Rare Rare Rare-earth Rare-earth earth-based half earth-based - - PowerPoint PPT Presentation
Rare Rare Rare-earth Rare-earth earth-based half earth-based half based half-Heusler based half-Heusler Heusler Heusler compounds as prospective materials compounds as prospective materials p p p p p p for thermoelectric
Institute of Low Temperature and Structure Research, Polish Academy of Sciences, Wrocław, Poland
Institute of Low Temperature and Structure Research, Polish Academy of Sciences, Wrocław
Max Planck Institut Max-Planck-Institut für Chemische Physik fester Stoffe, Dresden
Er Er Pd Pd Er Er Sb Sb
Compound Structure Space group Atomic positions
ErSb ErPdSb ErPd2Sb
Co pou d St uctu e type Space g oup to c pos t o s Er Sb Pd ErSb NaCl m Fm3 4b, (½ ½ ½) 4a, (0 0 0) - ErPdSb MgAgAs m F 3 4 4b, (½ ½ ½) 4a, (0 0 0) 4c, (¼ ¼ ¼) E Pd Sb M C Al 4b (½ ½ ½) 4 (0 0 0) 8 (¼ ¼ ¼) ErPd2Sb MnCu2Al m Fm3 4b, (½ ½ ½) 4a, (0 0 0) 8c, (¼ ¼ ¼)
Pierre 1997
Pierre, 1997
Kondo effect
p
h ll
Seebeck effect
hybrid automobile applications, power generation from waste heat (catalytic bl k h hi h
converters, motor blocks, heaters, high temperature furnaces, power plants) …
P lti ff t Peltier effect
spot cooling of electronic equipment, infrared detectors, car air-conditioners, refrigerators solar powered coolers
S.Williams, www.thermoelectrics.com
h i l t )
refrigerators, solar-powered coolers …
spot cooling electric power generation
Th Tc
p n p n p n
I
Tc
+ I
Th
coefficient of performance (COP)
+
coefficient of efficiency (COE)
γ γ φ + − − = 1
c h h c
T T T T
h c c h
T T T T γ γ η + − − = 1
1
ZT + 1 = γ
γ
c h h c
γ
ZT + 1 γ
S T S
2 2
2
L S T S ZT
2 2
= = κ σ
S = L1/2 = 157 µV/K ⇒ ZT = 1 S = (2L)1/2 = 225 µV/K ⇒ ZT = 2 state-of-the-art commercial devices
S – Seebeck coefficient κ – thermal conductivity
e.g. p-type BixSb2-xTe3-ySey
ρ – electrical resistivity
RECORD VALUES p-type alloy Bi2Te3/Sb2Te3/Sb2Se3 : ZT = 1.14 at T = 300 K
2 3 2 3 2 3
quantum dots lattice PbTe/PbSe0.98Te0.02 : ZT = 2.0 at T = 550 K thin-film superlattice Bi2Te3/Sb2Te3 : ZT = 2.4 at T = 300 K
ErPdSb
dSb ErPdSb ErPdSb
111 002 113 222 004 224 024 133 044 222 024 133 333 044
ErPdSb
single phase samples homogeneous stoichiometry homogeneous stoichiometry atomic disorder not detectable
Compound TN (K) θp (K) µeff (µB)
weak AF at low temp.
