Semiclassical Electron Dynamics
Under electric and magnetic fields, the wave vector of & ,E B
g ,electrons obey .ek eE r B
c
& ,E B
c
l h l i k l i fi ld
In the following, we will prove this equation separately.
Bloch electrons in a weak electric field
Puzzle: Since the electrostatic potential is in a V r E r Puzzle: Since the electrostatic potential is in a
constant E field, grows linearly in space. One must abandon periodic boundary conditions.
V r E r V r
One way to solve this puzzle is introducing a time‐dependent vector potential instead of V since the electric field isvector potential, instead of V, since the electric field is
1 AE V
12
E Vc t
To illustrate this trick, we use the following 1D geometry in which the vector potential is .A cEt p .A cEt
2
The Hamiltonian is21ˆ ˆ ˆ ˆ( )
2eP A U R
m c
H
The eigenfunction and eigenenergy li itl ti d d t iare explicitly time‐dependent, i.e.,
21 ˆ ˆ ˆe
1 ˆ ˆ ˆ( ) , ,2 t
eP A U R x t x tm c
E
Assuming that the circumference is L, the periodic conduction is
, , .x L t x t
13
21 ˆ ˆ ˆ( ) , ,
2 teP U R x t xA tt
E
We can simplify the Schrodinger by the following transformation
2m c
, , ,i xx t e x t called the Houston function.
i xei A e rc
i xAc
e re i x i xe r i e r
0e Ac
eAhc
demanding
2P̂
The simplified equation is
, .ik t xnkx t e u x ˆ ˆ( ) , , ,
2 tP U R x t x tm
E
14
Where did the effect of field go?E
The periodic conduction gives , ,x L t x t
/ ik ti A /i A L ik t L / .ik t xieAx cnke e u x
/ieAx cx t e x t AL
/ieA x L c ik t x Lnke e u x L
, , .x t e x t 2eAL k t L l
c
2eEt lk tL
A cEt
k eE The index obeys classical equations of motion for an electron in an electric field.
k
15
Alternatively (HW 16.7), we can prove this equation of motion by considering the evolution of the Bloch wave functions through
1 ˆt i
H
2ˆˆ ˆ ˆ ˆ( )2P U R eE R
H
t i ( )
2m
?dt
dtThen we apply the translation operator to .
?dt
† ik RRT e
† ik dt RRT dt e dt
?k dt
k eE (to the 1st order in dt)
16
Bloch oscillationsIn the semiclassical picture , many QM effects are k eE
because the energy of Bloch electrons is a
periodic function of and electrons obey Fermi statistics.nkE
kretained
If there were no damping, electrons would oscillate in timerather than travel, and metals would become insulators.
2 cos .k t ka EFor example, the 1D tight‐binding model of a lattice constant agives
17
1rk
E
2 sinta ka 2 cos .k t ka E
In a uniform electric field E, k
k eE
/k eEt 2 sinta aeEtr
2 cost aeEtreE
Bloch oscillations for Cs atoms trapped in potentials created by standing waves f l li hof laser light.
ben Dahan et al. PRL (1996)
18
Wave packetThe concept of “particles” refers to wave packets. The dynamics of semiclassical electrons is involved with the evolution of wave packets.
Recall that a “wave packet” in 1D can be described as
ikxf dk k
ikxf x dk g k e
2k k 20 / 4ik x xf 0 ,k kg k e 0 / 4ik x xf x e e
If
~ 1k x O
f( ) can be strongl locali ed or broad depending pon thef(x) can be strongly localized or broad, depending upon the width of g(k).
19
For Bloch electrons, we can define a wave packet centered in space at and dominated by wave vector .ck
cr
c cr kW r
1
cc
In the presence of a vector potential the wave vector is,A
/1 cc
c c c
ieA r ik rr
rkk k
ck
k
W r w e rN
20
1W r W r The wave packet must be normalized; hence
1c c c cr k r kW r W r
.rIn addition, the wave packet is expected to be centered at .cr, p p
,c kci k k
kk kkw w e
R
3 * ,k k ki d r u r u r
RIf ,c ckk kk ,
c c ck k kc
i d r u r u rk R
h then .c c c c cr k r kW r r W r r
*3 * . .
