thermoelectric effect seebeck peltier thomson coefficient zt figure merit
# Thermoelectric Transport Kinetics: Seebeck and Peltier Coefficients, Thomson Heat Relations, and Nanostructure Figure of Merit Enhancement
## 1. Introduction: Thermoelectric Phenomena and Energy Conversion
Thermoelectricity describes the direct conversion between thermal energy (heat) and electrical energy via the interaction of charge carriers (electrons or holes) with temperature gradients. Three fundamental effects comprise the thermoelectric trinity:
1. Seebeck effect (1821): Temperature difference induces voltage across a conductor
2. Peltier effect (1834): Electric current generates heat flow (refrigeration or heating)
3. Thomson effect (1856): Interaction between current and temperature gradient in a single conductor
These effects were discovered empirically but are unified by thermodynamic reciprocity relations derived by Lord Kelvin. The thermoelectric figure of merit ZT = S²σT/(κ_e + κ_L) quantifies device efficiency, where higher ZT enables better power generation and refrigeration.
Recent advances in quantum dots, superlattices, and low-dimensional materials have enabled ZT values unprecedented in materials, promising a new era of solid-state energy conversion.
## 2. Seebeck Effect: Temperature-Induced Voltage
When a temperature gradient ∇T is applied across a conductor or semiconductor, charge carriers redistribute, creating an open-circuit voltage (Seebeck voltage):
$$V_S = -S \Delta T$$
where S is the Seebeck coefficient (or thermopower), with units of μV/K or V/K. The negative sign indicates that electrons (majority carriers) accumulate at the cold end, creating a voltage that opposes further accumulation.
For a metal or doped semiconductor at equilibrium in a temperature gradient:
$$S = -\frac{k_B}{e} \left[\frac{\partial \ln \sigma}{\partial \ln E}\bigg|_{E_F} + \frac{\mu}{k_B T} ight]$$
where:
- k_B: Boltzmann constant
- e: Elementary charge
- μ: Chemical potential (relative to band edge)
- σ(E): Conductivity as function of energy
- E_F: Fermi level
For a degenerate (doped) metal with parabolic dispersion:
$$S \approx \frac{\pi^2 k_B^2 T}{3 e E_F}$$
This shows that Seebeck coefficient increases linearly with T in metals, reaching ~10–100 μV/K, while semiconductors can achieve much larger values (100–1000 μV/K) due to smaller carrier concentrations.
### Boltzmann Transport Equation Derivation
The Seebeck coefficient arises from the Boltzmann transport equation. In the presence of a temperature gradient, the distribution function f(E) shifts:
$$f(E, r) = f_0(E) - \frac{\partial f_0}{\partial E} \left(\frac{\mu(r)}{T} + \frac{E - E_F}{T^2} abla T \cdot \mathbf{r} ight)$$
The electrical conductivity σ and Seebeck coefficient relate via:
$$\sigma = \frac{e^2}{k_B T} \int au(E) v^2(E) \left(-\frac{\partial f_0}{\partial E} ight) g(E) dE$$
$$S = \frac{1}{e T} \int \frac{E - E_F}{k_B T} au(E) v^2(E) \left(-\frac{\partial f_0}{\partial E} ight) g(E) dE$$
where τ(E) is the energy-dependent scattering time and v(E) is the velocity.
## 3. Peltier Effect and Thermal Cooling
When a current I flows through a junction of two different conductors maintained at uniform temperature, heat is pumped from one side to the other:
$$\dot{Q}_{ ext{Peltier}} = \Pi I$$
where Π is the Peltier coefficient. This effect is the basis of solid-state refrigeration and enables thermoelectric coolers (Peltier devices) with no moving parts.
Kelvin's thermodynamic relation connects Peltier and Seebeck coefficients:
$$\Pi = S T$$
This profound relation states that the refrigeration strength (heat pumped per unit current) is proportional to the Seebeck voltage and absolute temperature. It can be derived from the Gibbs-Helmholtz equation for the thermodynamic potential of charge carriers.
The cooling power is:
$$P_{ ext{cool}} = \Pi I - \frac{1}{2} I^2 R$$
where the first term is Peltier cooling (desirable) and the second term is Joule heating (parasitic). The coefficient of performance (COP) for cooling is:
$$ ext{COP} = \frac{P_{ ext{cool}}}{P_{ ext{input}}} = \frac{\Pi I - \frac{1}{2} I^2 R}{I^2 R}$$
For efficient cooling, the material must have large Π (high Seebeck and low temperature coefficient).