YPdSb D
3.3
10.5
Curie-Weiss behavior
µeff ≈ µ ≈ µteo for RE3+ small negative θ DyPdSb 3.3 11.5 10.5 HoPdSb 2.0
10.7 small negative θp
weak CEF effect
20
1.2 1.3
HoPdSb u)
u/mol)
ErPdSb P
9.4 YPdBi D
15
60 80 100 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.1
mol/emu
g)
TN = 2 K
χ (emu T (K)
GdPdBi 13.5
8.0
5
1 2 3 4 5 20 40 60
B = 0.1 T
χ
T = 1.7 K
σ (emu/g B (T)
DyPdBi 3.5
10.7 HoPdBi 2.2
10.6
50 100 150 200 250 300 B 0.1 T
T (K)
B (T)
ErPdBi P
9.2
25 30
1.5 1.8 B = 0.1 T
20 25
0.3 0.6 0.9 1.2
l/emu)
χ (emu/mo
µeff (µB) θp (K) E PdSb 9 43 4 2
10 15
5 10 15 20 25 30
χ
ErPdBi
T (K)
ErPdSb 9.43
ErPdBi 9.20
50 100 150 200 250 300 5
χ
B = 0.1 T
no magnetic ordering down to 1 72 K T (K) no magnetic ordering down to 1.72 K Curie-Weiss behaviour:
µ ≈ µ for Er3+ (9 58 µ ) small negative θ µeff ≈ µteo for Er3 (9.58 µB) , small negative θp
weak CEF effect
80
ErPdSb
e
C C C T C + + ) (
60
l K)
6 8 10
mol K)
CEF ph el p
C C C T C + + = ) (
T T C γ = ) (
20 40
YPdSb (J/mol
4 8 12 16 20 2 4
Cp (J/ T (K)
Θ
⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Θ =
T x x B ph
D
dx x e T Nk T C
2 4 3
) 1 ( 9 ) ( T T Cel γ = ) (
50 100 150 200 250 300 20 Fit : ΘD = 264 K
γ = 0.22 mJ/molK
2
Cp
T (K)
− ⎠ ⎜ ⎝ Θ
x D B ph
e
2
) 1 ( ) (
/ 2
1 ) ( ⎞ ⎜ ⎜ ⎛ =
−
T k E n
B i
e E R T C
50 100 150 200 250 300
T (K)
2 / 1 2 2
1 ) ( ⎞ ⎜ ⎛ ⎠ ⎜ ⎜ ⎝ =
− =
T k E n i i B CEF
B i i
E R e E Z T k T C
no phase transition down to 2 K upturn below 6 K
1 2 2
⎠ ⎜ ⎜ ⎝ −
=
i i B
B i
e E Z T k
n T k E
pronounced CEF Schottky effect
= −
=
i T k E
B i
e Z
ErNiSb
220 K 166 K
doublet ground state
108 K 92 K
CEF scheme: doublet-quartet- doublet-quartet-quartet t t l litti f 186 K
Karla et al., 1999
total splitting of 186 K first excited state at 61K
10
ErPdSb
6 8
mol K)
5
4 6
C (J/m
3 4 /mol K)
Schottky ?
2
∆C
1 2
∆C (J
10 100
T (K)
0.0 0.1 0.2 0.3 T
magnetic ordering at T < 2 K ? l ib i
nuclear contribution
unlikely
2.1
E PdSb
1.5 1.8
ErPdSb
B = 0 T B = 1 T B = 2 T B = 4 T
2)
upturn transforms
0.9 1.2
B = 6 T B = 9 T
T (J/m
B
into maximum Tmax increases
0.3 0.6
Cp/T
max
for rising B
5 10 15 20 25 30 0.0
T (K)
clear Zeeman effect e.g. local distortion, internal-field distribution, …
semimetallic character
t l t t anomalies at low temperatures
for both AF and P systems !!!