2c
c c c c
kck k k k
uL d r u k u c c
irik
R2 c irik
k eeE r B
21
ck ceE r Bc
12
c ck kc c
Ler B k kk k
E
k
k k
ui u
k
R
2cc
ccmck k
R
is called Berry curvature; Berry phase d
“Berry potential”
Anomalous velocity and change sign as ,k k
d h i h i l i h i i
k k R
ckL
is called Berry curvature; Berry phase d
and hence vanish in any crystal with inversion symmetry. The 2nd two terms can be neglected for weak and .E B
Limitations of semiclassical dynamics:
The spatial ranges of and must be larger thanE B
The spatial ranges of and must be larger than atomic interspacing.Tunneling between bands should be avoided; and
E B
E B
fields can not be too large.For AC fields of frequency , can not be larger than the
22
energy gap.
Hamiltonian Dynamics Similar to classical dynamics, one
P
LQ P H Lcan obtain the Hamiltonian from .ll
PQ
,l ll
Q PH L
The Hamiltonian .2k k
e B L eVmc
H E
is a constant of the motion.
If and V vanish, travels along contours on thek
L
If and V vanish, travels along contours on the energy surface .
knkE
L
23
The 1st Brillouin zone of an fcc latticeFermi surface of Cu
1 1 12 2 2( , , )L 2 2 2( , , )L
24
Closed orbits areClosed orbits are contours on the Fermi surface for which isk
surface for which is periodic in the repeated zone scheme.
k
Open orbits are those for which continually k
increases.
Fermi surface of Cu
25
Quantization of Semiclassical Dynamics
The wave packets do not exactly obey the Schrodinger equation; they, however, approximately follow
H
W
/i H,i W Wt
H / ...... .i tW e H
After some time t=T the dynamics of the wave packets leads toAfter some time t=T, the dynamics of the wave packets leads to
0 ,k k K
T 2 .j HT j
The formal quantization condition is
H 2 .ll l
dt P jP
H 2l l
ldQ P j
l ll
Q P H L l
ll l
l l
QP d
P P dt
H is a constant;
often =1/2.
26
The quantization condition: 2l ll
dQ P j
ck
cr
The generalized coordinates Ql are and . l
L R
The canonical momenta Pl are
ep k AL and .kk
R p k Acr
and
A
integrating by parts2k
eAdc
kdk r j R
drdk kdk
dr k
2k
eAdk r dr j R
r k dk r
k c
27
Wannier‐Stark Ladders
For a system with a uniform electric field and no magnetic field,
2dk r j R
O Berry phase 2kdk r j R O k k d R
Kdk r K r
O dk R
0
dk r K r
is defined to be time average of r .r 2 j
Ok dk R
2 jrK
28
de Hass‐van Alphen Effect
In 1930, de Hass and van Alphen discovered that magnetization , p gof Bi in a magnetic field oscillates as a function of magnetic field, as a consequence of the quantization of electrons’ closed orbits.
MAu
M
1/B
29
1 .2
A B r
To understand the de Hass‐van Alphen effect, we recall the quantization condition, and adopt
2
2 .k
eAddk k r dr jc
R
O
c
2eAdt k r r j
12r A r B r
1 B
ek r Br r
2dt k r r jc
12 r r B
edt B k r B
cr r
2dt r r B
c
W i t bt iek We again use to obtain .ek r B
c
r
0 0ek t k r t r Bc
30
0 0B Bek t k r t r Bc
2 0 0e eB r t r rc c
B r t B
c c A B C A C B A B C
e r Bc
r
2 B kc kB e
0B B rr B
c B e
2edt r r B j 2
2dt r r B j
c
c B
T c B
0 2c Bdt k k
eB B
T
2c Bdk k
eB B
31
Because the change in k is normal to the direction of
,eck r B
.B
Thus must be confined to k t
.
the orbit defined by the intersection of the Fermi surface ith l l t
B
ˆ 2dk n k A
with a plane normal to .B
ˆ Bn
With we then have 2dk n k AA is the area in k‐space enclosed by the orbit .k t
Bn With , we then have .