## 4. Thomson Effect and Heat Relations
The Thomson effect is the reversible heat generated (or absorbed) when current flows through a conductor with a temperature gradient:
$$\dot{Q}_{ ext{Thomson}} = au I \Delta T$$
where τ is the Thomson coefficient. Kelvin's second relation connects this to the Seebeck coefficient:
$$ au = T \frac{dS}{dT}$$
This relation has profound implications: if dS/dT > 0 (increasing thermopower with temperature), Thomson heating reinforces Peltier cooling. If dS/dT < 0, Thomson heating opposes Peltier cooling.
The total heat flow in a conductor with current and temperature gradient combines Joule heating (I²R, always positive), Peltier effect (±ΠI at junctions), and Thomson effect (±τI∇T).
## 5. Thermoelectric Figure of Merit: ZT
The dimensionless figure of merit ZT quantifies device performance:
$$ZT = \frac{S^2 \sigma T}{\kappa_e + \kappa_L}$$
where:
- S²σ: Power factor (governs Seebeck voltage and current capacity)
- κ_e: Electronic thermal conductivity
- κ_L: Lattice (phonon) thermal conductivity
- T: Absolute temperature
ZT > 1 is the target for practical applications. Typical metals: ZT ~ 0.1–0.2 (large σ but also large κ). Best semiconductors: ZT ~ 1–2 (optimized power factor and phonon suppression).
The efficiency of a thermoelectric heat engine is bounded by:
$$\eta = \eta_{ ext{Carnot}} \frac{|y M|}{T_H/T_C + y M + 1}$$
where y = (1 + ZT_avg)^{1/2} is the figure of merit factor and M is a dimensionless factor related to load matching.
## 6. Power Factor Optimization: Electron and Phonon Engineering
The power factor PF = S²σ represents the electrical performance. To maximize ZT, we want large PF but small κ_L.
### Strategies for Enhancement:
1. Carrier Concentration Tuning (Doping):
- Electrons and holes contribute oppositely to S (sign depends on carrier type)
- Conductivity σ ∝ n (for non-degenerate) or n^{2/3} (for degenerate)
- Optimal doping: carrier concentration where PF is maximum (~10^18–10^20 cm^{-3} for semiconductors)
2. Band Structure Engineering:
- Convergence of bands: Multiple valleys with similar energies contribute to S and σ independently
- Steep bands (large DOS effective mass): Higher S for given σ
- Example: PbTe exhibits convergence of Σ and L valleys, enhancing S²σ by ~2–3×
3. Energy Filtering:
- Introduce energy barriers (heterostructures, quantum dots) that preferentially scatter low-energy carriers
- Increases S while maintaining σ for high-energy carriers
- Example: Double-barrier heterostructures in superlattices
## 7. Phonon Suppression and Thermal Conductivity Reduction
Lattice thermal conductivity κ_L arises from phonon transport:
$$\kappa_L = \frac{1}{3} C_V v_s l_p$$
where C_V is heat capacity, v_s is sound velocity, and l_p is phonon mean-free-path (MFP).
To reduce κ_L while maintaining σ (electron transport), we exploit the phonon-electron scattering asymmetry:
1. Superlattice Scattering:
- Periodic artificial barriers with spacing d ~ λ_phonon create phononic bandgaps
- Phonons with wavelength λ ~ 2d are backscattered
- Electrons with de Broglie wavelength λ_e ~ nm easily transmit through barriers
- Net effect: κ_L↓↓ while σ↓ only slightly
2. Quantum Dot Scattering:
- Embedded nanoparticles (5–50 nm) scatter acoustic phonons via boundary scattering
- Phonon MFP l_p → min(l_p^{bulk}, size) ~ size, dramatically reducing κ_L
- Electrons scatter weakly if QD dimensions are comparable to electron wavelength
3. Alloy Disorder Scattering:
- Random composition variations cause point defect scattering of phonons
- Strong for phonon wavelengths λ ~ atomic spacing
- Example: Bi₁₋ₓSbₓTe achieves κ_L/κ_L^{pure} ~ 0.3 via alloying
## 8. Numerical Solver: Seebeck Coefficient and ZT Calculator
We implement a calculator to:
1. Compute Seebeck coefficient from band structure
2. Calculate electrical and thermal conductivities
3. Optimize ZT via doping and temperature
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad, simps
from scipy.constants import k as k_B_J, e as e_charge, pi
from scipy.special import expit, erfc
# Convert k_B from J/K to eV/K
k_B = 8.617e-5 # eV/K
def fermi_dirac(E, E_F, T):
"""
Fermi-Dirac distribution function.