B g a
indirect gap Γ – X : ∆ ≈ 0 1 eV
∆ ≈ 0.1 eV
direct gap Γ – Γ : ca. 0.4 eV valence bands at Γ : parabolic with different curvature conduction band at X : nonparabolic
EF
→ heavy and light holes in p-type material → different effective masses
Lu 4f
different effective masses
and doped holes bands near E :
Mastronardi et al., 1999
bands near EF : strongly hybridized Pd-d and Lu-d states
DOS narrow gap E slightly above E total resistivity
Berger, 2003
narrow gap Eg slightly above EF
metallic conductivity at LT activation behaviour at HT
) ( ) ( ) ( T n T T
ph
ρ ρ ρ + =
) ( ) ( ) ( n T n T n T n + =
) (T n
Fermi Dirac distribution
) ( ) ( ) ( n T n T n T n
p n
+ =
exp 1 ln ) ( E T Nk NE T n
g B g n
⎥ ⎤ ⎢ ⎡ ⎞ ⎜ ⎜ ⎛ + + − =
Fermi-Dirac distribution
1
1 exp ) (
−
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = T k E E E f
F
2 ln ) ( p ) ( T Nk T n T k
B p B B g n
− = ⎥ ⎦ ⎢ ⎣ ⎠ ⎜ ⎝ ⎦ ⎣ ⎠ ⎝ T kB
carrier concentration
∞
=
F F
E E n
dE T E f E N T n ) , ( ) ( ) (
Θ
⎞ ⎛
T
D
dz z T
5 5
Bloch-Grüneisen law
[ ]
−
− =
F
E p
dE T E f E N T n ) , ( 1 ) ( ) (
−
− − ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Θ =
z z D ph
e e dz z T R T ) 1 )( 1 ( 4 ) ( ρ
1000 1100
R = 0.73 µΩcm/K Eg = 26 meV n0 = 0.25
800 900
R = 10.2 µΩcm/K Eg = 1066 K n0 = 0.08
Ω cm)
n0 0.25 N = 8.04 eV-1 for ΘD = 270 K 1 86 Ω
600 700
N = 6.7 eV
for
ΘD = 208 K ρ0 = 980 µΩcm
ErPdBi ρ (µΩ
ρ0 = 1.86 mΩcm
200 240
HoPdBi
160 200
R = 9.6 µΩ cm / K Eg = 165.2 K n = 0 045
Ω cm)
HoPdBi
T
ph
ρ ρ ρ + = Τ
120
n0 = 0.045 N = 2.42 eV
ΘD = 210 K ρ0 = 120 µΩ cm
ρ (µ
T n ρ = Τ
50 100 150 200 250 300 80
T (K)
200 250 YPdSb dSb
150 DyPdSb HoPdSb ErPdSb
V/K)
50 100
S (µV
50 100 150 200 250 300
T (K)
100 120
DyPdBi ErPdBi GdPdBi
60 80
GdPdBi HodBi YPdBi
µV/K)
F B
eE T k T S 3
2 2
π =
20 40
S (µ
REPdSb: EF = 30-60 meV n ≈ 1019 cm-3
F
50 100 150 200 250 300
T (K)
REPdBi: EF = 50-140 meV n ≈ 1020 cm-3
80 100
ErPdBi
positive
60 80
3.5 4.0
∆ = 27 meV Γ = 44.73 meV
V/K)
)
20 40
2.0 2.5 3.0
S (µV
T/S (K
2/µV)
large magnitude
50 100 150 200 250 300 20
2 4 6 8 10 1.5
T
2 (10 4 K 2)
phonon drag? crystal field?
T (K)
two band model: 4f band
crystal field?