22
c Bdk k jB B
2c jB
A2
jeB B
j
eBThe area of the orbital in k‐space is quantized.
32
Electron orbits are in the xy plane,zand occupy the intersection of the pyFermi surface with a set concentric “cylinders” which are quantized.
Consider the cross section of the Fermi sphere in the xz plane.
B
p pIf the Fermi sphere is of zero thickness, for a given j, two “circles” jin the xy plane satisfy the condition of two different values of B.F F i h i f fi iFor a Fermi sphere is of finite thickness, two “belts” satisfy the quantization conditionquantization condition.The width of the “belt” increases as electron orbits approach to lie along
33
pp gan extremal point of the surface.
For a closed orbit on the Fermi surface, as B increases, the area enclosed by the electron orbit in k space also increases.
At finite temperatures, whenever electron orbits lie along anAt finite temperatures, whenever electron orbits lie along an extremal point of the surface, i.e., the density of state is maximal.
0zk
A
In other words, the electrons resonantly oscillate on the Fermi surface
34
Fermi surface.
The oscillations in magnetization result from changes in the density of states at the Fermi surface.
One can accurately probe of the Fermi psurface by applying the magnetic field along diff di i
Other quantities such as specific heat and thermal conductivity
different directions.
Other quantities such as specific heat and thermal conductivity also exhibit oscillations with the same period in 1/B.
A i t f th i t b d t b th F iA variety of other experiments can be used to probe the Fermi surfaces. In particularly, angle‐resolved photoemission spectroscopy is an very important method.
35
spectroscopy is an very important method.
In the de Haas‐van Alphen oscillation of magnetization, large‐scale oscillations are due to extremal orbits around the thin neck, while the small‐scale ones are due to extremal orbits about the thick belly.
MAu
1/B 2c jeB
A
The magnetic field points 8.5° away from (111).
36
To further illustrate the quantization condition, we discuss
Landau quantization
q ,electrons in a uniform magnetic field along the z‐direction Choose the gauge
B
(0, ,0),A Bx
then ˆ.B Bz
The Schrodinger equation becomes ( , , ) (0, ,0)x y zB Bx
22 2 2
2 2 .2
ieBx r rm x y c z
E
21
2e A r r
m i c
E
2m x y c z
Because the Hamiltonian is not involved with y and z explicitly,the wave function can be written as
, , .y zi k y k zx y z e x
37
The equation for is
222 1d x B
x
2
2
1 .2 2 y
d x eBm x k x xm dx mc m
E
This is no more than the Schrodinger equation for the 1D simple harmonic oscillator of cyclotron frequency
centered on the point
y q y
,ceBmc
0
1 y
c
kx
m
.y
c keB
The energy thus is
c
12 ,cj E
2 212 .
2z
ckjm
Eand
The energy of the electron state is the sum of a translational energy along the magnetic field, together with the quantized energy of the cyclotron motion in the plane perpendicular to the field. The discrete energies are called Landau levels.
38
The magnetic field just simply drives the electron around with change of energy. The oscillation period is 2 2 mcT
.cc
TeB
Since is confined to the orbit perpendicular to k t
,B
Since is confined to the orbit perpendicular to
,ek v B dk e Bv
1 dvdk E
k t ,B
,c Bv
dt c dk
2c dk2c dk c dkdk c dkd dkeB dt
Ec dk c dkdkdt
eB v eB d
E
dkdk
2
2 m AE
2c dd
eB T
AE
2 m
2
1 eBj A 12c j A
ceB T
39
2 2j
mc m A 22 j
eB A
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