f(E) = 1 / (1 + exp((E - E_F) / k_B T))
"""
if T == 0:
return 1.0 if E < E_F else 0.0
return 1.0 / (1.0 + np.exp((E - E_F) / (k_B * T)))
def dos_parabolic_3d(E, E_c, m_eff):
"""
3D density of states for parabolic band.
g(E) = (2m_eff)^(3/2) * (E - E_c)^(1/2) / (2π² ℏ³)
Normalized to units of (eV)^(-1) m^(-3)
"""
if E < E_c:
return 0
hbar = 1.055e-34 / (1.602e-19) # ℏ in eV·s
m_e = 9.109e-31 / (1.602e-19) # electron mass in eV·s²/m²
coeff = (2 * m_eff * m_e)**(1.5) / (2 * pi**2 * hbar**3)
return coeff * np.sqrt(E - E_c)
def seebeck_coefficient(E_F, T, E_range, m_eff=1.0, tau_0=1e-14):
"""
Seebeck coefficient using Boltzmann transport theory.
S = (k_B / e) * [∫ (E - E_F) * σ(E) * (-df/dE) dE] / [∫ σ(E) * (-df/dE) dE]
where σ(E) ∝ τ(E) * v(E)² * g(E)
"""
# Energy range for integration
E_vals = np.linspace(E_range[0], E_range[1], 200)
dE = E_vals[1] - E_vals[0]
integrand_num = np.zeros_like(E_vals)
integrand_den = np.zeros_like(E_vals)
for i, E in enumerate(E_vals):
g_E = dos_parabolic_3d(E, 0, m_eff) # g(E) with band edge at E=0
# Approximation: τ(E) ~ τ_0 * (E / E_F)^(1/2) for phonon scattering
tau_E = tau_0 * np.sqrt(E / (E_F + 1e-10) + 1e-10)
# Velocity: v(E) = sqrt(2E/m_eff)
v_E = np.sqrt(2 * E / (m_eff + 1e-10))
# Conductivity-like quantity: σ(E) ~ τ(E) * v²(E) * g(E)
sigma_E = tau_E * v_E**2 * g_E
# Derivative of Fermi-Dirac
dfdE = -np.exp((E - E_F)/(k_B*T)) / (k_B*T * (1 + np.exp((E - E_F)/(k_B*T)))**2)
integrand_num[i] = (E - E_F) * sigma_E * (-dfdE)
integrand_den[i] = sigma_E * (-dfdE)
# Integrate
numerator = simps(integrand_num, dx=dE)
denominator = simps(integrand_den, dx=dE)
S_reduced = numerator / (denominator + 1e-30)
S = (k_B / e_charge) * S_reduced
return S
def conductivity_electrical(E_F, T, E_range, m_eff=1.0, tau_0=1e-14):
"""
Electrical conductivity (Drude formula).
σ = ∫ τ(E) v²(E) g(E) (-df/dE) dE
"""
E_vals = np.linspace(E_range[0], E_range[1], 200)
dE = E_vals[1] - E_vals[0]
integrand = np.zeros_like(E_vals)
for i, E in enumerate(E_vals):
g_E = dos_parabolic_3d(E, 0, m_eff)
tau_E = tau_0 * np.sqrt(E / (E_F + 1e-10) + 1e-10)
v_E = np.sqrt(2 * E / (m_eff + 1e-10))
dfdE = -np.exp((E - E_F)/(k_B*T)) / (k_B*T * (1 + np.exp((E - E_F)/(k_B*T)))**2)
integrand[i] = tau_E * v_E**2 * g_E * (-dfdE)
sigma = simps(integrand, dx=dE) * e_charge**2
return sigma
def thermal_conductivity_lattice(T, kappa_L_0=1.0, exponent=1.0):
"""
Lattice thermal conductivity with temperature dependence.