2 2
2 2
Γ + ∆
position peak 4f E E
F f
− − = ∆
4
two-band model: - 4f band
Gottwick et al., 1985
e A ∆ = 2
2 2 2 2 2
3
B
k B π Γ + ∆ =
bandwidth peak 4f position peak 4f E E
F f
− Γ ∆
4
100
three-band model:
80
∆W = 33.2 meV
) DyPdBi
three band model:
conduction band
40 60
W
ΓW = 47 meV
(µV/K)
20
∆N = 0.63 meV ΓN = 8.1 meV
S (
position peak wide E E bandwidth peak narrow position peak narrow E E
F W W N F N N
− − = ∆ − Γ − − = ∆
50 100 150 200 250 300
T (K)
bandwidth peak wide
W F W W
− Γ
1 A 1 ' A
( )
2 2 2 2
T B T A T B T A S S S
W N W N
+ + + = + =
2∆
2 2
Γ + ∆
2 2
1 ) (
N N N
A N Γ + ∆ = π ε
2 2
1 ' ) (
W W W
A N Γ + ∆ = π ε
T B T B
W N
+ +
e A
W N W N , ,
2∆ =
2 2 , , 2 ,
3
B W N W N W N
k B π Γ + ∆ = Bando et al., 2000
d i t positive dominant holes low carrier large low carrier concentration multiple strongly dependent on temperature and field p electrons and holes bands
) ( ) ( ) (
1 1 1
T T T
ac im − − −
+ = µ µ µ
3 2 3 ; ) ( = = p aT T
p im
µ
( ) ( )
2 / 3 2 4 2 / 1
8 ) ( d h e T ν π µ = (
)
2 / 5 2 2 / 3
3 ) (
eff ac B ac
m E T k T µ
2
k
D B
ν Θ =
3 2
6 n h π
l e
electronic : T
Wiedemann-Franz law
l
κ
ρ κ T L
e
=
e
κ
lattice :
Callaway model
lattice :
Callaway model
Θ
⎠ ⎞ ⎜ ⎝ ⎛ =
T x B B l
D
dx e x T k k
/ 2 4
κ
−
− ⎠ ⎜ ⎝
x p l
e
2 1
1 2 τ πυ h
1 1 1 1 1 − − − − −
+ + + = τ τ τ τ τ
input : ΘD = 270 K (υ = 2400 m/s) ⎞ ⎛ Θ
−
+ + + =
ph el boundary defect Umklapp p
τ τ τ τ τ
CTx
ph el
=
− − 1
τ B
boundary = −1
τ
4 4 1
T Dx
defect = −
τ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Θ − =
−
T x AT
D Umplapp
2 exp
2 3 1
τ
discrepancies for T > 170 K discrepancies for T > 170 K
d sc ep c es o 70 d sc ep c es o 70
radiation losses T-dependent Lorenz number ? p error in ΘD input value ? bipolaron contribution ?
Gofryk et al., PRB 75 (2007) 224426
dx e x T k n
D
x B L
Τ Θ
⎠ ⎞ ⎜ ⎝ ⎛ =
2 3 2 2 3 1
3 κ
ex
D L
− Θ ⎠ ⎜ ⎝
2
1 4
min
h π
3 28
10 37 . 4
−
⋅ = m n
Cahill, 1989
l
L el tot
ρ κ L
el =
Wiedemann Wiedemann-
Franz law
2
10 2.45
−
Ω ⋅ = K W L
T k k
D
x Τ Θ
⎞ ⎛
4 3
( )
dx e e x T k v k
x x P B s B L
Τ 2
− ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =
2 4
1 2 τ π κ h
D
Θ 1
( )
s m n k v
D B s
1947 6
3 1 2
= Θ = π h
ΘD= 208 K
( )
dx e e x T k n
D
x x D B L
Τ
− Θ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =
2 3 2 2 3 1
1 4 3
min
h π κ
3 28
10 37 . 4
−
⋅ = m n
figure of merit :
state-of-the-art commercial devices
κρ
2
TS ZT =
e.g. p-type BixSb2-xTe3-ySey
κρ
gy, 2002 d Technolog Science and f Materials:
cyclopedia of
Credit: Enc
comp.: 3d-metal half-Heusler phases, skutterudites, clathrates, …
novel compounds : REPdSb and REPdBi RE = Y Gd Dy Ho Er RE = Y, Gd, Dy, Ho, Er structural, magnetic, electrical and thermal properties : cubic (MgAgAs type) cubic (MgAgAs-type) paramagnetic (Er), antiferromagnetic (Gd,Dy,Ho); RE3+ ions semimetallic (narrow band semiconductors) ( )
electron and hole bands low concentrations of carriers (REPdSb) strongly T-dependent concentrations and mobilities strongly T-dependent concentrations and mobilities electronic structure very sensitive to magnetic field (LT)
promising thermoelectric characteristics (ErPdSb, DyPdBi) electronic band structure (role of disorder) high temperature behaviour (LT f
E PdX)
Open problems
high-temperature behaviour (LT for ErPdX)