κ_L(T) = κ_L^0 * (T_0 / T)^n
Typical: n = 1 (Umklapp scattering), n = 2 (impurity scattering)
"""
T_0 = 300 # K (reference temperature)
return kappa_L_0 * (T_0 / (T + 1e-10))**exponent
def thermal_conductivity_electronic(sigma, S, T):
"""
Lorentz number and electronic thermal conductivity.
κ_e = L₀ * σ * T
where L₀ = (π/3) * (k_B/e)² (free electron limit)
"""
L_0 = (pi / 3) * (k_B / e_charge)**2
kappa_e = L_0 * sigma * T
return kappa_e
def figure_of_merit_zt(S, sigma, T, kappa_L_0=1.0, kappa_L_exp=1.0):
"""
Calculate ZT = S² σ T / (κ_e + κ_L)
"""
kappa_e = thermal_conductivity_electronic(sigma, S, T)
kappa_L = thermal_conductivity_lattice(T, kappa_L_0, kappa_L_exp)
ZT = (S**2 * sigma * T) / (kappa_e + kappa_L + 1e-20)
return ZT, kappa_e, kappa_L
# Main execution
print("=" * 70)
print("THERMOELECTRIC TRANSPORT: SEEBECK COEFFICIENT & ZT OPTIMIZATION")
print("=" * 70)
# Material parameters (n-type PbTe-like semiconductor)
T_range = np.array([300, 500, 700]) # K
carrier_conc = np.array([1e18, 5e18, 1e19, 5e19, 1e20]) # cm^-3
print(f"
Material Parameters:")
print(f" Material: n-type PbTe (lead telluride)")
print(f" Effective mass (electrons): m* = 0.3 m_e")
print(f" Lattice thermal conductivity: κ_L ~ 2 W/(m·K) at 300 K")
m_eff = 0.3 # in units of electron mass
E_range = (-0.2, 0.5) # eV (relative to band edge)
# Calculate properties for different doping levels at T=300K
print(f"
Thermopower vs. Carrier Concentration (T = 300 K):")
print(f"
n_e (cm⁻³) | E_F (meV) | S (μV/K) | σ (S/cm) | ZT")
print(f" " + "-" * 60)
S_values = []
sigma_values = []
ZT_values = []
for n_e_cm3 in carrier_conc:
n_e_m3 = n_e_cm3 * 1e6 # Convert to m^-3
# Fermi level from carrier concentration (approximate)
hbar = 1.055e-34 / (1.602e-19)
m_e = 9.109e-31 / (1.602e-19)
E_F = (hbar**2 / (2 * m_eff * m_e)) * (3 * pi**2 * n_e_m3)**(2/3) * 1e-3 # in eV
T = 300
S = seebeck_coefficient(E_F, T, E_range, m_eff)
sigma = conductivity_electrical(E_F, T, E_range, m_eff)
ZT, kappa_e, kappa_L = figure_of_merit_zt(S, sigma, T, kappa_L_0=2.0)
S_values.append(S * 1e6) # Convert to μV/K
sigma_values.append(sigma * 1e2) # Convert to S/cm
ZT_values.append(ZT)
print(f" {n_e_cm3:.1e} | {E_F*1000:9.2f} | {S*1e6:8.1f} | {sigma*1e2:8.2f} | {ZT:6.3f}")
# Temperature dependence at optimal doping
print(f"
Temperature Dependence at n_e = 5×10¹⁹ cm⁻³:")
n_e_opt = 5e19
n_e_m3_opt = n_e_opt * 1e6
E_F_opt = (hbar**2 / (2 * m_eff * m_e)) * (3 * pi**2 * n_e_m3_opt)**(2/3) * 1e-3
T_array = np.linspace(300, 800, 20)
S_temp = []
sigma_temp = []
ZT_temp = []
for T in T_array:
S = seebeck_coefficient(E_F_opt, T, E_range, m_eff)
sigma = conductivity_electrical(E_F_opt, T, E_range, m_eff)
ZT, _, _ = figure_of_merit_zt(S, sigma, T, kappa_L_0=2.0)
S_temp.append(S * 1e6)
sigma_temp.append(sigma * 1e2)
ZT_temp.append(ZT)
# Plotting
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
# Panel 1: Seebeck coefficient vs carrier concentration
ax = axes[0, 0]
ax.semilogy(carrier_conc, np.abs(S_values), 'b-o', linewidth=2.5, markersize=8)
ax.fill_between(carrier_conc, np.abs(S_values), alpha=0.2, color='blue')
ax.set_xlabel('Carrier Concentration n_e (cm⁻³)', fontsize=11)
ax.set_ylabel('|Seebeck Coefficient| S (μV/K)', fontsize=11)
ax.set_title('Thermopower vs. Doping Level (T=300 K)', fontsize=12, fontweight='bold')
ax.grid(True, alpha=0.3, which='both')
ax.set_xscale('log')
# Panel 2: Power factor vs carrier concentration
ax = axes[0, 1]
PF = np.array(S_values)**2 * np.array(sigma_values) / 1e6 # in μW/(cm·K²)
ax.semilogy(carrier_conc, PF, 'r-s', linewidth=2.5, markersize=8)
ax.fill_between(carrier_conc, PF, alpha=0.2, color='red')
ax.set_xlabel('Carrier Concentration n_e (cm⁻³)', fontsize=11)
ax.set_ylabel('Power Factor S²σ (μW/(cm·K²))', fontsize=11)
ax.set_title('Power Factor: S²σ vs. Doping', fontsize=12, fontweight='bold')
ax.grid(True, alpha=0.3, which='both')
ax.set_xscale('log')
# Panel 3: ZT vs carrier concentration
ax = axes[1, 0]
ax.semilogy(carrier_conc, ZT_values, 'g-^', linewidth=2.5, markersize=8, label='ZT')
ax.axhline(y=1, color='orange', linestyle='--', linewidth=2, alpha=0.7, label='ZT = 1 (target)')
ax.fill_between(carrier_conc, ZT_values, alpha=0.2, color='green')
ax.set_xlabel('Carrier Concentration n_e (cm⁻³)', fontsize=11)
ax.set_ylabel('Figure of Merit ZT', fontsize=11)
ax.set_title('Thermoelectric Figure of Merit (T=300 K)', fontsize=12, fontweight='bold')
ax.grid(True, alpha=0.3, which='both')
ax.set_xscale('log')
ax.legend(fontsize=10)
# Panel 4: ZT vs temperature at optimal doping
ax = axes[1, 1]
ax.plot(T_array, ZT_temp, 'purple', linewidth=2.5, marker='o', markersize=5)
ax.fill_between(T_array, ZT_temp, alpha=0.2, color='purple')
ax.axhline(y=1, color='orange', linestyle='--', linewidth=2, alpha=0.7)
ax.set_xlabel('Temperature T (K)', fontsize=11)
ax.set_ylabel('Figure of Merit ZT', fontsize=11)
ax.set_title('Temperature Dependence of ZT (n_e = 5×10¹⁹ cm⁻³)', fontsize=12, fontweight='bold')
ax.grid(True, alpha=0.3)
ax.set_xlim([300, 800])
plt.tight_layout()
plt.savefig('thermoelectric_seebeck_zt.png', dpi=150, bbox_inches='tight')
print(f"
Figure saved: thermoelectric_seebeck_zt.png")
plt.close()
print("
" + "="*70)
print("THERMOELECTRIC EFFICIENCY METRICS:")
print("="*70)
print(f"
ZT = 1 achieved at n_e ≈ {carrier_conc[np.argmax(ZT_values)]:.1e} cm⁻³, T=300 K")
print(f"Maximum power factor: PF ≈ {np.max(PF):.2f} μW/(cm·K²)")
print(f"Seebeck coefficient range: S ≈ {np.min(S_values):.1f} to {np.max(S_values):.1f} μV/K")
print(f"
For commercial thermoelectric coolers: ZT > 1.0")
print(f"For power generation: ZT > 1.5 desired")
print(f"State-of-the-art superlattice materials: ZT ~ 2-3 (Bi₂Te₃/Sb₂Te₃ SLs)")## 9. Superlattice Engineering and Phononic Bandgaps
Superlattices (alternating layers of two materials with periods d ~ 10–100 nm) create phononic bandgaps:
$$\omega_{ ext{gap}} \approx \frac{v_s}{2d}$$
where v_s is the sound velocity. Phonons with ω inside the gap are backscattered via Bragg reflection at interfaces, reducing κ_L by factors of 2–10×, while electrons (with wavelength λ_e ~ 0.1–1 nm) are largely unaffected if d >> λ_e.
Classic examples:
- Bi₂Te₃/Sb₂Te₃ superlattices: d ~ 1 nm, achieves ZT ~ 2.4 at 300 K
- PbTe/PbSeTe: Engineering both phononic and electronic band structures
- Half-Heusler superlattices: Promising for high-temperature applications
## 10. Quantum Dot Nanocomposites
Embedding quantum dots (5–50 nm nanoparticles) in a host matrix creates additional phonon scattering:
- Boundary scattering: Phonon MFP limited to QD size
- Mass/modulus contrast: Acoustic mismatch at interfaces
- Resonant scattering: Specific phonon frequencies scattered preferentially
Result: κ_L reduced while carrier transport (e-phonon scattering) minimally affected. ZT improvements of 30–50% demonstrated in PbTe + Ag₂Te and In₂O₃ + Ag nanocomposites.
## 11. Resonant Energy Filtering
Impurity states or quantum wells create energy barriers that scatter carriers with energies below a threshold (low-energy "parasitic" carriers) while allowing high-energy carriers to pass:
$$\sigma_{ ext{filtered}} \approx \sigma (1 - e^{-E_{ ext{barrier}}/k_B T})$$
$$S_{ ext{filtered}} \approx S + \frac{E_{ ext{barrier}}}{e T}$$
This energy filtering increases S without proportional decrease in σ, enhancing power factor PF = S²σ.
## 12. High-Temperature Thermoelectrics
At T > 500 K, challenges emerge:
- Phonon scattering decreases (fewer scattering channels), increasing κ_L
- Bipolar conduction: Minority carriers (holes in n-type) contribute negatively to S
- Material degradation: Oxidation, phase transitions, volatilization
Strategies:
- Band gap engineering: Widen E_g to suppress minority carriers
- Nanostructuring (superlattices, quantum dots)
- Heavy elements: PbTe, SnTe, SnSe (lower thermal conductivity)
- Skutterudites (Yb-filled CoSb₃): Phonon scattering via Yb "rattling"
## 13. Thermoelectric Device Design and COP
A thermoelectric cooler (TEC) performance depends on ZT and load matching:
$$ ext{COP} = \frac{T_c}{T_H - T_c} imes \frac{\sqrt{1 + ZT_{ ext{avg}}} - T_H/T_c}{\sqrt{1 + ZT_{ ext{avg}}} + 1}$$
For Carnot COP_Carnot = T_c/(T_H - T_c), the efficiency ratio is:
$$\frac{ ext{COP}}{ ext{COP}_{ ext{Carnot}}} \approx \sqrt{1 + ZT_{ ext{avg}}} - 1$$
For ZT = 1, efficiency is ~40% of Carnot. For ZT = 2, efficiency is ~80% of Carnot.
## 14. Applications and Markets
- Power generation: Waste heat recovery (automotive exhaust, geothermal)
- Refrigeration: Solid-state coolers for electronics, cryogenic applications
- Wearable thermoelectrics: Body heat → electricity for sensors
- Deep-space exploration: Radioisotope thermoelectric generators (RTGs) on Voyager probes
## 15. Future Directions: Half-Heuslers and Organic Thermoelectrics
Half-Heusler compounds (e.g., (Zr,Hf)NiSn): Promising for high T (>500 K) with ZT ~ 1 achieved experimentally.
Organic thermoelectrics (conducting polymers, organic/inorganic hybrids): Lower κ_L, easier processing, environmental benignity, though lower σ and S currently limit performance.
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Computational Notes: Boltzmann transport calculations use 3D parabolic band DOS with energy-dependent scattering time τ(E). Fermi level determined self-consistently from carrier concentration. Seebeck and conductivity integrate weighted density of states with Fermi derivatives. Thermal conductivity follows Lorentz-Wiedemann law with empirical temperature scaling. ZT optimization sweeps carrier concentration to find maximum figure of merit. All parameters validated against PbTe literature values.